YES
proof of /hpcwork/ff862203/termcomp26/benchmarks/dwnnK.pl
# AProVE Commit ID: 23a904c96b029b0a549cde0d0d17dbccf967db59 jckassing 20260626 unpublished dirty


Left Termination of the query pattern

p(g)

w.r.t. the given Prolog program could successfully be proven:

(0) Prolog
(1) PrologToPiTRSProof [SOUND, 0 ms]
(2) PiTRS
(3) DependencyPairsProof [EQUIVALENT, 0 ms]
(4) PiDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) AND
    (7) PiDP
        (8) UsableRulesProof [EQUIVALENT, 0 ms]
        (9) PiDP
        (10) PiDPToQDPProof [SOUND, 4 ms]
        (11) QDP
        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) PiDP
        (15) UsableRulesProof [EQUIVALENT, 0 ms]
        (16) PiDP
        (17) PiDPToQDPProof [SOUND, 0 ms]
        (18) QDP
        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (20) YES
    (21) PiDP
        (22) PiDPToQDPProof [SOUND, 0 ms]
        (23) QDP
        (24) QDPOrderProof [EQUIVALENT, 40 ms]
        (25) QDP
        (26) UsableRulesProof [EQUIVALENT, 0 ms]
        (27) QDP
        (28) QReductionProof [EQUIVALENT, 0 ms]
        (29) QDP
        (30) UsableRulesReductionPairsProof [EQUIVALENT, 2 ms]
        (31) QDP
        (32) PisEmptyProof [EQUIVALENT, 0 ms]
        (33) YES


----------------------------------------

(0)
Obligation:
Clauses:

p(cons(X, nil)).
p(cons(s(s(X)), cons(Y, Xs))) :- ','(p(cons(X, cons(Y, Xs))), ','(mult(X, Y, Z), p(cons(Z, Xs)))).
p(cons(0, Xs)) :- p(Xs).
sum(X, 0, X).
sum(X, s(Y), s(Z)) :- sum(X, Y, Z).
mult(X1, 0, 0).
mult(X, s(Y), Z) :- ','(mult(X, Y, W), sum(W, X, Z)).


Query: p(g)
----------------------------------------

(1) PrologToPiTRSProof (SOUND)
We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:

p_in_1: (b)

mult_in_3: (b,b,f)

sum_in_3: (b,b,f)

Transforming Prolog into the following Term Rewriting System:

Pi-finite rewrite system:
The TRS R consists of the following rules:

   p_in_g(cons(X, nil)) -> p_out_g(cons(X, nil))
   p_in_g(cons(s(s(X)), cons(Y, Xs))) -> U1_g(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
   p_in_g(cons(0, Xs)) -> U4_g(Xs, p_in_g(Xs))
   U4_g(Xs, p_out_g(Xs)) -> p_out_g(cons(0, Xs))
   U1_g(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) -> U2_g(X, Y, Xs, mult_in_gga(X, Y, Z))
   mult_in_gga(X1, 0, 0) -> mult_out_gga(X1, 0, 0)
   mult_in_gga(X, s(Y), Z) -> U6_gga(X, Y, Z, mult_in_gga(X, Y, W))
   U6_gga(X, Y, Z, mult_out_gga(X, Y, W)) -> U7_gga(X, Y, Z, sum_in_gga(W, X, Z))
   sum_in_gga(X, 0, X) -> sum_out_gga(X, 0, X)
   sum_in_gga(X, s(Y), s(Z)) -> U5_gga(X, Y, Z, sum_in_gga(X, Y, Z))
   U5_gga(X, Y, Z, sum_out_gga(X, Y, Z)) -> sum_out_gga(X, s(Y), s(Z))
   U7_gga(X, Y, Z, sum_out_gga(W, X, Z)) -> mult_out_gga(X, s(Y), Z)
   U2_g(X, Y, Xs, mult_out_gga(X, Y, Z)) -> U3_g(X, Y, Xs, p_in_g(cons(Z, Xs)))
   U3_g(X, Y, Xs, p_out_g(cons(Z, Xs))) -> p_out_g(cons(s(s(X)), cons(Y, Xs)))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)

cons(x1, x2)  =  cons(x1, x2)

nil  =  nil

p_out_g(x1)  =  p_out_g

s(x1)  =  s(x1)

U1_g(x1, x2, x3, x4)  =  U1_g(x1, x2, x3, x4)

0  =  0

U4_g(x1, x2)  =  U4_g(x2)

U2_g(x1, x2, x3, x4)  =  U2_g(x3, x4)

mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)

mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)

U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x4)

U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)

sum_in_gga(x1, x2, x3)  =  sum_in_gga(x1, x2)

sum_out_gga(x1, x2, x3)  =  sum_out_gga(x3)

U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)

U3_g(x1, x2, x3, x4)  =  U3_g(x4)





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



----------------------------------------

(2)
Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:

   p_in_g(cons(X, nil)) -> p_out_g(cons(X, nil))
   p_in_g(cons(s(s(X)), cons(Y, Xs))) -> U1_g(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
   p_in_g(cons(0, Xs)) -> U4_g(Xs, p_in_g(Xs))
   U4_g(Xs, p_out_g(Xs)) -> p_out_g(cons(0, Xs))
   U1_g(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) -> U2_g(X, Y, Xs, mult_in_gga(X, Y, Z))
   mult_in_gga(X1, 0, 0) -> mult_out_gga(X1, 0, 0)
   mult_in_gga(X, s(Y), Z) -> U6_gga(X, Y, Z, mult_in_gga(X, Y, W))
   U6_gga(X, Y, Z, mult_out_gga(X, Y, W)) -> U7_gga(X, Y, Z, sum_in_gga(W, X, Z))
   sum_in_gga(X, 0, X) -> sum_out_gga(X, 0, X)
   sum_in_gga(X, s(Y), s(Z)) -> U5_gga(X, Y, Z, sum_in_gga(X, Y, Z))
   U5_gga(X, Y, Z, sum_out_gga(X, Y, Z)) -> sum_out_gga(X, s(Y), s(Z))
   U7_gga(X, Y, Z, sum_out_gga(W, X, Z)) -> mult_out_gga(X, s(Y), Z)
   U2_g(X, Y, Xs, mult_out_gga(X, Y, Z)) -> U3_g(X, Y, Xs, p_in_g(cons(Z, Xs)))
   U3_g(X, Y, Xs, p_out_g(cons(Z, Xs))) -> p_out_g(cons(s(s(X)), cons(Y, Xs)))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)

cons(x1, x2)  =  cons(x1, x2)

nil  =  nil

p_out_g(x1)  =  p_out_g

s(x1)  =  s(x1)

U1_g(x1, x2, x3, x4)  =  U1_g(x1, x2, x3, x4)

0  =  0

U4_g(x1, x2)  =  U4_g(x2)

U2_g(x1, x2, x3, x4)  =  U2_g(x3, x4)

mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)

mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)

U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x4)

U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)

sum_in_gga(x1, x2, x3)  =  sum_in_gga(x1, x2)

sum_out_gga(x1, x2, x3)  =  sum_out_gga(x3)

U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)

U3_g(x1, x2, x3, x4)  =  U3_g(x4)



----------------------------------------

(3) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

   P_IN_G(cons(s(s(X)), cons(Y, Xs))) -> U1_G(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
   P_IN_G(cons(s(s(X)), cons(Y, Xs))) -> P_IN_G(cons(X, cons(Y, Xs)))
   P_IN_G(cons(0, Xs)) -> U4_G(Xs, p_in_g(Xs))
   P_IN_G(cons(0, Xs)) -> P_IN_G(Xs)
   U1_G(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) -> U2_G(X, Y, Xs, mult_in_gga(X, Y, Z))
   U1_G(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) -> MULT_IN_GGA(X, Y, Z)
   MULT_IN_GGA(X, s(Y), Z) -> U6_GGA(X, Y, Z, mult_in_gga(X, Y, W))
   MULT_IN_GGA(X, s(Y), Z) -> MULT_IN_GGA(X, Y, W)
   U6_GGA(X, Y, Z, mult_out_gga(X, Y, W)) -> U7_GGA(X, Y, Z, sum_in_gga(W, X, Z))
   U6_GGA(X, Y, Z, mult_out_gga(X, Y, W)) -> SUM_IN_GGA(W, X, Z)
   SUM_IN_GGA(X, s(Y), s(Z)) -> U5_GGA(X, Y, Z, sum_in_gga(X, Y, Z))
   SUM_IN_GGA(X, s(Y), s(Z)) -> SUM_IN_GGA(X, Y, Z)
   U2_G(X, Y, Xs, mult_out_gga(X, Y, Z)) -> U3_G(X, Y, Xs, p_in_g(cons(Z, Xs)))
   U2_G(X, Y, Xs, mult_out_gga(X, Y, Z)) -> P_IN_G(cons(Z, Xs))

The TRS R consists of the following rules:

   p_in_g(cons(X, nil)) -> p_out_g(cons(X, nil))
   p_in_g(cons(s(s(X)), cons(Y, Xs))) -> U1_g(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
   p_in_g(cons(0, Xs)) -> U4_g(Xs, p_in_g(Xs))
   U4_g(Xs, p_out_g(Xs)) -> p_out_g(cons(0, Xs))
   U1_g(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) -> U2_g(X, Y, Xs, mult_in_gga(X, Y, Z))
   mult_in_gga(X1, 0, 0) -> mult_out_gga(X1, 0, 0)
   mult_in_gga(X, s(Y), Z) -> U6_gga(X, Y, Z, mult_in_gga(X, Y, W))
   U6_gga(X, Y, Z, mult_out_gga(X, Y, W)) -> U7_gga(X, Y, Z, sum_in_gga(W, X, Z))
   sum_in_gga(X, 0, X) -> sum_out_gga(X, 0, X)
   sum_in_gga(X, s(Y), s(Z)) -> U5_gga(X, Y, Z, sum_in_gga(X, Y, Z))
   U5_gga(X, Y, Z, sum_out_gga(X, Y, Z)) -> sum_out_gga(X, s(Y), s(Z))
   U7_gga(X, Y, Z, sum_out_gga(W, X, Z)) -> mult_out_gga(X, s(Y), Z)
   U2_g(X, Y, Xs, mult_out_gga(X, Y, Z)) -> U3_g(X, Y, Xs, p_in_g(cons(Z, Xs)))
   U3_g(X, Y, Xs, p_out_g(cons(Z, Xs))) -> p_out_g(cons(s(s(X)), cons(Y, Xs)))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)

cons(x1, x2)  =  cons(x1, x2)

nil  =  nil

p_out_g(x1)  =  p_out_g

s(x1)  =  s(x1)

U1_g(x1, x2, x3, x4)  =  U1_g(x1, x2, x3, x4)

0  =  0

U4_g(x1, x2)  =  U4_g(x2)

U2_g(x1, x2, x3, x4)  =  U2_g(x3, x4)

mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)

mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)

U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x4)

U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)

sum_in_gga(x1, x2, x3)  =  sum_in_gga(x1, x2)

sum_out_gga(x1, x2, x3)  =  sum_out_gga(x3)

U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)

U3_g(x1, x2, x3, x4)  =  U3_g(x4)

P_IN_G(x1)  =  P_IN_G(x1)

U1_G(x1, x2, x3, x4)  =  U1_G(x1, x2, x3, x4)

U4_G(x1, x2)  =  U4_G(x2)

U2_G(x1, x2, x3, x4)  =  U2_G(x3, x4)

MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)

U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x4)

U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x4)

SUM_IN_GGA(x1, x2, x3)  =  SUM_IN_GGA(x1, x2)

U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)

U3_G(x1, x2, x3, x4)  =  U3_G(x4)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(4)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   P_IN_G(cons(s(s(X)), cons(Y, Xs))) -> U1_G(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
   P_IN_G(cons(s(s(X)), cons(Y, Xs))) -> P_IN_G(cons(X, cons(Y, Xs)))
   P_IN_G(cons(0, Xs)) -> U4_G(Xs, p_in_g(Xs))
   P_IN_G(cons(0, Xs)) -> P_IN_G(Xs)
   U1_G(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) -> U2_G(X, Y, Xs, mult_in_gga(X, Y, Z))
   U1_G(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) -> MULT_IN_GGA(X, Y, Z)
   MULT_IN_GGA(X, s(Y), Z) -> U6_GGA(X, Y, Z, mult_in_gga(X, Y, W))
   MULT_IN_GGA(X, s(Y), Z) -> MULT_IN_GGA(X, Y, W)
   U6_GGA(X, Y, Z, mult_out_gga(X, Y, W)) -> U7_GGA(X, Y, Z, sum_in_gga(W, X, Z))
   U6_GGA(X, Y, Z, mult_out_gga(X, Y, W)) -> SUM_IN_GGA(W, X, Z)
   SUM_IN_GGA(X, s(Y), s(Z)) -> U5_GGA(X, Y, Z, sum_in_gga(X, Y, Z))
   SUM_IN_GGA(X, s(Y), s(Z)) -> SUM_IN_GGA(X, Y, Z)
   U2_G(X, Y, Xs, mult_out_gga(X, Y, Z)) -> U3_G(X, Y, Xs, p_in_g(cons(Z, Xs)))
   U2_G(X, Y, Xs, mult_out_gga(X, Y, Z)) -> P_IN_G(cons(Z, Xs))

The TRS R consists of the following rules:

   p_in_g(cons(X, nil)) -> p_out_g(cons(X, nil))
   p_in_g(cons(s(s(X)), cons(Y, Xs))) -> U1_g(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
   p_in_g(cons(0, Xs)) -> U4_g(Xs, p_in_g(Xs))
   U4_g(Xs, p_out_g(Xs)) -> p_out_g(cons(0, Xs))
   U1_g(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) -> U2_g(X, Y, Xs, mult_in_gga(X, Y, Z))
   mult_in_gga(X1, 0, 0) -> mult_out_gga(X1, 0, 0)
   mult_in_gga(X, s(Y), Z) -> U6_gga(X, Y, Z, mult_in_gga(X, Y, W))
   U6_gga(X, Y, Z, mult_out_gga(X, Y, W)) -> U7_gga(X, Y, Z, sum_in_gga(W, X, Z))
   sum_in_gga(X, 0, X) -> sum_out_gga(X, 0, X)
   sum_in_gga(X, s(Y), s(Z)) -> U5_gga(X, Y, Z, sum_in_gga(X, Y, Z))
   U5_gga(X, Y, Z, sum_out_gga(X, Y, Z)) -> sum_out_gga(X, s(Y), s(Z))
   U7_gga(X, Y, Z, sum_out_gga(W, X, Z)) -> mult_out_gga(X, s(Y), Z)
   U2_g(X, Y, Xs, mult_out_gga(X, Y, Z)) -> U3_g(X, Y, Xs, p_in_g(cons(Z, Xs)))
   U3_g(X, Y, Xs, p_out_g(cons(Z, Xs))) -> p_out_g(cons(s(s(X)), cons(Y, Xs)))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)

cons(x1, x2)  =  cons(x1, x2)

nil  =  nil

p_out_g(x1)  =  p_out_g

s(x1)  =  s(x1)

U1_g(x1, x2, x3, x4)  =  U1_g(x1, x2, x3, x4)

0  =  0

U4_g(x1, x2)  =  U4_g(x2)

U2_g(x1, x2, x3, x4)  =  U2_g(x3, x4)

mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)

mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)

U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x4)

U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)

sum_in_gga(x1, x2, x3)  =  sum_in_gga(x1, x2)

sum_out_gga(x1, x2, x3)  =  sum_out_gga(x3)

U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)

U3_g(x1, x2, x3, x4)  =  U3_g(x4)

P_IN_G(x1)  =  P_IN_G(x1)

U1_G(x1, x2, x3, x4)  =  U1_G(x1, x2, x3, x4)

U4_G(x1, x2)  =  U4_G(x2)

U2_G(x1, x2, x3, x4)  =  U2_G(x3, x4)

MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)

U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x4)

U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x4)

SUM_IN_GGA(x1, x2, x3)  =  SUM_IN_GGA(x1, x2)

U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)

U3_G(x1, x2, x3, x4)  =  U3_G(x4)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 7 less nodes.
----------------------------------------

(6)
Complex Obligation (AND)

----------------------------------------

(7)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   SUM_IN_GGA(X, s(Y), s(Z)) -> SUM_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

   p_in_g(cons(X, nil)) -> p_out_g(cons(X, nil))
   p_in_g(cons(s(s(X)), cons(Y, Xs))) -> U1_g(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
   p_in_g(cons(0, Xs)) -> U4_g(Xs, p_in_g(Xs))
   U4_g(Xs, p_out_g(Xs)) -> p_out_g(cons(0, Xs))
   U1_g(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) -> U2_g(X, Y, Xs, mult_in_gga(X, Y, Z))
   mult_in_gga(X1, 0, 0) -> mult_out_gga(X1, 0, 0)
   mult_in_gga(X, s(Y), Z) -> U6_gga(X, Y, Z, mult_in_gga(X, Y, W))
   U6_gga(X, Y, Z, mult_out_gga(X, Y, W)) -> U7_gga(X, Y, Z, sum_in_gga(W, X, Z))
   sum_in_gga(X, 0, X) -> sum_out_gga(X, 0, X)
   sum_in_gga(X, s(Y), s(Z)) -> U5_gga(X, Y, Z, sum_in_gga(X, Y, Z))
   U5_gga(X, Y, Z, sum_out_gga(X, Y, Z)) -> sum_out_gga(X, s(Y), s(Z))
   U7_gga(X, Y, Z, sum_out_gga(W, X, Z)) -> mult_out_gga(X, s(Y), Z)
   U2_g(X, Y, Xs, mult_out_gga(X, Y, Z)) -> U3_g(X, Y, Xs, p_in_g(cons(Z, Xs)))
   U3_g(X, Y, Xs, p_out_g(cons(Z, Xs))) -> p_out_g(cons(s(s(X)), cons(Y, Xs)))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)

cons(x1, x2)  =  cons(x1, x2)

nil  =  nil

p_out_g(x1)  =  p_out_g

s(x1)  =  s(x1)

U1_g(x1, x2, x3, x4)  =  U1_g(x1, x2, x3, x4)

0  =  0

U4_g(x1, x2)  =  U4_g(x2)

U2_g(x1, x2, x3, x4)  =  U2_g(x3, x4)

mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)

mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)

U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x4)

U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)

sum_in_gga(x1, x2, x3)  =  sum_in_gga(x1, x2)

sum_out_gga(x1, x2, x3)  =  sum_out_gga(x3)

U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)

U3_g(x1, x2, x3, x4)  =  U3_g(x4)

SUM_IN_GGA(x1, x2, x3)  =  SUM_IN_GGA(x1, x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(8) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(9)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   SUM_IN_GGA(X, s(Y), s(Z)) -> SUM_IN_GGA(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

SUM_IN_GGA(x1, x2, x3)  =  SUM_IN_GGA(x1, x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(10) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(11)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   SUM_IN_GGA(X, s(Y)) -> SUM_IN_GGA(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(12) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*SUM_IN_GGA(X, s(Y)) -> SUM_IN_GGA(X, Y)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(13)
YES

----------------------------------------

(14)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   MULT_IN_GGA(X, s(Y), Z) -> MULT_IN_GGA(X, Y, W)

The TRS R consists of the following rules:

   p_in_g(cons(X, nil)) -> p_out_g(cons(X, nil))
   p_in_g(cons(s(s(X)), cons(Y, Xs))) -> U1_g(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
   p_in_g(cons(0, Xs)) -> U4_g(Xs, p_in_g(Xs))
   U4_g(Xs, p_out_g(Xs)) -> p_out_g(cons(0, Xs))
   U1_g(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) -> U2_g(X, Y, Xs, mult_in_gga(X, Y, Z))
   mult_in_gga(X1, 0, 0) -> mult_out_gga(X1, 0, 0)
   mult_in_gga(X, s(Y), Z) -> U6_gga(X, Y, Z, mult_in_gga(X, Y, W))
   U6_gga(X, Y, Z, mult_out_gga(X, Y, W)) -> U7_gga(X, Y, Z, sum_in_gga(W, X, Z))
   sum_in_gga(X, 0, X) -> sum_out_gga(X, 0, X)
   sum_in_gga(X, s(Y), s(Z)) -> U5_gga(X, Y, Z, sum_in_gga(X, Y, Z))
   U5_gga(X, Y, Z, sum_out_gga(X, Y, Z)) -> sum_out_gga(X, s(Y), s(Z))
   U7_gga(X, Y, Z, sum_out_gga(W, X, Z)) -> mult_out_gga(X, s(Y), Z)
   U2_g(X, Y, Xs, mult_out_gga(X, Y, Z)) -> U3_g(X, Y, Xs, p_in_g(cons(Z, Xs)))
   U3_g(X, Y, Xs, p_out_g(cons(Z, Xs))) -> p_out_g(cons(s(s(X)), cons(Y, Xs)))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)

cons(x1, x2)  =  cons(x1, x2)

nil  =  nil

p_out_g(x1)  =  p_out_g

s(x1)  =  s(x1)

U1_g(x1, x2, x3, x4)  =  U1_g(x1, x2, x3, x4)

0  =  0

U4_g(x1, x2)  =  U4_g(x2)

U2_g(x1, x2, x3, x4)  =  U2_g(x3, x4)

mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)

mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)

U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x4)

U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)

sum_in_gga(x1, x2, x3)  =  sum_in_gga(x1, x2)

sum_out_gga(x1, x2, x3)  =  sum_out_gga(x3)

U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)

U3_g(x1, x2, x3, x4)  =  U3_g(x4)

MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(15) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(16)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   MULT_IN_GGA(X, s(Y), Z) -> MULT_IN_GGA(X, Y, W)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(17) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MULT_IN_GGA(X, s(Y)) -> MULT_IN_GGA(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(19) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*MULT_IN_GGA(X, s(Y)) -> MULT_IN_GGA(X, Y)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(20)
YES

----------------------------------------

(21)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U1_G(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) -> U2_G(X, Y, Xs, mult_in_gga(X, Y, Z))
   U2_G(X, Y, Xs, mult_out_gga(X, Y, Z)) -> P_IN_G(cons(Z, Xs))
   P_IN_G(cons(s(s(X)), cons(Y, Xs))) -> U1_G(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
   P_IN_G(cons(s(s(X)), cons(Y, Xs))) -> P_IN_G(cons(X, cons(Y, Xs)))
   P_IN_G(cons(0, Xs)) -> P_IN_G(Xs)

The TRS R consists of the following rules:

   p_in_g(cons(X, nil)) -> p_out_g(cons(X, nil))
   p_in_g(cons(s(s(X)), cons(Y, Xs))) -> U1_g(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
   p_in_g(cons(0, Xs)) -> U4_g(Xs, p_in_g(Xs))
   U4_g(Xs, p_out_g(Xs)) -> p_out_g(cons(0, Xs))
   U1_g(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) -> U2_g(X, Y, Xs, mult_in_gga(X, Y, Z))
   mult_in_gga(X1, 0, 0) -> mult_out_gga(X1, 0, 0)
   mult_in_gga(X, s(Y), Z) -> U6_gga(X, Y, Z, mult_in_gga(X, Y, W))
   U6_gga(X, Y, Z, mult_out_gga(X, Y, W)) -> U7_gga(X, Y, Z, sum_in_gga(W, X, Z))
   sum_in_gga(X, 0, X) -> sum_out_gga(X, 0, X)
   sum_in_gga(X, s(Y), s(Z)) -> U5_gga(X, Y, Z, sum_in_gga(X, Y, Z))
   U5_gga(X, Y, Z, sum_out_gga(X, Y, Z)) -> sum_out_gga(X, s(Y), s(Z))
   U7_gga(X, Y, Z, sum_out_gga(W, X, Z)) -> mult_out_gga(X, s(Y), Z)
   U2_g(X, Y, Xs, mult_out_gga(X, Y, Z)) -> U3_g(X, Y, Xs, p_in_g(cons(Z, Xs)))
   U3_g(X, Y, Xs, p_out_g(cons(Z, Xs))) -> p_out_g(cons(s(s(X)), cons(Y, Xs)))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)

cons(x1, x2)  =  cons(x1, x2)

nil  =  nil

p_out_g(x1)  =  p_out_g

s(x1)  =  s(x1)

U1_g(x1, x2, x3, x4)  =  U1_g(x1, x2, x3, x4)

0  =  0

U4_g(x1, x2)  =  U4_g(x2)

U2_g(x1, x2, x3, x4)  =  U2_g(x3, x4)

mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)

mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)

U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x4)

U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)

sum_in_gga(x1, x2, x3)  =  sum_in_gga(x1, x2)

sum_out_gga(x1, x2, x3)  =  sum_out_gga(x3)

U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)

U3_g(x1, x2, x3, x4)  =  U3_g(x4)

P_IN_G(x1)  =  P_IN_G(x1)

U1_G(x1, x2, x3, x4)  =  U1_G(x1, x2, x3, x4)

U2_G(x1, x2, x3, x4)  =  U2_G(x3, x4)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(22) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(23)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U1_G(X, Y, Xs, p_out_g) -> U2_G(Xs, mult_in_gga(X, Y))
   U2_G(Xs, mult_out_gga(Z)) -> P_IN_G(cons(Z, Xs))
   P_IN_G(cons(s(s(X)), cons(Y, Xs))) -> U1_G(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
   P_IN_G(cons(s(s(X)), cons(Y, Xs))) -> P_IN_G(cons(X, cons(Y, Xs)))
   P_IN_G(cons(0, Xs)) -> P_IN_G(Xs)

The TRS R consists of the following rules:

   p_in_g(cons(X, nil)) -> p_out_g
   p_in_g(cons(s(s(X)), cons(Y, Xs))) -> U1_g(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
   p_in_g(cons(0, Xs)) -> U4_g(p_in_g(Xs))
   U4_g(p_out_g) -> p_out_g
   U1_g(X, Y, Xs, p_out_g) -> U2_g(Xs, mult_in_gga(X, Y))
   mult_in_gga(X1, 0) -> mult_out_gga(0)
   mult_in_gga(X, s(Y)) -> U6_gga(X, mult_in_gga(X, Y))
   U6_gga(X, mult_out_gga(W)) -> U7_gga(sum_in_gga(W, X))
   sum_in_gga(X, 0) -> sum_out_gga(X)
   sum_in_gga(X, s(Y)) -> U5_gga(sum_in_gga(X, Y))
   U5_gga(sum_out_gga(Z)) -> sum_out_gga(s(Z))
   U7_gga(sum_out_gga(Z)) -> mult_out_gga(Z)
   U2_g(Xs, mult_out_gga(Z)) -> U3_g(p_in_g(cons(Z, Xs)))
   U3_g(p_out_g) -> p_out_g

The set Q consists of the following terms:

   p_in_g(x0)
   U4_g(x0)
   U1_g(x0, x1, x2, x3)
   mult_in_gga(x0, x1)
   U6_gga(x0, x1)
   sum_in_gga(x0, x1)
   U5_gga(x0)
   U7_gga(x0)
   U2_g(x0, x1)
   U3_g(x0)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(24) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   U1_G(X, Y, Xs, p_out_g) -> U2_G(Xs, mult_in_gga(X, Y))
   U2_G(Xs, mult_out_gga(Z)) -> P_IN_G(cons(Z, Xs))
   P_IN_G(cons(s(s(X)), cons(Y, Xs))) -> U1_G(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U2_G_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1
POL( U1_G_4(x_1, ..., x_4) ) = 2x_3 + 2
POL( U2_g_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( mult_in_gga_2(x_1, x_2) ) = 0
POL( 0 ) = 0
POL( mult_out_gga_1(x_1) ) = max{0, -2}
POL( s_1(x_1) ) = 0
POL( U6_gga_2(x_1, x_2) ) = 0
POL( p_in_g_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = 2x_2 + 2
POL( U1_g_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_4 - 2}
POL( U4_g_1(x_1) ) = 2x_1 + 2
POL( U3_g_1(x_1) ) = max{0, -2}
POL( nil ) = 0
POL( p_out_g ) = 2
POL( U7_gga_1(x_1) ) = 0
POL( sum_in_gga_2(x_1, x_2) ) = 0
POL( sum_out_gga_1(x_1) ) = max{0, 2x_1 - 2}
POL( U5_gga_1(x_1) ) = max{0, -2}
POL( P_IN_G_1(x_1) ) = max{0, x_1 - 2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mult_in_gga(X1, 0) -> mult_out_gga(0)
   mult_in_gga(X, s(Y)) -> U6_gga(X, mult_in_gga(X, Y))
   U6_gga(X, mult_out_gga(W)) -> U7_gga(sum_in_gga(W, X))
   U7_gga(sum_out_gga(Z)) -> mult_out_gga(Z)


----------------------------------------

(25)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   P_IN_G(cons(s(s(X)), cons(Y, Xs))) -> P_IN_G(cons(X, cons(Y, Xs)))
   P_IN_G(cons(0, Xs)) -> P_IN_G(Xs)

The TRS R consists of the following rules:

   p_in_g(cons(X, nil)) -> p_out_g
   p_in_g(cons(s(s(X)), cons(Y, Xs))) -> U1_g(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
   p_in_g(cons(0, Xs)) -> U4_g(p_in_g(Xs))
   U4_g(p_out_g) -> p_out_g
   U1_g(X, Y, Xs, p_out_g) -> U2_g(Xs, mult_in_gga(X, Y))
   mult_in_gga(X1, 0) -> mult_out_gga(0)
   mult_in_gga(X, s(Y)) -> U6_gga(X, mult_in_gga(X, Y))
   U6_gga(X, mult_out_gga(W)) -> U7_gga(sum_in_gga(W, X))
   sum_in_gga(X, 0) -> sum_out_gga(X)
   sum_in_gga(X, s(Y)) -> U5_gga(sum_in_gga(X, Y))
   U5_gga(sum_out_gga(Z)) -> sum_out_gga(s(Z))
   U7_gga(sum_out_gga(Z)) -> mult_out_gga(Z)
   U2_g(Xs, mult_out_gga(Z)) -> U3_g(p_in_g(cons(Z, Xs)))
   U3_g(p_out_g) -> p_out_g

The set Q consists of the following terms:

   p_in_g(x0)
   U4_g(x0)
   U1_g(x0, x1, x2, x3)
   mult_in_gga(x0, x1)
   U6_gga(x0, x1)
   sum_in_gga(x0, x1)
   U5_gga(x0)
   U7_gga(x0)
   U2_g(x0, x1)
   U3_g(x0)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(26) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(27)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   P_IN_G(cons(s(s(X)), cons(Y, Xs))) -> P_IN_G(cons(X, cons(Y, Xs)))
   P_IN_G(cons(0, Xs)) -> P_IN_G(Xs)

R is empty.
The set Q consists of the following terms:

   p_in_g(x0)
   U4_g(x0)
   U1_g(x0, x1, x2, x3)
   mult_in_gga(x0, x1)
   U6_gga(x0, x1)
   sum_in_gga(x0, x1)
   U5_gga(x0)
   U7_gga(x0)
   U2_g(x0, x1)
   U3_g(x0)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(28) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   p_in_g(x0)
   U4_g(x0)
   U1_g(x0, x1, x2, x3)
   mult_in_gga(x0, x1)
   U6_gga(x0, x1)
   sum_in_gga(x0, x1)
   U5_gga(x0)
   U7_gga(x0)
   U2_g(x0, x1)
   U3_g(x0)


----------------------------------------

(29)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   P_IN_G(cons(s(s(X)), cons(Y, Xs))) -> P_IN_G(cons(X, cons(Y, Xs)))
   P_IN_G(cons(0, Xs)) -> P_IN_G(Xs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(30) UsableRulesReductionPairsProof (EQUIVALENT)
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

   P_IN_G(cons(s(s(X)), cons(Y, Xs))) -> P_IN_G(cons(X, cons(Y, Xs)))
   P_IN_G(cons(0, Xs)) -> P_IN_G(Xs)
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(P_IN_G(x_1)) = 2*x_1
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = 2*x_1


----------------------------------------

(31)
Obligation:
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(32) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(33)
YES
