Left Termination of the query pattern myis_in_2(a, g) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

myis(Z, X) :- evaluate(X, Z).
evaluate(+(X, Y), Z) :- ','(evaluate(X, X1), ','(evaluate(Y, Y1), add(X1, Y1, Z))).
evaluate(-(X, Y), Z) :- ','(evaluate(X, X1), ','(evaluate(Y, Y1), sub(X1, Y1, Z))).
evaluate(*(X, Y), Z) :- ','(evaluate(X, X1), ','(evaluate(Y, Y1), mult(X1, Y1, Z))).
evaluate(X, X) :- myinteger(X).
myinteger(s(X)) :- myinteger(X).
myinteger(0).
add(s(X), Y, s(Z)) :- add(X, Y, Z).
add(0, X, X).
sub(s(X), s(Y), Z) :- sub(X, Y, Z).
sub(X, 0, X).
mult(s(X), Y, R) :- ','(mult(X, Y, Z), add(Y, Z, R)).
mult(0, Y, 0).
notEq(s(X), s(Y)) :- notEq(X, Y).
notEq(s(X), 0).
notEq(0, s(X)).
lt(s(X), s(Y)) :- lt(X, Y).
lt(0, s(Y)).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(Y)).
le(0, 0).

Queries:

myis(a,g).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
myis_in: (f,b)
evaluate_in: (b,f)
myinteger_in: (b)
mult_in: (b,b,f)
add_in: (b,b,f)
sub_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

myis_in_ag(Z, X) → U1_ag(Z, X, evaluate_in_ga(X, Z))
evaluate_in_ga(+(X, Y), Z) → U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(-(X, Y), Z) → U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(*(X, Y), Z) → U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(X, X) → U11_ga(X, myinteger_in_g(X))
myinteger_in_g(s(X)) → U12_g(X, myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g(0)
U12_g(X, myinteger_out_g(X)) → myinteger_out_g(s(X))
U11_ga(X, myinteger_out_g(X)) → evaluate_out_ga(X, X)
U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
mult_in_gga(s(X), Y, R) → U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U16_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U16_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) → evaluate_out_ga(*(X, Y), Z)
U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
sub_in_gga(s(X), s(Y), Z) → U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) → evaluate_out_ga(-(X, Y), Z)
U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) → evaluate_out_ga(+(X, Y), Z)
U1_ag(Z, X, evaluate_out_ga(X, Z)) → myis_out_ag(Z, X)

The argument filtering Pi contains the following mapping:
myis_in_ag(x1, x2)  =  myis_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)
+(x1, x2)  =  +(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)
-(x1, x2)  =  -(x1, x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x2, x4)
*(x1, x2)  =  *(x1, x2)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x2, x4)
U11_ga(x1, x2)  =  U11_ga(x1, x2)
myinteger_in_g(x1)  =  myinteger_in_g(x1)
s(x1)  =  s(x1)
U12_g(x1, x2)  =  U12_g(x2)
0  =  0
myinteger_out_g(x1)  =  myinteger_out_g
evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x2)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x4, x5)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x4, x5)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
myis_out_ag(x1, x2)  =  myis_out_ag(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

myis_in_ag(Z, X) → U1_ag(Z, X, evaluate_in_ga(X, Z))
evaluate_in_ga(+(X, Y), Z) → U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(-(X, Y), Z) → U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(*(X, Y), Z) → U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(X, X) → U11_ga(X, myinteger_in_g(X))
myinteger_in_g(s(X)) → U12_g(X, myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g(0)
U12_g(X, myinteger_out_g(X)) → myinteger_out_g(s(X))
U11_ga(X, myinteger_out_g(X)) → evaluate_out_ga(X, X)
U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
mult_in_gga(s(X), Y, R) → U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U16_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U16_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) → evaluate_out_ga(*(X, Y), Z)
U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
sub_in_gga(s(X), s(Y), Z) → U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) → evaluate_out_ga(-(X, Y), Z)
U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) → evaluate_out_ga(+(X, Y), Z)
U1_ag(Z, X, evaluate_out_ga(X, Z)) → myis_out_ag(Z, X)

The argument filtering Pi contains the following mapping:
myis_in_ag(x1, x2)  =  myis_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)
+(x1, x2)  =  +(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)
-(x1, x2)  =  -(x1, x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x2, x4)
*(x1, x2)  =  *(x1, x2)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x2, x4)
U11_ga(x1, x2)  =  U11_ga(x1, x2)
myinteger_in_g(x1)  =  myinteger_in_g(x1)
s(x1)  =  s(x1)
U12_g(x1, x2)  =  U12_g(x2)
0  =  0
myinteger_out_g(x1)  =  myinteger_out_g
evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x2)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x4, x5)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x4, x5)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
myis_out_ag(x1, x2)  =  myis_out_ag(x1)

(3) DependencyPairsProof (EQUIVALENT transformation)

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

MYIS_IN_AG(Z, X) → U1_AG(Z, X, evaluate_in_ga(X, Z))
MYIS_IN_AG(Z, X) → EVALUATE_IN_GA(X, Z)
EVALUATE_IN_GA(+(X, Y), Z) → U2_GA(X, Y, Z, evaluate_in_ga(X, X1))
EVALUATE_IN_GA(+(X, Y), Z) → EVALUATE_IN_GA(X, X1)
EVALUATE_IN_GA(-(X, Y), Z) → U5_GA(X, Y, Z, evaluate_in_ga(X, X1))
EVALUATE_IN_GA(-(X, Y), Z) → EVALUATE_IN_GA(X, X1)
EVALUATE_IN_GA(*(X, Y), Z) → U8_GA(X, Y, Z, evaluate_in_ga(X, X1))
EVALUATE_IN_GA(*(X, Y), Z) → EVALUATE_IN_GA(X, X1)
EVALUATE_IN_GA(X, X) → U11_GA(X, myinteger_in_g(X))
EVALUATE_IN_GA(X, X) → MYINTEGER_IN_G(X)
MYINTEGER_IN_G(s(X)) → U12_G(X, myinteger_in_g(X))
MYINTEGER_IN_G(s(X)) → MYINTEGER_IN_G(X)
U8_GA(X, Y, Z, evaluate_out_ga(X, X1)) → U9_GA(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U8_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
U9_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_GA(X, Y, Z, mult_in_gga(X1, Y1, Z))
U9_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → MULT_IN_GGA(X1, Y1, Z)
MULT_IN_GGA(s(X), Y, R) → U15_GGA(X, Y, R, mult_in_gga(X, Y, Z))
MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)
U15_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → U16_GGA(X, Y, R, add_in_gga(Y, Z, R))
U15_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → ADD_IN_GGA(Y, Z, R)
ADD_IN_GGA(s(X), Y, s(Z)) → U13_GGA(X, Y, Z, add_in_gga(X, Y, Z))
ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)
U5_GA(X, Y, Z, evaluate_out_ga(X, X1)) → U6_GA(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U5_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
U6_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_GA(X, Y, Z, sub_in_gga(X1, Y1, Z))
U6_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → SUB_IN_GGA(X1, Y1, Z)
SUB_IN_GGA(s(X), s(Y), Z) → U14_GGA(X, Y, Z, sub_in_gga(X, Y, Z))
SUB_IN_GGA(s(X), s(Y), Z) → SUB_IN_GGA(X, Y, Z)
U2_GA(X, Y, Z, evaluate_out_ga(X, X1)) → U3_GA(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U2_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
U3_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_GA(X, Y, Z, add_in_gga(X1, Y1, Z))
U3_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → ADD_IN_GGA(X1, Y1, Z)

The TRS R consists of the following rules:

myis_in_ag(Z, X) → U1_ag(Z, X, evaluate_in_ga(X, Z))
evaluate_in_ga(+(X, Y), Z) → U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(-(X, Y), Z) → U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(*(X, Y), Z) → U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(X, X) → U11_ga(X, myinteger_in_g(X))
myinteger_in_g(s(X)) → U12_g(X, myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g(0)
U12_g(X, myinteger_out_g(X)) → myinteger_out_g(s(X))
U11_ga(X, myinteger_out_g(X)) → evaluate_out_ga(X, X)
U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
mult_in_gga(s(X), Y, R) → U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U16_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U16_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) → evaluate_out_ga(*(X, Y), Z)
U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
sub_in_gga(s(X), s(Y), Z) → U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) → evaluate_out_ga(-(X, Y), Z)
U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) → evaluate_out_ga(+(X, Y), Z)
U1_ag(Z, X, evaluate_out_ga(X, Z)) → myis_out_ag(Z, X)

The argument filtering Pi contains the following mapping:
myis_in_ag(x1, x2)  =  myis_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)
+(x1, x2)  =  +(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)
-(x1, x2)  =  -(x1, x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x2, x4)
*(x1, x2)  =  *(x1, x2)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x2, x4)
U11_ga(x1, x2)  =  U11_ga(x1, x2)
myinteger_in_g(x1)  =  myinteger_in_g(x1)
s(x1)  =  s(x1)
U12_g(x1, x2)  =  U12_g(x2)
0  =  0
myinteger_out_g(x1)  =  myinteger_out_g
evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x2)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x4, x5)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x4, x5)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
myis_out_ag(x1, x2)  =  myis_out_ag(x1)
MYIS_IN_AG(x1, x2)  =  MYIS_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x3)
EVALUATE_IN_GA(x1, x2)  =  EVALUATE_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x2, x4)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x2, x4)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x2, x4)
U11_GA(x1, x2)  =  U11_GA(x1, x2)
MYINTEGER_IN_G(x1)  =  MYINTEGER_IN_G(x1)
U12_G(x1, x2)  =  U12_G(x2)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x4, x5)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x4)
MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)
U15_GGA(x1, x2, x3, x4)  =  U15_GGA(x2, x4)
U16_GGA(x1, x2, x3, x4)  =  U16_GGA(x4)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x4)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x4, x5)
U7_GA(x1, x2, x3, x4)  =  U7_GA(x4)
SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)
U14_GGA(x1, x2, x3, x4)  =  U14_GGA(x4)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x4, x5)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x4)

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

(4) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MYIS_IN_AG(Z, X) → U1_AG(Z, X, evaluate_in_ga(X, Z))
MYIS_IN_AG(Z, X) → EVALUATE_IN_GA(X, Z)
EVALUATE_IN_GA(+(X, Y), Z) → U2_GA(X, Y, Z, evaluate_in_ga(X, X1))
EVALUATE_IN_GA(+(X, Y), Z) → EVALUATE_IN_GA(X, X1)
EVALUATE_IN_GA(-(X, Y), Z) → U5_GA(X, Y, Z, evaluate_in_ga(X, X1))
EVALUATE_IN_GA(-(X, Y), Z) → EVALUATE_IN_GA(X, X1)
EVALUATE_IN_GA(*(X, Y), Z) → U8_GA(X, Y, Z, evaluate_in_ga(X, X1))
EVALUATE_IN_GA(*(X, Y), Z) → EVALUATE_IN_GA(X, X1)
EVALUATE_IN_GA(X, X) → U11_GA(X, myinteger_in_g(X))
EVALUATE_IN_GA(X, X) → MYINTEGER_IN_G(X)
MYINTEGER_IN_G(s(X)) → U12_G(X, myinteger_in_g(X))
MYINTEGER_IN_G(s(X)) → MYINTEGER_IN_G(X)
U8_GA(X, Y, Z, evaluate_out_ga(X, X1)) → U9_GA(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U8_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
U9_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_GA(X, Y, Z, mult_in_gga(X1, Y1, Z))
U9_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → MULT_IN_GGA(X1, Y1, Z)
MULT_IN_GGA(s(X), Y, R) → U15_GGA(X, Y, R, mult_in_gga(X, Y, Z))
MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)
U15_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → U16_GGA(X, Y, R, add_in_gga(Y, Z, R))
U15_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → ADD_IN_GGA(Y, Z, R)
ADD_IN_GGA(s(X), Y, s(Z)) → U13_GGA(X, Y, Z, add_in_gga(X, Y, Z))
ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)
U5_GA(X, Y, Z, evaluate_out_ga(X, X1)) → U6_GA(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U5_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
U6_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_GA(X, Y, Z, sub_in_gga(X1, Y1, Z))
U6_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → SUB_IN_GGA(X1, Y1, Z)
SUB_IN_GGA(s(X), s(Y), Z) → U14_GGA(X, Y, Z, sub_in_gga(X, Y, Z))
SUB_IN_GGA(s(X), s(Y), Z) → SUB_IN_GGA(X, Y, Z)
U2_GA(X, Y, Z, evaluate_out_ga(X, X1)) → U3_GA(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U2_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
U3_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_GA(X, Y, Z, add_in_gga(X1, Y1, Z))
U3_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → ADD_IN_GGA(X1, Y1, Z)

The TRS R consists of the following rules:

myis_in_ag(Z, X) → U1_ag(Z, X, evaluate_in_ga(X, Z))
evaluate_in_ga(+(X, Y), Z) → U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(-(X, Y), Z) → U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(*(X, Y), Z) → U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(X, X) → U11_ga(X, myinteger_in_g(X))
myinteger_in_g(s(X)) → U12_g(X, myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g(0)
U12_g(X, myinteger_out_g(X)) → myinteger_out_g(s(X))
U11_ga(X, myinteger_out_g(X)) → evaluate_out_ga(X, X)
U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
mult_in_gga(s(X), Y, R) → U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U16_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U16_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) → evaluate_out_ga(*(X, Y), Z)
U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
sub_in_gga(s(X), s(Y), Z) → U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) → evaluate_out_ga(-(X, Y), Z)
U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) → evaluate_out_ga(+(X, Y), Z)
U1_ag(Z, X, evaluate_out_ga(X, Z)) → myis_out_ag(Z, X)

The argument filtering Pi contains the following mapping:
myis_in_ag(x1, x2)  =  myis_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)
+(x1, x2)  =  +(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)
-(x1, x2)  =  -(x1, x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x2, x4)
*(x1, x2)  =  *(x1, x2)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x2, x4)
U11_ga(x1, x2)  =  U11_ga(x1, x2)
myinteger_in_g(x1)  =  myinteger_in_g(x1)
s(x1)  =  s(x1)
U12_g(x1, x2)  =  U12_g(x2)
0  =  0
myinteger_out_g(x1)  =  myinteger_out_g
evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x2)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x4, x5)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x4, x5)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
myis_out_ag(x1, x2)  =  myis_out_ag(x1)
MYIS_IN_AG(x1, x2)  =  MYIS_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x3)
EVALUATE_IN_GA(x1, x2)  =  EVALUATE_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x2, x4)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x2, x4)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x2, x4)
U11_GA(x1, x2)  =  U11_GA(x1, x2)
MYINTEGER_IN_G(x1)  =  MYINTEGER_IN_G(x1)
U12_G(x1, x2)  =  U12_G(x2)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x4, x5)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x4)
MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)
U15_GGA(x1, x2, x3, x4)  =  U15_GGA(x2, x4)
U16_GGA(x1, x2, x3, x4)  =  U16_GGA(x4)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x4)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x4, x5)
U7_GA(x1, x2, x3, x4)  =  U7_GA(x4)
SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)
U14_GGA(x1, x2, x3, x4)  =  U14_GGA(x4)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x4, x5)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 19 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SUB_IN_GGA(s(X), s(Y), Z) → SUB_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

myis_in_ag(Z, X) → U1_ag(Z, X, evaluate_in_ga(X, Z))
evaluate_in_ga(+(X, Y), Z) → U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(-(X, Y), Z) → U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(*(X, Y), Z) → U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(X, X) → U11_ga(X, myinteger_in_g(X))
myinteger_in_g(s(X)) → U12_g(X, myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g(0)
U12_g(X, myinteger_out_g(X)) → myinteger_out_g(s(X))
U11_ga(X, myinteger_out_g(X)) → evaluate_out_ga(X, X)
U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
mult_in_gga(s(X), Y, R) → U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U16_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U16_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) → evaluate_out_ga(*(X, Y), Z)
U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
sub_in_gga(s(X), s(Y), Z) → U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) → evaluate_out_ga(-(X, Y), Z)
U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) → evaluate_out_ga(+(X, Y), Z)
U1_ag(Z, X, evaluate_out_ga(X, Z)) → myis_out_ag(Z, X)

The argument filtering Pi contains the following mapping:
myis_in_ag(x1, x2)  =  myis_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)
+(x1, x2)  =  +(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)
-(x1, x2)  =  -(x1, x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x2, x4)
*(x1, x2)  =  *(x1, x2)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x2, x4)
U11_ga(x1, x2)  =  U11_ga(x1, x2)
myinteger_in_g(x1)  =  myinteger_in_g(x1)
s(x1)  =  s(x1)
U12_g(x1, x2)  =  U12_g(x2)
0  =  0
myinteger_out_g(x1)  =  myinteger_out_g
evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x2)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x4, x5)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x4, x5)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
myis_out_ag(x1, x2)  =  myis_out_ag(x1)
SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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:

SUB_IN_GGA(s(X), s(Y), Z) → SUB_IN_GGA(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)

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

(10) PiDPToQDPProof (SOUND transformation)

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:

SUB_IN_GGA(s(X), s(Y)) → SUB_IN_GGA(X, Y)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

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:

• SUB_IN_GGA(s(X), s(Y)) → SUB_IN_GGA(X, Y)
  The graph contains the following edges 1 > 1, 2 > 2

(13) TRUE

(14) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

myis_in_ag(Z, X) → U1_ag(Z, X, evaluate_in_ga(X, Z))
evaluate_in_ga(+(X, Y), Z) → U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(-(X, Y), Z) → U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(*(X, Y), Z) → U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(X, X) → U11_ga(X, myinteger_in_g(X))
myinteger_in_g(s(X)) → U12_g(X, myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g(0)
U12_g(X, myinteger_out_g(X)) → myinteger_out_g(s(X))
U11_ga(X, myinteger_out_g(X)) → evaluate_out_ga(X, X)
U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
mult_in_gga(s(X), Y, R) → U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U16_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U16_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) → evaluate_out_ga(*(X, Y), Z)
U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
sub_in_gga(s(X), s(Y), Z) → U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) → evaluate_out_ga(-(X, Y), Z)
U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) → evaluate_out_ga(+(X, Y), Z)
U1_ag(Z, X, evaluate_out_ga(X, Z)) → myis_out_ag(Z, X)

The argument filtering Pi contains the following mapping:
myis_in_ag(x1, x2)  =  myis_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)
+(x1, x2)  =  +(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)
-(x1, x2)  =  -(x1, x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x2, x4)
*(x1, x2)  =  *(x1, x2)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x2, x4)
U11_ga(x1, x2)  =  U11_ga(x1, x2)
myinteger_in_g(x1)  =  myinteger_in_g(x1)
s(x1)  =  s(x1)
U12_g(x1, x2)  =  U12_g(x2)
0  =  0
myinteger_out_g(x1)  =  myinteger_out_g
evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x2)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x4, x5)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x4, x5)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
myis_out_ag(x1, x2)  =  myis_out_ag(x1)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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:

ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)

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

(17) PiDPToQDPProof (SOUND transformation)

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:

ADD_IN_GGA(s(X), Y) → ADD_IN_GGA(X, Y)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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:

• ADD_IN_GGA(s(X), Y) → ADD_IN_GGA(X, Y)
  The graph contains the following edges 1 > 1, 2 >= 2

(20) TRUE

(21) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

myis_in_ag(Z, X) → U1_ag(Z, X, evaluate_in_ga(X, Z))
evaluate_in_ga(+(X, Y), Z) → U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(-(X, Y), Z) → U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(*(X, Y), Z) → U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(X, X) → U11_ga(X, myinteger_in_g(X))
myinteger_in_g(s(X)) → U12_g(X, myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g(0)
U12_g(X, myinteger_out_g(X)) → myinteger_out_g(s(X))
U11_ga(X, myinteger_out_g(X)) → evaluate_out_ga(X, X)
U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
mult_in_gga(s(X), Y, R) → U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U16_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U16_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) → evaluate_out_ga(*(X, Y), Z)
U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
sub_in_gga(s(X), s(Y), Z) → U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) → evaluate_out_ga(-(X, Y), Z)
U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) → evaluate_out_ga(+(X, Y), Z)
U1_ag(Z, X, evaluate_out_ga(X, Z)) → myis_out_ag(Z, X)

The argument filtering Pi contains the following mapping:
myis_in_ag(x1, x2)  =  myis_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)
+(x1, x2)  =  +(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)
-(x1, x2)  =  -(x1, x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x2, x4)
*(x1, x2)  =  *(x1, x2)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x2, x4)
U11_ga(x1, x2)  =  U11_ga(x1, x2)
myinteger_in_g(x1)  =  myinteger_in_g(x1)
s(x1)  =  s(x1)
U12_g(x1, x2)  =  U12_g(x2)
0  =  0
myinteger_out_g(x1)  =  myinteger_out_g
evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x2)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x4, x5)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x4, x5)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
myis_out_ag(x1, x2)  =  myis_out_ag(x1)
MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MULT_IN_GGA(s(X), Y) → MULT_IN_GGA(X, Y)

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

(26) QDPSizeChangeProof (EQUIVALENT transformation)

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(s(X), Y) → MULT_IN_GGA(X, Y)
  The graph contains the following edges 1 > 1, 2 >= 2

(27) TRUE

(28) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MYINTEGER_IN_G(s(X)) → MYINTEGER_IN_G(X)

The TRS R consists of the following rules:

myis_in_ag(Z, X) → U1_ag(Z, X, evaluate_in_ga(X, Z))
evaluate_in_ga(+(X, Y), Z) → U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(-(X, Y), Z) → U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(*(X, Y), Z) → U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(X, X) → U11_ga(X, myinteger_in_g(X))
myinteger_in_g(s(X)) → U12_g(X, myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g(0)
U12_g(X, myinteger_out_g(X)) → myinteger_out_g(s(X))
U11_ga(X, myinteger_out_g(X)) → evaluate_out_ga(X, X)
U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
mult_in_gga(s(X), Y, R) → U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U16_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U16_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) → evaluate_out_ga(*(X, Y), Z)
U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
sub_in_gga(s(X), s(Y), Z) → U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) → evaluate_out_ga(-(X, Y), Z)
U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) → evaluate_out_ga(+(X, Y), Z)
U1_ag(Z, X, evaluate_out_ga(X, Z)) → myis_out_ag(Z, X)

The argument filtering Pi contains the following mapping:
myis_in_ag(x1, x2)  =  myis_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)
+(x1, x2)  =  +(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)
-(x1, x2)  =  -(x1, x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x2, x4)
*(x1, x2)  =  *(x1, x2)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x2, x4)
U11_ga(x1, x2)  =  U11_ga(x1, x2)
myinteger_in_g(x1)  =  myinteger_in_g(x1)
s(x1)  =  s(x1)
U12_g(x1, x2)  =  U12_g(x2)
0  =  0
myinteger_out_g(x1)  =  myinteger_out_g
evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x2)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x4, x5)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x4, x5)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
myis_out_ag(x1, x2)  =  myis_out_ag(x1)
MYINTEGER_IN_G(x1)  =  MYINTEGER_IN_G(x1)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MYINTEGER_IN_G(s(X)) → MYINTEGER_IN_G(X)

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

(31) PiDPToQDPProof (EQUIVALENT transformation)

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

(32) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MYINTEGER_IN_G(s(X)) → MYINTEGER_IN_G(X)

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

(33) QDPSizeChangeProof (EQUIVALENT transformation)

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:

• MYINTEGER_IN_G(s(X)) → MYINTEGER_IN_G(X)
  The graph contains the following edges 1 > 1

(34) TRUE

(35) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U2_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
EVALUATE_IN_GA(+(X, Y), Z) → U2_GA(X, Y, Z, evaluate_in_ga(X, X1))
EVALUATE_IN_GA(+(X, Y), Z) → EVALUATE_IN_GA(X, X1)
EVALUATE_IN_GA(-(X, Y), Z) → U5_GA(X, Y, Z, evaluate_in_ga(X, X1))
U5_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
EVALUATE_IN_GA(-(X, Y), Z) → EVALUATE_IN_GA(X, X1)
EVALUATE_IN_GA(*(X, Y), Z) → U8_GA(X, Y, Z, evaluate_in_ga(X, X1))
U8_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
EVALUATE_IN_GA(*(X, Y), Z) → EVALUATE_IN_GA(X, X1)

The TRS R consists of the following rules:

myis_in_ag(Z, X) → U1_ag(Z, X, evaluate_in_ga(X, Z))
evaluate_in_ga(+(X, Y), Z) → U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(-(X, Y), Z) → U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(*(X, Y), Z) → U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(X, X) → U11_ga(X, myinteger_in_g(X))
myinteger_in_g(s(X)) → U12_g(X, myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g(0)
U12_g(X, myinteger_out_g(X)) → myinteger_out_g(s(X))
U11_ga(X, myinteger_out_g(X)) → evaluate_out_ga(X, X)
U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
mult_in_gga(s(X), Y, R) → U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U16_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U16_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) → evaluate_out_ga(*(X, Y), Z)
U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
sub_in_gga(s(X), s(Y), Z) → U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) → evaluate_out_ga(-(X, Y), Z)
U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) → evaluate_out_ga(+(X, Y), Z)
U1_ag(Z, X, evaluate_out_ga(X, Z)) → myis_out_ag(Z, X)

The argument filtering Pi contains the following mapping:
myis_in_ag(x1, x2)  =  myis_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)
+(x1, x2)  =  +(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)
-(x1, x2)  =  -(x1, x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x2, x4)
*(x1, x2)  =  *(x1, x2)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x2, x4)
U11_ga(x1, x2)  =  U11_ga(x1, x2)
myinteger_in_g(x1)  =  myinteger_in_g(x1)
s(x1)  =  s(x1)
U12_g(x1, x2)  =  U12_g(x2)
0  =  0
myinteger_out_g(x1)  =  myinteger_out_g
evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x2)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x4, x5)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x4, x5)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
myis_out_ag(x1, x2)  =  myis_out_ag(x1)
EVALUATE_IN_GA(x1, x2)  =  EVALUATE_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x2, x4)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x2, x4)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x2, x4)

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

(36) UsableRulesProof (EQUIVALENT transformation)

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

(37) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U2_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
EVALUATE_IN_GA(+(X, Y), Z) → U2_GA(X, Y, Z, evaluate_in_ga(X, X1))
EVALUATE_IN_GA(+(X, Y), Z) → EVALUATE_IN_GA(X, X1)
EVALUATE_IN_GA(-(X, Y), Z) → U5_GA(X, Y, Z, evaluate_in_ga(X, X1))
U5_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
EVALUATE_IN_GA(-(X, Y), Z) → EVALUATE_IN_GA(X, X1)
EVALUATE_IN_GA(*(X, Y), Z) → U8_GA(X, Y, Z, evaluate_in_ga(X, X1))
U8_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
EVALUATE_IN_GA(*(X, Y), Z) → EVALUATE_IN_GA(X, X1)

The TRS R consists of the following rules:

evaluate_in_ga(+(X, Y), Z) → U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(-(X, Y), Z) → U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(*(X, Y), Z) → U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(X, X) → U11_ga(X, myinteger_in_g(X))
U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U11_ga(X, myinteger_out_g(X)) → evaluate_out_ga(X, X)
U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
myinteger_in_g(s(X)) → U12_g(X, myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g(0)
U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) → evaluate_out_ga(+(X, Y), Z)
U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) → evaluate_out_ga(-(X, Y), Z)
U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) → evaluate_out_ga(*(X, Y), Z)
U12_g(X, myinteger_out_g(X)) → myinteger_out_g(s(X))
add_in_gga(s(X), Y, s(Z)) → U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
sub_in_gga(s(X), s(Y), Z) → U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
mult_in_gga(s(X), Y, R) → U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U16_gga(X, Y, R, add_in_gga(Y, Z, R))
U16_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)

The argument filtering Pi contains the following mapping:
evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)
+(x1, x2)  =  +(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)
-(x1, x2)  =  -(x1, x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x2, x4)
*(x1, x2)  =  *(x1, x2)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x2, x4)
U11_ga(x1, x2)  =  U11_ga(x1, x2)
myinteger_in_g(x1)  =  myinteger_in_g(x1)
s(x1)  =  s(x1)
U12_g(x1, x2)  =  U12_g(x2)
0  =  0
myinteger_out_g(x1)  =  myinteger_out_g
evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x2)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x4, x5)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x4, x5)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
EVALUATE_IN_GA(x1, x2)  =  EVALUATE_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x2, x4)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x2, x4)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x2, x4)

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

(38) PiDPToQDPProof (SOUND transformation)

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

(39) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U2_GA(Y, evaluate_out_ga(X1)) → EVALUATE_IN_GA(Y)
EVALUATE_IN_GA(+(X, Y)) → U2_GA(Y, evaluate_in_ga(X))
EVALUATE_IN_GA(+(X, Y)) → EVALUATE_IN_GA(X)
EVALUATE_IN_GA(-(X, Y)) → U5_GA(Y, evaluate_in_ga(X))
U5_GA(Y, evaluate_out_ga(X1)) → EVALUATE_IN_GA(Y)
EVALUATE_IN_GA(-(X, Y)) → EVALUATE_IN_GA(X)
EVALUATE_IN_GA(*(X, Y)) → U8_GA(Y, evaluate_in_ga(X))
U8_GA(Y, evaluate_out_ga(X1)) → EVALUATE_IN_GA(Y)
EVALUATE_IN_GA(*(X, Y)) → EVALUATE_IN_GA(X)

The TRS R consists of the following rules:

evaluate_in_ga(+(X, Y)) → U2_ga(Y, evaluate_in_ga(X))
evaluate_in_ga(-(X, Y)) → U5_ga(Y, evaluate_in_ga(X))
evaluate_in_ga(*(X, Y)) → U8_ga(Y, evaluate_in_ga(X))
evaluate_in_ga(X) → U11_ga(X, myinteger_in_g(X))
U2_ga(Y, evaluate_out_ga(X1)) → U3_ga(X1, evaluate_in_ga(Y))
U5_ga(Y, evaluate_out_ga(X1)) → U6_ga(X1, evaluate_in_ga(Y))
U8_ga(Y, evaluate_out_ga(X1)) → U9_ga(X1, evaluate_in_ga(Y))
U11_ga(X, myinteger_out_g) → evaluate_out_ga(X)
U3_ga(X1, evaluate_out_ga(Y1)) → U4_ga(add_in_gga(X1, Y1))
U6_ga(X1, evaluate_out_ga(Y1)) → U7_ga(sub_in_gga(X1, Y1))
U9_ga(X1, evaluate_out_ga(Y1)) → U10_ga(mult_in_gga(X1, Y1))
myinteger_in_g(s(X)) → U12_g(myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g
U4_ga(add_out_gga(Z)) → evaluate_out_ga(Z)
U7_ga(sub_out_gga(Z)) → evaluate_out_ga(Z)
U10_ga(mult_out_gga(Z)) → evaluate_out_ga(Z)
U12_g(myinteger_out_g) → myinteger_out_g
add_in_gga(s(X), Y) → U13_gga(add_in_gga(X, Y))
add_in_gga(0, X) → add_out_gga(X)
sub_in_gga(s(X), s(Y)) → U14_gga(sub_in_gga(X, Y))
sub_in_gga(X, 0) → sub_out_gga(X)
mult_in_gga(s(X), Y) → U15_gga(Y, mult_in_gga(X, Y))
mult_in_gga(0, Y) → mult_out_gga(0)
U13_gga(add_out_gga(Z)) → add_out_gga(s(Z))
U14_gga(sub_out_gga(Z)) → sub_out_gga(Z)
U15_gga(Y, mult_out_gga(Z)) → U16_gga(add_in_gga(Y, Z))
U16_gga(add_out_gga(R)) → mult_out_gga(R)

The set Q consists of the following terms:

evaluate_in_ga(x0)
U2_ga(x0, x1)
U5_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U3_ga(x0, x1)
U6_ga(x0, x1)
U9_ga(x0, x1)
myinteger_in_g(x0)
U4_ga(x0)
U7_ga(x0)
U10_ga(x0)
U12_g(x0)
add_in_gga(x0, x1)
sub_in_gga(x0, x1)
mult_in_gga(x0, x1)
U13_gga(x0)
U14_gga(x0)
U15_gga(x0, x1)
U16_gga(x0)

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

(40) QDPSizeChangeProof (EQUIVALENT transformation)

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:

• EVALUATE_IN_GA(+(X, Y)) → U2_GA(Y, evaluate_in_ga(X))
  The graph contains the following edges 1 > 1

• U2_GA(Y, evaluate_out_ga(X1)) → EVALUATE_IN_GA(Y)
  The graph contains the following edges 1 >= 1

• U5_GA(Y, evaluate_out_ga(X1)) → EVALUATE_IN_GA(Y)
  The graph contains the following edges 1 >= 1

• U8_GA(Y, evaluate_out_ga(X1)) → EVALUATE_IN_GA(Y)
  The graph contains the following edges 1 >= 1

• EVALUATE_IN_GA(-(X, Y)) → U5_GA(Y, evaluate_in_ga(X))
  The graph contains the following edges 1 > 1

• EVALUATE_IN_GA(*(X, Y)) → U8_GA(Y, evaluate_in_ga(X))
  The graph contains the following edges 1 > 1

• EVALUATE_IN_GA(+(X, Y)) → EVALUATE_IN_GA(X)
  The graph contains the following edges 1 > 1

• EVALUATE_IN_GA(-(X, Y)) → EVALUATE_IN_GA(X)
  The graph contains the following edges 1 > 1

• EVALUATE_IN_GA(*(X, Y)) → EVALUATE_IN_GA(X)
  The graph contains the following edges 1 > 1

(41) TRUE

(42) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
myis_in: (f,b)
evaluate_in: (b,f)
myinteger_in: (b)
mult_in: (b,b,f)
add_in: (b,b,f)
sub_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

myis_in_ag(Z, X) → U1_ag(Z, X, evaluate_in_ga(X, Z))
evaluate_in_ga(+(X, Y), Z) → U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(-(X, Y), Z) → U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(*(X, Y), Z) → U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(X, X) → U11_ga(X, myinteger_in_g(X))
myinteger_in_g(s(X)) → U12_g(X, myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g(0)
U12_g(X, myinteger_out_g(X)) → myinteger_out_g(s(X))
U11_ga(X, myinteger_out_g(X)) → evaluate_out_ga(X, X)
U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
mult_in_gga(s(X), Y, R) → U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U16_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U16_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) → evaluate_out_ga(*(X, Y), Z)
U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
sub_in_gga(s(X), s(Y), Z) → U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) → evaluate_out_ga(-(X, Y), Z)
U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) → evaluate_out_ga(+(X, Y), Z)
U1_ag(Z, X, evaluate_out_ga(X, Z)) → myis_out_ag(Z, X)

The argument filtering Pi contains the following mapping:
myis_in_ag(x1, x2)  =  myis_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)
+(x1, x2)  =  +(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
-(x1, x2)  =  -(x1, x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x2, x4)
*(x1, x2)  =  *(x1, x2)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
U11_ga(x1, x2)  =  U11_ga(x1, x2)
myinteger_in_g(x1)  =  myinteger_in_g(x1)
s(x1)  =  s(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
0  =  0
myinteger_out_g(x1)  =  myinteger_out_g(x1)
evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x1, x2)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x4, x5)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x2, x4)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x1, x2, x4, x5)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x2, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x1, x2, x3)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
myis_out_ag(x1, x2)  =  myis_out_ag(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(43) Obligation:

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

myis_in_ag(Z, X) → U1_ag(Z, X, evaluate_in_ga(X, Z))
evaluate_in_ga(+(X, Y), Z) → U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(-(X, Y), Z) → U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(*(X, Y), Z) → U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(X, X) → U11_ga(X, myinteger_in_g(X))
myinteger_in_g(s(X)) → U12_g(X, myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g(0)
U12_g(X, myinteger_out_g(X)) → myinteger_out_g(s(X))
U11_ga(X, myinteger_out_g(X)) → evaluate_out_ga(X, X)
U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
mult_in_gga(s(X), Y, R) → U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U16_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U16_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) → evaluate_out_ga(*(X, Y), Z)
U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
sub_in_gga(s(X), s(Y), Z) → U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) → evaluate_out_ga(-(X, Y), Z)
U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) → evaluate_out_ga(+(X, Y), Z)
U1_ag(Z, X, evaluate_out_ga(X, Z)) → myis_out_ag(Z, X)

The argument filtering Pi contains the following mapping:
myis_in_ag(x1, x2)  =  myis_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)
+(x1, x2)  =  +(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
-(x1, x2)  =  -(x1, x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x2, x4)
*(x1, x2)  =  *(x1, x2)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
U11_ga(x1, x2)  =  U11_ga(x1, x2)
myinteger_in_g(x1)  =  myinteger_in_g(x1)
s(x1)  =  s(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
0  =  0
myinteger_out_g(x1)  =  myinteger_out_g(x1)
evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x1, x2)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x4, x5)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x2, x4)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x1, x2, x4, x5)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x2, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x1, x2, x3)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
myis_out_ag(x1, x2)  =  myis_out_ag(x1, x2)

(44) DependencyPairsProof (EQUIVALENT transformation)

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

MYIS_IN_AG(Z, X) → U1_AG(Z, X, evaluate_in_ga(X, Z))
MYIS_IN_AG(Z, X) → EVALUATE_IN_GA(X, Z)
EVALUATE_IN_GA(+(X, Y), Z) → U2_GA(X, Y, Z, evaluate_in_ga(X, X1))
EVALUATE_IN_GA(+(X, Y), Z) → EVALUATE_IN_GA(X, X1)
EVALUATE_IN_GA(-(X, Y), Z) → U5_GA(X, Y, Z, evaluate_in_ga(X, X1))
EVALUATE_IN_GA(-(X, Y), Z) → EVALUATE_IN_GA(X, X1)
EVALUATE_IN_GA(*(X, Y), Z) → U8_GA(X, Y, Z, evaluate_in_ga(X, X1))
EVALUATE_IN_GA(*(X, Y), Z) → EVALUATE_IN_GA(X, X1)
EVALUATE_IN_GA(X, X) → U11_GA(X, myinteger_in_g(X))
EVALUATE_IN_GA(X, X) → MYINTEGER_IN_G(X)
MYINTEGER_IN_G(s(X)) → U12_G(X, myinteger_in_g(X))
MYINTEGER_IN_G(s(X)) → MYINTEGER_IN_G(X)
U8_GA(X, Y, Z, evaluate_out_ga(X, X1)) → U9_GA(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U8_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
U9_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_GA(X, Y, Z, mult_in_gga(X1, Y1, Z))
U9_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → MULT_IN_GGA(X1, Y1, Z)
MULT_IN_GGA(s(X), Y, R) → U15_GGA(X, Y, R, mult_in_gga(X, Y, Z))
MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)
U15_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → U16_GGA(X, Y, R, add_in_gga(Y, Z, R))
U15_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → ADD_IN_GGA(Y, Z, R)
ADD_IN_GGA(s(X), Y, s(Z)) → U13_GGA(X, Y, Z, add_in_gga(X, Y, Z))
ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)
U5_GA(X, Y, Z, evaluate_out_ga(X, X1)) → U6_GA(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U5_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
U6_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_GA(X, Y, Z, sub_in_gga(X1, Y1, Z))
U6_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → SUB_IN_GGA(X1, Y1, Z)
SUB_IN_GGA(s(X), s(Y), Z) → U14_GGA(X, Y, Z, sub_in_gga(X, Y, Z))
SUB_IN_GGA(s(X), s(Y), Z) → SUB_IN_GGA(X, Y, Z)
U2_GA(X, Y, Z, evaluate_out_ga(X, X1)) → U3_GA(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U2_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
U3_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_GA(X, Y, Z, add_in_gga(X1, Y1, Z))
U3_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → ADD_IN_GGA(X1, Y1, Z)

The TRS R consists of the following rules:

myis_in_ag(Z, X) → U1_ag(Z, X, evaluate_in_ga(X, Z))
evaluate_in_ga(+(X, Y), Z) → U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(-(X, Y), Z) → U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(*(X, Y), Z) → U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(X, X) → U11_ga(X, myinteger_in_g(X))
myinteger_in_g(s(X)) → U12_g(X, myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g(0)
U12_g(X, myinteger_out_g(X)) → myinteger_out_g(s(X))
U11_ga(X, myinteger_out_g(X)) → evaluate_out_ga(X, X)
U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
mult_in_gga(s(X), Y, R) → U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U16_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U16_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) → evaluate_out_ga(*(X, Y), Z)
U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
sub_in_gga(s(X), s(Y), Z) → U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) → evaluate_out_ga(-(X, Y), Z)
U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) → evaluate_out_ga(+(X, Y), Z)
U1_ag(Z, X, evaluate_out_ga(X, Z)) → myis_out_ag(Z, X)

The argument filtering Pi contains the following mapping:
myis_in_ag(x1, x2)  =  myis_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)
+(x1, x2)  =  +(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
-(x1, x2)  =  -(x1, x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x2, x4)
*(x1, x2)  =  *(x1, x2)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
U11_ga(x1, x2)  =  U11_ga(x1, x2)
myinteger_in_g(x1)  =  myinteger_in_g(x1)
s(x1)  =  s(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
0  =  0
myinteger_out_g(x1)  =  myinteger_out_g(x1)
evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x1, x2)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x4, x5)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x2, x4)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x1, x2, x4, x5)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x2, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x1, x2, x3)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
myis_out_ag(x1, x2)  =  myis_out_ag(x1, x2)
MYIS_IN_AG(x1, x2)  =  MYIS_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
EVALUATE_IN_GA(x1, x2)  =  EVALUATE_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x2, x4)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x1, x2, x4)
U11_GA(x1, x2)  =  U11_GA(x1, x2)
MYINTEGER_IN_G(x1)  =  MYINTEGER_IN_G(x1)
U12_G(x1, x2)  =  U12_G(x1, x2)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x4, x5)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x1, x2, x4)
MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)
U15_GGA(x1, x2, x3, x4)  =  U15_GGA(x1, x2, x4)
U16_GGA(x1, x2, x3, x4)  =  U16_GGA(x1, x2, x4)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x1, x2, x4)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x4, x5)
U7_GA(x1, x2, x3, x4)  =  U7_GA(x1, x2, x4)
SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)
U14_GGA(x1, x2, x3, x4)  =  U14_GGA(x1, x2, x4)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x4, x5)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x2, x4)

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

(45) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MYIS_IN_AG(Z, X) → U1_AG(Z, X, evaluate_in_ga(X, Z))
MYIS_IN_AG(Z, X) → EVALUATE_IN_GA(X, Z)
EVALUATE_IN_GA(+(X, Y), Z) → U2_GA(X, Y, Z, evaluate_in_ga(X, X1))
EVALUATE_IN_GA(+(X, Y), Z) → EVALUATE_IN_GA(X, X1)
EVALUATE_IN_GA(-(X, Y), Z) → U5_GA(X, Y, Z, evaluate_in_ga(X, X1))
EVALUATE_IN_GA(-(X, Y), Z) → EVALUATE_IN_GA(X, X1)
EVALUATE_IN_GA(*(X, Y), Z) → U8_GA(X, Y, Z, evaluate_in_ga(X, X1))
EVALUATE_IN_GA(*(X, Y), Z) → EVALUATE_IN_GA(X, X1)
EVALUATE_IN_GA(X, X) → U11_GA(X, myinteger_in_g(X))
EVALUATE_IN_GA(X, X) → MYINTEGER_IN_G(X)
MYINTEGER_IN_G(s(X)) → U12_G(X, myinteger_in_g(X))
MYINTEGER_IN_G(s(X)) → MYINTEGER_IN_G(X)
U8_GA(X, Y, Z, evaluate_out_ga(X, X1)) → U9_GA(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U8_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
U9_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_GA(X, Y, Z, mult_in_gga(X1, Y1, Z))
U9_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → MULT_IN_GGA(X1, Y1, Z)
MULT_IN_GGA(s(X), Y, R) → U15_GGA(X, Y, R, mult_in_gga(X, Y, Z))
MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)
U15_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → U16_GGA(X, Y, R, add_in_gga(Y, Z, R))
U15_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → ADD_IN_GGA(Y, Z, R)
ADD_IN_GGA(s(X), Y, s(Z)) → U13_GGA(X, Y, Z, add_in_gga(X, Y, Z))
ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)
U5_GA(X, Y, Z, evaluate_out_ga(X, X1)) → U6_GA(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U5_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
U6_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_GA(X, Y, Z, sub_in_gga(X1, Y1, Z))
U6_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → SUB_IN_GGA(X1, Y1, Z)
SUB_IN_GGA(s(X), s(Y), Z) → U14_GGA(X, Y, Z, sub_in_gga(X, Y, Z))
SUB_IN_GGA(s(X), s(Y), Z) → SUB_IN_GGA(X, Y, Z)
U2_GA(X, Y, Z, evaluate_out_ga(X, X1)) → U3_GA(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U2_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
U3_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_GA(X, Y, Z, add_in_gga(X1, Y1, Z))
U3_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → ADD_IN_GGA(X1, Y1, Z)

The TRS R consists of the following rules:

myis_in_ag(Z, X) → U1_ag(Z, X, evaluate_in_ga(X, Z))
evaluate_in_ga(+(X, Y), Z) → U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(-(X, Y), Z) → U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(*(X, Y), Z) → U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(X, X) → U11_ga(X, myinteger_in_g(X))
myinteger_in_g(s(X)) → U12_g(X, myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g(0)
U12_g(X, myinteger_out_g(X)) → myinteger_out_g(s(X))
U11_ga(X, myinteger_out_g(X)) → evaluate_out_ga(X, X)
U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
mult_in_gga(s(X), Y, R) → U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U16_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U16_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) → evaluate_out_ga(*(X, Y), Z)
U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
sub_in_gga(s(X), s(Y), Z) → U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) → evaluate_out_ga(-(X, Y), Z)
U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) → evaluate_out_ga(+(X, Y), Z)
U1_ag(Z, X, evaluate_out_ga(X, Z)) → myis_out_ag(Z, X)

The argument filtering Pi contains the following mapping:
myis_in_ag(x1, x2)  =  myis_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)
+(x1, x2)  =  +(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
-(x1, x2)  =  -(x1, x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x2, x4)
*(x1, x2)  =  *(x1, x2)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
U11_ga(x1, x2)  =  U11_ga(x1, x2)
myinteger_in_g(x1)  =  myinteger_in_g(x1)
s(x1)  =  s(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
0  =  0
myinteger_out_g(x1)  =  myinteger_out_g(x1)
evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x1, x2)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x4, x5)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x2, x4)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x1, x2, x4, x5)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x2, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x1, x2, x3)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
myis_out_ag(x1, x2)  =  myis_out_ag(x1, x2)
MYIS_IN_AG(x1, x2)  =  MYIS_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
EVALUATE_IN_GA(x1, x2)  =  EVALUATE_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x2, x4)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x1, x2, x4)
U11_GA(x1, x2)  =  U11_GA(x1, x2)
MYINTEGER_IN_G(x1)  =  MYINTEGER_IN_G(x1)
U12_G(x1, x2)  =  U12_G(x1, x2)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x4, x5)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x1, x2, x4)
MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)
U15_GGA(x1, x2, x3, x4)  =  U15_GGA(x1, x2, x4)
U16_GGA(x1, x2, x3, x4)  =  U16_GGA(x1, x2, x4)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x1, x2, x4)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x4, x5)
U7_GA(x1, x2, x3, x4)  =  U7_GA(x1, x2, x4)
SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)
U14_GGA(x1, x2, x3, x4)  =  U14_GGA(x1, x2, x4)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x4, x5)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x2, x4)

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

(46) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 19 less nodes.

(47) Complex Obligation (AND)

(48) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SUB_IN_GGA(s(X), s(Y), Z) → SUB_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

myis_in_ag(Z, X) → U1_ag(Z, X, evaluate_in_ga(X, Z))
evaluate_in_ga(+(X, Y), Z) → U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(-(X, Y), Z) → U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(*(X, Y), Z) → U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(X, X) → U11_ga(X, myinteger_in_g(X))
myinteger_in_g(s(X)) → U12_g(X, myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g(0)
U12_g(X, myinteger_out_g(X)) → myinteger_out_g(s(X))
U11_ga(X, myinteger_out_g(X)) → evaluate_out_ga(X, X)
U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
mult_in_gga(s(X), Y, R) → U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U16_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U16_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) → evaluate_out_ga(*(X, Y), Z)
U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
sub_in_gga(s(X), s(Y), Z) → U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) → evaluate_out_ga(-(X, Y), Z)
U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) → evaluate_out_ga(+(X, Y), Z)
U1_ag(Z, X, evaluate_out_ga(X, Z)) → myis_out_ag(Z, X)

The argument filtering Pi contains the following mapping:
myis_in_ag(x1, x2)  =  myis_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)
+(x1, x2)  =  +(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
-(x1, x2)  =  -(x1, x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x2, x4)
*(x1, x2)  =  *(x1, x2)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
U11_ga(x1, x2)  =  U11_ga(x1, x2)
myinteger_in_g(x1)  =  myinteger_in_g(x1)
s(x1)  =  s(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
0  =  0
myinteger_out_g(x1)  =  myinteger_out_g(x1)
evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x1, x2)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x4, x5)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x2, x4)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x1, x2, x4, x5)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x2, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x1, x2, x3)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
myis_out_ag(x1, x2)  =  myis_out_ag(x1, x2)
SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)

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

(49) UsableRulesProof (EQUIVALENT transformation)

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

(50) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SUB_IN_GGA(s(X), s(Y), Z) → SUB_IN_GGA(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)

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

(51) PiDPToQDPProof (SOUND transformation)

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

(52) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SUB_IN_GGA(s(X), s(Y)) → SUB_IN_GGA(X, Y)

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

(53) QDPSizeChangeProof (EQUIVALENT transformation)

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:

• SUB_IN_GGA(s(X), s(Y)) → SUB_IN_GGA(X, Y)
  The graph contains the following edges 1 > 1, 2 > 2

(54) TRUE

(55) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

myis_in_ag(Z, X) → U1_ag(Z, X, evaluate_in_ga(X, Z))
evaluate_in_ga(+(X, Y), Z) → U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(-(X, Y), Z) → U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(*(X, Y), Z) → U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(X, X) → U11_ga(X, myinteger_in_g(X))
myinteger_in_g(s(X)) → U12_g(X, myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g(0)
U12_g(X, myinteger_out_g(X)) → myinteger_out_g(s(X))
U11_ga(X, myinteger_out_g(X)) → evaluate_out_ga(X, X)
U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
mult_in_gga(s(X), Y, R) → U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U16_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U16_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) → evaluate_out_ga(*(X, Y), Z)
U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
sub_in_gga(s(X), s(Y), Z) → U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) → evaluate_out_ga(-(X, Y), Z)
U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) → evaluate_out_ga(+(X, Y), Z)
U1_ag(Z, X, evaluate_out_ga(X, Z)) → myis_out_ag(Z, X)

The argument filtering Pi contains the following mapping:
myis_in_ag(x1, x2)  =  myis_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)
+(x1, x2)  =  +(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
-(x1, x2)  =  -(x1, x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x2, x4)
*(x1, x2)  =  *(x1, x2)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
U11_ga(x1, x2)  =  U11_ga(x1, x2)
myinteger_in_g(x1)  =  myinteger_in_g(x1)
s(x1)  =  s(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
0  =  0
myinteger_out_g(x1)  =  myinteger_out_g(x1)
evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x1, x2)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x4, x5)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x2, x4)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x1, x2, x4, x5)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x2, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x1, x2, x3)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
myis_out_ag(x1, x2)  =  myis_out_ag(x1, x2)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)

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

(56) UsableRulesProof (EQUIVALENT transformation)

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

(57) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)

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

(58) PiDPToQDPProof (SOUND transformation)

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

(59) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ADD_IN_GGA(s(X), Y) → ADD_IN_GGA(X, Y)

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

(60) QDPSizeChangeProof (EQUIVALENT transformation)

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:

• ADD_IN_GGA(s(X), Y) → ADD_IN_GGA(X, Y)
  The graph contains the following edges 1 > 1, 2 >= 2

(61) TRUE

(62) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

myis_in_ag(Z, X) → U1_ag(Z, X, evaluate_in_ga(X, Z))
evaluate_in_ga(+(X, Y), Z) → U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(-(X, Y), Z) → U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(*(X, Y), Z) → U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(X, X) → U11_ga(X, myinteger_in_g(X))
myinteger_in_g(s(X)) → U12_g(X, myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g(0)
U12_g(X, myinteger_out_g(X)) → myinteger_out_g(s(X))
U11_ga(X, myinteger_out_g(X)) → evaluate_out_ga(X, X)
U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
mult_in_gga(s(X), Y, R) → U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U16_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U16_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) → evaluate_out_ga(*(X, Y), Z)
U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
sub_in_gga(s(X), s(Y), Z) → U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) → evaluate_out_ga(-(X, Y), Z)
U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) → evaluate_out_ga(+(X, Y), Z)
U1_ag(Z, X, evaluate_out_ga(X, Z)) → myis_out_ag(Z, X)

The argument filtering Pi contains the following mapping:
myis_in_ag(x1, x2)  =  myis_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)
+(x1, x2)  =  +(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
-(x1, x2)  =  -(x1, x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x2, x4)
*(x1, x2)  =  *(x1, x2)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
U11_ga(x1, x2)  =  U11_ga(x1, x2)
myinteger_in_g(x1)  =  myinteger_in_g(x1)
s(x1)  =  s(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
0  =  0
myinteger_out_g(x1)  =  myinteger_out_g(x1)
evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x1, x2)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x4, x5)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x2, x4)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x1, x2, x4, x5)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x2, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x1, x2, x3)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
myis_out_ag(x1, x2)  =  myis_out_ag(x1, x2)
MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)

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

(63) UsableRulesProof (EQUIVALENT transformation)

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

(64) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)

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

(65) PiDPToQDPProof (SOUND transformation)

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

(66) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MULT_IN_GGA(s(X), Y) → MULT_IN_GGA(X, Y)

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

(67) QDPSizeChangeProof (EQUIVALENT transformation)

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(s(X), Y) → MULT_IN_GGA(X, Y)
  The graph contains the following edges 1 > 1, 2 >= 2

(68) TRUE

(69) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MYINTEGER_IN_G(s(X)) → MYINTEGER_IN_G(X)

The TRS R consists of the following rules:

myis_in_ag(Z, X) → U1_ag(Z, X, evaluate_in_ga(X, Z))
evaluate_in_ga(+(X, Y), Z) → U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(-(X, Y), Z) → U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(*(X, Y), Z) → U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(X, X) → U11_ga(X, myinteger_in_g(X))
myinteger_in_g(s(X)) → U12_g(X, myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g(0)
U12_g(X, myinteger_out_g(X)) → myinteger_out_g(s(X))
U11_ga(X, myinteger_out_g(X)) → evaluate_out_ga(X, X)
U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
mult_in_gga(s(X), Y, R) → U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U16_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U16_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) → evaluate_out_ga(*(X, Y), Z)
U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
sub_in_gga(s(X), s(Y), Z) → U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) → evaluate_out_ga(-(X, Y), Z)
U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) → evaluate_out_ga(+(X, Y), Z)
U1_ag(Z, X, evaluate_out_ga(X, Z)) → myis_out_ag(Z, X)

The argument filtering Pi contains the following mapping:
myis_in_ag(x1, x2)  =  myis_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)
+(x1, x2)  =  +(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
-(x1, x2)  =  -(x1, x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x2, x4)
*(x1, x2)  =  *(x1, x2)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
U11_ga(x1, x2)  =  U11_ga(x1, x2)
myinteger_in_g(x1)  =  myinteger_in_g(x1)
s(x1)  =  s(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
0  =  0
myinteger_out_g(x1)  =  myinteger_out_g(x1)
evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x1, x2)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x4, x5)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x2, x4)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x1, x2, x4, x5)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x2, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x1, x2, x3)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
myis_out_ag(x1, x2)  =  myis_out_ag(x1, x2)
MYINTEGER_IN_G(x1)  =  MYINTEGER_IN_G(x1)

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

(70) UsableRulesProof (EQUIVALENT transformation)

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

(71) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MYINTEGER_IN_G(s(X)) → MYINTEGER_IN_G(X)

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

(72) PiDPToQDPProof (EQUIVALENT transformation)

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

(73) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MYINTEGER_IN_G(s(X)) → MYINTEGER_IN_G(X)

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

(74) QDPSizeChangeProof (EQUIVALENT transformation)

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:

• MYINTEGER_IN_G(s(X)) → MYINTEGER_IN_G(X)
  The graph contains the following edges 1 > 1

(75) TRUE

(76) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U2_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
EVALUATE_IN_GA(+(X, Y), Z) → U2_GA(X, Y, Z, evaluate_in_ga(X, X1))
EVALUATE_IN_GA(+(X, Y), Z) → EVALUATE_IN_GA(X, X1)
EVALUATE_IN_GA(-(X, Y), Z) → U5_GA(X, Y, Z, evaluate_in_ga(X, X1))
U5_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
EVALUATE_IN_GA(-(X, Y), Z) → EVALUATE_IN_GA(X, X1)
EVALUATE_IN_GA(*(X, Y), Z) → U8_GA(X, Y, Z, evaluate_in_ga(X, X1))
U8_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
EVALUATE_IN_GA(*(X, Y), Z) → EVALUATE_IN_GA(X, X1)

The TRS R consists of the following rules:

myis_in_ag(Z, X) → U1_ag(Z, X, evaluate_in_ga(X, Z))
evaluate_in_ga(+(X, Y), Z) → U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(-(X, Y), Z) → U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(*(X, Y), Z) → U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(X, X) → U11_ga(X, myinteger_in_g(X))
myinteger_in_g(s(X)) → U12_g(X, myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g(0)
U12_g(X, myinteger_out_g(X)) → myinteger_out_g(s(X))
U11_ga(X, myinteger_out_g(X)) → evaluate_out_ga(X, X)
U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
mult_in_gga(s(X), Y, R) → U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U16_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U16_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) → evaluate_out_ga(*(X, Y), Z)
U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
sub_in_gga(s(X), s(Y), Z) → U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) → evaluate_out_ga(-(X, Y), Z)
U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) → evaluate_out_ga(+(X, Y), Z)
U1_ag(Z, X, evaluate_out_ga(X, Z)) → myis_out_ag(Z, X)

The argument filtering Pi contains the following mapping:
myis_in_ag(x1, x2)  =  myis_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)
+(x1, x2)  =  +(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
-(x1, x2)  =  -(x1, x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x2, x4)
*(x1, x2)  =  *(x1, x2)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
U11_ga(x1, x2)  =  U11_ga(x1, x2)
myinteger_in_g(x1)  =  myinteger_in_g(x1)
s(x1)  =  s(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
0  =  0
myinteger_out_g(x1)  =  myinteger_out_g(x1)
evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x1, x2)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x4, x5)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x2, x4)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x1, x2, x4, x5)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x2, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x1, x2, x3)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
myis_out_ag(x1, x2)  =  myis_out_ag(x1, x2)
EVALUATE_IN_GA(x1, x2)  =  EVALUATE_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x2, x4)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x1, x2, x4)

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

(77) UsableRulesProof (EQUIVALENT transformation)

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

(78) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U2_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
EVALUATE_IN_GA(+(X, Y), Z) → U2_GA(X, Y, Z, evaluate_in_ga(X, X1))
EVALUATE_IN_GA(+(X, Y), Z) → EVALUATE_IN_GA(X, X1)
EVALUATE_IN_GA(-(X, Y), Z) → U5_GA(X, Y, Z, evaluate_in_ga(X, X1))
U5_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
EVALUATE_IN_GA(-(X, Y), Z) → EVALUATE_IN_GA(X, X1)
EVALUATE_IN_GA(*(X, Y), Z) → U8_GA(X, Y, Z, evaluate_in_ga(X, X1))
U8_GA(X, Y, Z, evaluate_out_ga(X, X1)) → EVALUATE_IN_GA(Y, Y1)
EVALUATE_IN_GA(*(X, Y), Z) → EVALUATE_IN_GA(X, X1)

The TRS R consists of the following rules:

evaluate_in_ga(+(X, Y), Z) → U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(-(X, Y), Z) → U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(*(X, Y), Z) → U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
evaluate_in_ga(X, X) → U11_ga(X, myinteger_in_g(X))
U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) → U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
U11_ga(X, myinteger_out_g(X)) → evaluate_out_ga(X, X)
U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) → U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
myinteger_in_g(s(X)) → U12_g(X, myinteger_in_g(X))
myinteger_in_g(0) → myinteger_out_g(0)
U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) → evaluate_out_ga(+(X, Y), Z)
U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) → evaluate_out_ga(-(X, Y), Z)
U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) → evaluate_out_ga(*(X, Y), Z)
U12_g(X, myinteger_out_g(X)) → myinteger_out_g(s(X))
add_in_gga(s(X), Y, s(Z)) → U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
sub_in_gga(s(X), s(Y), Z) → U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
sub_in_gga(X, 0, X) → sub_out_gga(X, 0, X)
mult_in_gga(s(X), Y, R) → U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) → sub_out_gga(s(X), s(Y), Z)
U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U16_gga(X, Y, R, add_in_gga(Y, Z, R))
U16_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)

The argument filtering Pi contains the following mapping:
evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)
+(x1, x2)  =  +(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
-(x1, x2)  =  -(x1, x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x2, x4)
*(x1, x2)  =  *(x1, x2)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
U11_ga(x1, x2)  =  U11_ga(x1, x2)
myinteger_in_g(x1)  =  myinteger_in_g(x1)
s(x1)  =  s(x1)
U12_g(x1, x2)  =  U12_g(x1, x2)
0  =  0
myinteger_out_g(x1)  =  myinteger_out_g(x1)
evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x1, x2)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x4, x5)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x2, x4)
mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
mult_out_gga(x1, x2, x3)  =  mult_out_gga(x1, x2, x3)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x1, x2, x4, x5)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x2, x4)
sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
sub_out_gga(x1, x2, x3)  =  sub_out_gga(x1, x2, x3)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
EVALUATE_IN_GA(x1, x2)  =  EVALUATE_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x2, x4)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x1, x2, x4)

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