(0) Obligation:
Clauses:
isNat(s(X)) :- isNat(X).
isNat(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).
even(s(X)) :- odd(X).
even(0).
odd(s(X)) :- even(X).
odd(s(0)).
add(s(X), Y, s(Z)) :- add(X, Y, Z).
add(0, X, X).
mult(s(X), Y, R) :- ','(mult(X, Y, Z), add(Y, Z, R)).
mult(0, Y, 0).
factorial(s(X), R) :- ','(factorial(X, Y), mult(s(X), Y, R)).
factorial(0, s(0)).
Queries:
factorial(g,a).
(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:
factorial_in: (b,f)
mult_in: (b,b,f)
add_in: (b,b,f)
Transforming
Prolog into the following
Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)
The argument filtering Pi contains the following mapping:
factorial_in_ga(
x1,
x2) =
factorial_in_ga(
x1)
s(
x1) =
s(
x1)
U11_ga(
x1,
x2,
x3) =
U11_ga(
x1,
x3)
0 =
0
factorial_out_ga(
x1,
x2) =
factorial_out_ga(
x1,
x2)
U12_ga(
x1,
x2,
x3) =
U12_ga(
x1,
x3)
mult_in_gga(
x1,
x2,
x3) =
mult_in_gga(
x1,
x2)
U9_gga(
x1,
x2,
x3,
x4) =
U9_gga(
x1,
x2,
x4)
mult_out_gga(
x1,
x2,
x3) =
mult_out_gga(
x1,
x2,
x3)
U10_gga(
x1,
x2,
x3,
x4) =
U10_gga(
x1,
x2,
x4)
add_in_gga(
x1,
x2,
x3) =
add_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4) =
U8_gga(
x1,
x2,
x4)
add_out_gga(
x1,
x2,
x3) =
add_out_gga(
x1,
x2,
x3)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(2) Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:
factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)
The argument filtering Pi contains the following mapping:
factorial_in_ga(
x1,
x2) =
factorial_in_ga(
x1)
s(
x1) =
s(
x1)
U11_ga(
x1,
x2,
x3) =
U11_ga(
x1,
x3)
0 =
0
factorial_out_ga(
x1,
x2) =
factorial_out_ga(
x1,
x2)
U12_ga(
x1,
x2,
x3) =
U12_ga(
x1,
x3)
mult_in_gga(
x1,
x2,
x3) =
mult_in_gga(
x1,
x2)
U9_gga(
x1,
x2,
x3,
x4) =
U9_gga(
x1,
x2,
x4)
mult_out_gga(
x1,
x2,
x3) =
mult_out_gga(
x1,
x2,
x3)
U10_gga(
x1,
x2,
x3,
x4) =
U10_gga(
x1,
x2,
x4)
add_in_gga(
x1,
x2,
x3) =
add_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4) =
U8_gga(
x1,
x2,
x4)
add_out_gga(
x1,
x2,
x3) =
add_out_gga(
x1,
x2,
x3)
(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:
FACTORIAL_IN_GA(s(X), R) → U11_GA(X, R, factorial_in_ga(X, Y))
FACTORIAL_IN_GA(s(X), R) → FACTORIAL_IN_GA(X, Y)
U11_GA(X, R, factorial_out_ga(X, Y)) → U12_GA(X, R, mult_in_gga(s(X), Y, R))
U11_GA(X, R, factorial_out_ga(X, Y)) → MULT_IN_GGA(s(X), Y, R)
MULT_IN_GGA(s(X), Y, R) → U9_GGA(X, Y, R, mult_in_gga(X, Y, Z))
MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)
U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → U10_GGA(X, Y, R, add_in_gga(Y, Z, R))
U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → ADD_IN_GGA(Y, Z, R)
ADD_IN_GGA(s(X), Y, s(Z)) → U8_GGA(X, Y, Z, add_in_gga(X, Y, Z))
ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)
The TRS R consists of the following rules:
factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)
The argument filtering Pi contains the following mapping:
factorial_in_ga(
x1,
x2) =
factorial_in_ga(
x1)
s(
x1) =
s(
x1)
U11_ga(
x1,
x2,
x3) =
U11_ga(
x1,
x3)
0 =
0
factorial_out_ga(
x1,
x2) =
factorial_out_ga(
x1,
x2)
U12_ga(
x1,
x2,
x3) =
U12_ga(
x1,
x3)
mult_in_gga(
x1,
x2,
x3) =
mult_in_gga(
x1,
x2)
U9_gga(
x1,
x2,
x3,
x4) =
U9_gga(
x1,
x2,
x4)
mult_out_gga(
x1,
x2,
x3) =
mult_out_gga(
x1,
x2,
x3)
U10_gga(
x1,
x2,
x3,
x4) =
U10_gga(
x1,
x2,
x4)
add_in_gga(
x1,
x2,
x3) =
add_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4) =
U8_gga(
x1,
x2,
x4)
add_out_gga(
x1,
x2,
x3) =
add_out_gga(
x1,
x2,
x3)
FACTORIAL_IN_GA(
x1,
x2) =
FACTORIAL_IN_GA(
x1)
U11_GA(
x1,
x2,
x3) =
U11_GA(
x1,
x3)
U12_GA(
x1,
x2,
x3) =
U12_GA(
x1,
x3)
MULT_IN_GGA(
x1,
x2,
x3) =
MULT_IN_GGA(
x1,
x2)
U9_GGA(
x1,
x2,
x3,
x4) =
U9_GGA(
x1,
x2,
x4)
U10_GGA(
x1,
x2,
x3,
x4) =
U10_GGA(
x1,
x2,
x4)
ADD_IN_GGA(
x1,
x2,
x3) =
ADD_IN_GGA(
x1,
x2)
U8_GGA(
x1,
x2,
x3,
x4) =
U8_GGA(
x1,
x2,
x4)
We have to consider all (P,R,Pi)-chains
(4) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
FACTORIAL_IN_GA(s(X), R) → U11_GA(X, R, factorial_in_ga(X, Y))
FACTORIAL_IN_GA(s(X), R) → FACTORIAL_IN_GA(X, Y)
U11_GA(X, R, factorial_out_ga(X, Y)) → U12_GA(X, R, mult_in_gga(s(X), Y, R))
U11_GA(X, R, factorial_out_ga(X, Y)) → MULT_IN_GGA(s(X), Y, R)
MULT_IN_GGA(s(X), Y, R) → U9_GGA(X, Y, R, mult_in_gga(X, Y, Z))
MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)
U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → U10_GGA(X, Y, R, add_in_gga(Y, Z, R))
U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → ADD_IN_GGA(Y, Z, R)
ADD_IN_GGA(s(X), Y, s(Z)) → U8_GGA(X, Y, Z, add_in_gga(X, Y, Z))
ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)
The TRS R consists of the following rules:
factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)
The argument filtering Pi contains the following mapping:
factorial_in_ga(
x1,
x2) =
factorial_in_ga(
x1)
s(
x1) =
s(
x1)
U11_ga(
x1,
x2,
x3) =
U11_ga(
x1,
x3)
0 =
0
factorial_out_ga(
x1,
x2) =
factorial_out_ga(
x1,
x2)
U12_ga(
x1,
x2,
x3) =
U12_ga(
x1,
x3)
mult_in_gga(
x1,
x2,
x3) =
mult_in_gga(
x1,
x2)
U9_gga(
x1,
x2,
x3,
x4) =
U9_gga(
x1,
x2,
x4)
mult_out_gga(
x1,
x2,
x3) =
mult_out_gga(
x1,
x2,
x3)
U10_gga(
x1,
x2,
x3,
x4) =
U10_gga(
x1,
x2,
x4)
add_in_gga(
x1,
x2,
x3) =
add_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4) =
U8_gga(
x1,
x2,
x4)
add_out_gga(
x1,
x2,
x3) =
add_out_gga(
x1,
x2,
x3)
FACTORIAL_IN_GA(
x1,
x2) =
FACTORIAL_IN_GA(
x1)
U11_GA(
x1,
x2,
x3) =
U11_GA(
x1,
x3)
U12_GA(
x1,
x2,
x3) =
U12_GA(
x1,
x3)
MULT_IN_GGA(
x1,
x2,
x3) =
MULT_IN_GGA(
x1,
x2)
U9_GGA(
x1,
x2,
x3,
x4) =
U9_GGA(
x1,
x2,
x4)
U10_GGA(
x1,
x2,
x3,
x4) =
U10_GGA(
x1,
x2,
x4)
ADD_IN_GGA(
x1,
x2,
x3) =
ADD_IN_GGA(
x1,
x2)
U8_GGA(
x1,
x2,
x3,
x4) =
U8_GGA(
x1,
x2,
x4)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 7 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)
The TRS R consists of the following rules:
factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)
The argument filtering Pi contains the following mapping:
factorial_in_ga(
x1,
x2) =
factorial_in_ga(
x1)
s(
x1) =
s(
x1)
U11_ga(
x1,
x2,
x3) =
U11_ga(
x1,
x3)
0 =
0
factorial_out_ga(
x1,
x2) =
factorial_out_ga(
x1,
x2)
U12_ga(
x1,
x2,
x3) =
U12_ga(
x1,
x3)
mult_in_gga(
x1,
x2,
x3) =
mult_in_gga(
x1,
x2)
U9_gga(
x1,
x2,
x3,
x4) =
U9_gga(
x1,
x2,
x4)
mult_out_gga(
x1,
x2,
x3) =
mult_out_gga(
x1,
x2,
x3)
U10_gga(
x1,
x2,
x3,
x4) =
U10_gga(
x1,
x2,
x4)
add_in_gga(
x1,
x2,
x3) =
add_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4) =
U8_gga(
x1,
x2,
x4)
add_out_gga(
x1,
x2,
x3) =
add_out_gga(
x1,
x2,
x3)
ADD_IN_GGA(
x1,
x2,
x3) =
ADD_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:
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
(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:
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.
(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:
- ADD_IN_GGA(s(X), Y) → ADD_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:
MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)
The TRS R consists of the following rules:
factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)
The argument filtering Pi contains the following mapping:
factorial_in_ga(
x1,
x2) =
factorial_in_ga(
x1)
s(
x1) =
s(
x1)
U11_ga(
x1,
x2,
x3) =
U11_ga(
x1,
x3)
0 =
0
factorial_out_ga(
x1,
x2) =
factorial_out_ga(
x1,
x2)
U12_ga(
x1,
x2,
x3) =
U12_ga(
x1,
x3)
mult_in_gga(
x1,
x2,
x3) =
mult_in_gga(
x1,
x2)
U9_gga(
x1,
x2,
x3,
x4) =
U9_gga(
x1,
x2,
x4)
mult_out_gga(
x1,
x2,
x3) =
mult_out_gga(
x1,
x2,
x3)
U10_gga(
x1,
x2,
x3,
x4) =
U10_gga(
x1,
x2,
x4)
add_in_gga(
x1,
x2,
x3) =
add_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4) =
U8_gga(
x1,
x2,
x4)
add_out_gga(
x1,
x2,
x3) =
add_out_gga(
x1,
x2,
x3)
MULT_IN_GGA(
x1,
x2,
x3) =
MULT_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:
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
(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:
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.
(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:
- MULT_IN_GGA(s(X), Y) → MULT_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:
FACTORIAL_IN_GA(s(X), R) → FACTORIAL_IN_GA(X, Y)
The TRS R consists of the following rules:
factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)
The argument filtering Pi contains the following mapping:
factorial_in_ga(
x1,
x2) =
factorial_in_ga(
x1)
s(
x1) =
s(
x1)
U11_ga(
x1,
x2,
x3) =
U11_ga(
x1,
x3)
0 =
0
factorial_out_ga(
x1,
x2) =
factorial_out_ga(
x1,
x2)
U12_ga(
x1,
x2,
x3) =
U12_ga(
x1,
x3)
mult_in_gga(
x1,
x2,
x3) =
mult_in_gga(
x1,
x2)
U9_gga(
x1,
x2,
x3,
x4) =
U9_gga(
x1,
x2,
x4)
mult_out_gga(
x1,
x2,
x3) =
mult_out_gga(
x1,
x2,
x3)
U10_gga(
x1,
x2,
x3,
x4) =
U10_gga(
x1,
x2,
x4)
add_in_gga(
x1,
x2,
x3) =
add_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4) =
U8_gga(
x1,
x2,
x4)
add_out_gga(
x1,
x2,
x3) =
add_out_gga(
x1,
x2,
x3)
FACTORIAL_IN_GA(
x1,
x2) =
FACTORIAL_IN_GA(
x1)
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:
FACTORIAL_IN_GA(s(X), R) → FACTORIAL_IN_GA(X, Y)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
FACTORIAL_IN_GA(
x1,
x2) =
FACTORIAL_IN_GA(
x1)
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:
FACTORIAL_IN_GA(s(X)) → FACTORIAL_IN_GA(X)
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:
- FACTORIAL_IN_GA(s(X)) → FACTORIAL_IN_GA(X)
The graph contains the following edges 1 > 1
(27) TRUE
(28) PrologToPiTRSProof (SOUND transformation)
We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
factorial_in: (b,f)
mult_in: (b,b,f)
add_in: (b,b,f)
Transforming
Prolog into the following
Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)
The argument filtering Pi contains the following mapping:
factorial_in_ga(
x1,
x2) =
factorial_in_ga(
x1)
s(
x1) =
s(
x1)
U11_ga(
x1,
x2,
x3) =
U11_ga(
x1,
x3)
0 =
0
factorial_out_ga(
x1,
x2) =
factorial_out_ga(
x2)
U12_ga(
x1,
x2,
x3) =
U12_ga(
x3)
mult_in_gga(
x1,
x2,
x3) =
mult_in_gga(
x1,
x2)
U9_gga(
x1,
x2,
x3,
x4) =
U9_gga(
x2,
x4)
mult_out_gga(
x1,
x2,
x3) =
mult_out_gga(
x3)
U10_gga(
x1,
x2,
x3,
x4) =
U10_gga(
x4)
add_in_gga(
x1,
x2,
x3) =
add_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4) =
U8_gga(
x4)
add_out_gga(
x1,
x2,
x3) =
add_out_gga(
x3)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(29) Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:
factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)
The argument filtering Pi contains the following mapping:
factorial_in_ga(
x1,
x2) =
factorial_in_ga(
x1)
s(
x1) =
s(
x1)
U11_ga(
x1,
x2,
x3) =
U11_ga(
x1,
x3)
0 =
0
factorial_out_ga(
x1,
x2) =
factorial_out_ga(
x2)
U12_ga(
x1,
x2,
x3) =
U12_ga(
x3)
mult_in_gga(
x1,
x2,
x3) =
mult_in_gga(
x1,
x2)
U9_gga(
x1,
x2,
x3,
x4) =
U9_gga(
x2,
x4)
mult_out_gga(
x1,
x2,
x3) =
mult_out_gga(
x3)
U10_gga(
x1,
x2,
x3,
x4) =
U10_gga(
x4)
add_in_gga(
x1,
x2,
x3) =
add_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4) =
U8_gga(
x4)
add_out_gga(
x1,
x2,
x3) =
add_out_gga(
x3)
(30) 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:
FACTORIAL_IN_GA(s(X), R) → U11_GA(X, R, factorial_in_ga(X, Y))
FACTORIAL_IN_GA(s(X), R) → FACTORIAL_IN_GA(X, Y)
U11_GA(X, R, factorial_out_ga(X, Y)) → U12_GA(X, R, mult_in_gga(s(X), Y, R))
U11_GA(X, R, factorial_out_ga(X, Y)) → MULT_IN_GGA(s(X), Y, R)
MULT_IN_GGA(s(X), Y, R) → U9_GGA(X, Y, R, mult_in_gga(X, Y, Z))
MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)
U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → U10_GGA(X, Y, R, add_in_gga(Y, Z, R))
U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → ADD_IN_GGA(Y, Z, R)
ADD_IN_GGA(s(X), Y, s(Z)) → U8_GGA(X, Y, Z, add_in_gga(X, Y, Z))
ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)
The TRS R consists of the following rules:
factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)
The argument filtering Pi contains the following mapping:
factorial_in_ga(
x1,
x2) =
factorial_in_ga(
x1)
s(
x1) =
s(
x1)
U11_ga(
x1,
x2,
x3) =
U11_ga(
x1,
x3)
0 =
0
factorial_out_ga(
x1,
x2) =
factorial_out_ga(
x2)
U12_ga(
x1,
x2,
x3) =
U12_ga(
x3)
mult_in_gga(
x1,
x2,
x3) =
mult_in_gga(
x1,
x2)
U9_gga(
x1,
x2,
x3,
x4) =
U9_gga(
x2,
x4)
mult_out_gga(
x1,
x2,
x3) =
mult_out_gga(
x3)
U10_gga(
x1,
x2,
x3,
x4) =
U10_gga(
x4)
add_in_gga(
x1,
x2,
x3) =
add_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4) =
U8_gga(
x4)
add_out_gga(
x1,
x2,
x3) =
add_out_gga(
x3)
FACTORIAL_IN_GA(
x1,
x2) =
FACTORIAL_IN_GA(
x1)
U11_GA(
x1,
x2,
x3) =
U11_GA(
x1,
x3)
U12_GA(
x1,
x2,
x3) =
U12_GA(
x3)
MULT_IN_GGA(
x1,
x2,
x3) =
MULT_IN_GGA(
x1,
x2)
U9_GGA(
x1,
x2,
x3,
x4) =
U9_GGA(
x2,
x4)
U10_GGA(
x1,
x2,
x3,
x4) =
U10_GGA(
x4)
ADD_IN_GGA(
x1,
x2,
x3) =
ADD_IN_GGA(
x1,
x2)
U8_GGA(
x1,
x2,
x3,
x4) =
U8_GGA(
x4)
We have to consider all (P,R,Pi)-chains
(31) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
FACTORIAL_IN_GA(s(X), R) → U11_GA(X, R, factorial_in_ga(X, Y))
FACTORIAL_IN_GA(s(X), R) → FACTORIAL_IN_GA(X, Y)
U11_GA(X, R, factorial_out_ga(X, Y)) → U12_GA(X, R, mult_in_gga(s(X), Y, R))
U11_GA(X, R, factorial_out_ga(X, Y)) → MULT_IN_GGA(s(X), Y, R)
MULT_IN_GGA(s(X), Y, R) → U9_GGA(X, Y, R, mult_in_gga(X, Y, Z))
MULT_IN_GGA(s(X), Y, R) → MULT_IN_GGA(X, Y, Z)
U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → U10_GGA(X, Y, R, add_in_gga(Y, Z, R))
U9_GGA(X, Y, R, mult_out_gga(X, Y, Z)) → ADD_IN_GGA(Y, Z, R)
ADD_IN_GGA(s(X), Y, s(Z)) → U8_GGA(X, Y, Z, add_in_gga(X, Y, Z))
ADD_IN_GGA(s(X), Y, s(Z)) → ADD_IN_GGA(X, Y, Z)
The TRS R consists of the following rules:
factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)
The argument filtering Pi contains the following mapping:
factorial_in_ga(
x1,
x2) =
factorial_in_ga(
x1)
s(
x1) =
s(
x1)
U11_ga(
x1,
x2,
x3) =
U11_ga(
x1,
x3)
0 =
0
factorial_out_ga(
x1,
x2) =
factorial_out_ga(
x2)
U12_ga(
x1,
x2,
x3) =
U12_ga(
x3)
mult_in_gga(
x1,
x2,
x3) =
mult_in_gga(
x1,
x2)
U9_gga(
x1,
x2,
x3,
x4) =
U9_gga(
x2,
x4)
mult_out_gga(
x1,
x2,
x3) =
mult_out_gga(
x3)
U10_gga(
x1,
x2,
x3,
x4) =
U10_gga(
x4)
add_in_gga(
x1,
x2,
x3) =
add_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4) =
U8_gga(
x4)
add_out_gga(
x1,
x2,
x3) =
add_out_gga(
x3)
FACTORIAL_IN_GA(
x1,
x2) =
FACTORIAL_IN_GA(
x1)
U11_GA(
x1,
x2,
x3) =
U11_GA(
x1,
x3)
U12_GA(
x1,
x2,
x3) =
U12_GA(
x3)
MULT_IN_GGA(
x1,
x2,
x3) =
MULT_IN_GGA(
x1,
x2)
U9_GGA(
x1,
x2,
x3,
x4) =
U9_GGA(
x2,
x4)
U10_GGA(
x1,
x2,
x3,
x4) =
U10_GGA(
x4)
ADD_IN_GGA(
x1,
x2,
x3) =
ADD_IN_GGA(
x1,
x2)
U8_GGA(
x1,
x2,
x3,
x4) =
U8_GGA(
x4)
We have to consider all (P,R,Pi)-chains
(32) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 7 less nodes.
(33) Complex Obligation (AND)
(34) 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:
factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)
The argument filtering Pi contains the following mapping:
factorial_in_ga(
x1,
x2) =
factorial_in_ga(
x1)
s(
x1) =
s(
x1)
U11_ga(
x1,
x2,
x3) =
U11_ga(
x1,
x3)
0 =
0
factorial_out_ga(
x1,
x2) =
factorial_out_ga(
x2)
U12_ga(
x1,
x2,
x3) =
U12_ga(
x3)
mult_in_gga(
x1,
x2,
x3) =
mult_in_gga(
x1,
x2)
U9_gga(
x1,
x2,
x3,
x4) =
U9_gga(
x2,
x4)
mult_out_gga(
x1,
x2,
x3) =
mult_out_gga(
x3)
U10_gga(
x1,
x2,
x3,
x4) =
U10_gga(
x4)
add_in_gga(
x1,
x2,
x3) =
add_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4) =
U8_gga(
x4)
add_out_gga(
x1,
x2,
x3) =
add_out_gga(
x3)
ADD_IN_GGA(
x1,
x2,
x3) =
ADD_IN_GGA(
x1,
x2)
We have to consider all (P,R,Pi)-chains
(35) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(36) 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
(37) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(38) 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.
(39) 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
(40) TRUE
(41) 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:
factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)
The argument filtering Pi contains the following mapping:
factorial_in_ga(
x1,
x2) =
factorial_in_ga(
x1)
s(
x1) =
s(
x1)
U11_ga(
x1,
x2,
x3) =
U11_ga(
x1,
x3)
0 =
0
factorial_out_ga(
x1,
x2) =
factorial_out_ga(
x2)
U12_ga(
x1,
x2,
x3) =
U12_ga(
x3)
mult_in_gga(
x1,
x2,
x3) =
mult_in_gga(
x1,
x2)
U9_gga(
x1,
x2,
x3,
x4) =
U9_gga(
x2,
x4)
mult_out_gga(
x1,
x2,
x3) =
mult_out_gga(
x3)
U10_gga(
x1,
x2,
x3,
x4) =
U10_gga(
x4)
add_in_gga(
x1,
x2,
x3) =
add_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4) =
U8_gga(
x4)
add_out_gga(
x1,
x2,
x3) =
add_out_gga(
x3)
MULT_IN_GGA(
x1,
x2,
x3) =
MULT_IN_GGA(
x1,
x2)
We have to consider all (P,R,Pi)-chains
(42) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(43) 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
(44) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(45) 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.
(46) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
FACTORIAL_IN_GA(s(X), R) → FACTORIAL_IN_GA(X, Y)
The TRS R consists of the following rules:
factorial_in_ga(s(X), R) → U11_ga(X, R, factorial_in_ga(X, Y))
factorial_in_ga(0, s(0)) → factorial_out_ga(0, s(0))
U11_ga(X, R, factorial_out_ga(X, Y)) → U12_ga(X, R, mult_in_gga(s(X), Y, R))
mult_in_gga(s(X), Y, R) → U9_gga(X, Y, R, mult_in_gga(X, Y, Z))
mult_in_gga(0, Y, 0) → mult_out_gga(0, Y, 0)
U9_gga(X, Y, R, mult_out_gga(X, Y, Z)) → U10_gga(X, Y, R, add_in_gga(Y, Z, R))
add_in_gga(s(X), Y, s(Z)) → U8_gga(X, Y, Z, add_in_gga(X, Y, Z))
add_in_gga(0, X, X) → add_out_gga(0, X, X)
U8_gga(X, Y, Z, add_out_gga(X, Y, Z)) → add_out_gga(s(X), Y, s(Z))
U10_gga(X, Y, R, add_out_gga(Y, Z, R)) → mult_out_gga(s(X), Y, R)
U12_ga(X, R, mult_out_gga(s(X), Y, R)) → factorial_out_ga(s(X), R)
The argument filtering Pi contains the following mapping:
factorial_in_ga(
x1,
x2) =
factorial_in_ga(
x1)
s(
x1) =
s(
x1)
U11_ga(
x1,
x2,
x3) =
U11_ga(
x1,
x3)
0 =
0
factorial_out_ga(
x1,
x2) =
factorial_out_ga(
x2)
U12_ga(
x1,
x2,
x3) =
U12_ga(
x3)
mult_in_gga(
x1,
x2,
x3) =
mult_in_gga(
x1,
x2)
U9_gga(
x1,
x2,
x3,
x4) =
U9_gga(
x2,
x4)
mult_out_gga(
x1,
x2,
x3) =
mult_out_gga(
x3)
U10_gga(
x1,
x2,
x3,
x4) =
U10_gga(
x4)
add_in_gga(
x1,
x2,
x3) =
add_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3,
x4) =
U8_gga(
x4)
add_out_gga(
x1,
x2,
x3) =
add_out_gga(
x3)
FACTORIAL_IN_GA(
x1,
x2) =
FACTORIAL_IN_GA(
x1)
We have to consider all (P,R,Pi)-chains
(47) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(48) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
FACTORIAL_IN_GA(s(X), R) → FACTORIAL_IN_GA(X, Y)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
FACTORIAL_IN_GA(
x1,
x2) =
FACTORIAL_IN_GA(
x1)
We have to consider all (P,R,Pi)-chains