Left Termination of the query pattern
factorial_in_2(g, a)
w.r.t. the given Prolog program could successfully be proven:
↳ Prolog
↳ PrologToPiTRSProof
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).
We use the technique of [30]. 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
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
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)
Using Dependency Pairs [1,30] 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)
ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2)
U10_GGA(x1, x2, x3, x4) = U10_GGA(x4)
U12_GA(x1, x2, x3) = U12_GA(x3)
U11_GA(x1, x2, x3) = U11_GA(x1, x3)
U9_GGA(x1, x2, x3, x4) = U9_GGA(x2, x4)
MULT_IN_GGA(x1, x2, x3) = MULT_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4) = U8_GGA(x4)
FACTORIAL_IN_GA(x1, x2) = FACTORIAL_IN_GA(x1)
We have to consider all (P,R,Pi)-chains
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
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)
ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2)
U10_GGA(x1, x2, x3, x4) = U10_GGA(x4)
U12_GA(x1, x2, x3) = U12_GA(x3)
U11_GA(x1, x2, x3) = U11_GA(x1, x3)
U9_GGA(x1, x2, x3, x4) = U9_GGA(x2, x4)
MULT_IN_GGA(x1, x2, x3) = MULT_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4) = U8_GGA(x4)
FACTORIAL_IN_GA(x1, x2) = FACTORIAL_IN_GA(x1)
We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 3 SCCs with 7 less nodes.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
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
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PiDP
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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
↳ PiDP
↳ PiDP
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.
By using the subterm criterion [20] together with the size-change analysis [32] 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
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
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
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
↳ PiDP
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.
By using the subterm criterion [20] together with the size-change analysis [32] 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
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
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
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
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.
By using the subterm criterion [20] together with the size-change analysis [32] 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