Left Termination proof of /tmp/tmppDHFXa/gcd.pl

Left Termination of the query pattern gcd_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog

  ↳1 PrologToPiTRSProof (⇐)

    ↳2 PiTRS

    ↳3 DependencyPairsProof (⇔)

    ↳4 PiDP

    ↳5 DependencyGraphProof (⇔)

    ↳6 AND

      ↳7 PiDP

        ↳8 UsableRulesProof (⇔)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇔)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔)

        ↳13 TRUE

      ↳14 PiDP

        ↳15 UsableRulesProof (⇔)

        ↳16 PiDP

        ↳17 PiDPToQDPProof (⇐)

        ↳18 QDP

        ↳19 QDPSizeChangeProof (⇔)

        ↳20 TRUE

      ↳21 PiDP

        ↳22 UsableRulesProof (⇔)

        ↳23 PiDP

        ↳24 PiDPToQDPProof (⇔)

        ↳25 QDP

        ↳26 QDPSizeChangeProof (⇔)

        ↳27 TRUE

      ↳28 PiDP

        ↳29 UsableRulesProof (⇔)

        ↳30 PiDP

        ↳31 PiDPToQDPProof (⇐)

        ↳32 QDP

        ↳33 QDPOrderProof (⇔)

        ↳34 QDP

        ↳35 DependencyGraphProof (⇔)

        ↳36 TRUE

  ↳37 PrologToPiTRSProof (⇐)

    ↳38 PiTRS

    ↳39 DependencyPairsProof (⇔)

    ↳40 PiDP

    ↳41 DependencyGraphProof (⇔)

    ↳42 AND

      ↳43 PiDP

        ↳44 UsableRulesProof (⇔)

        ↳45 PiDP

        ↳46 PiDPToQDPProof (⇔)

        ↳47 QDP

        ↳48 QDPSizeChangeProof (⇔)

        ↳49 TRUE

      ↳50 PiDP

        ↳51 UsableRulesProof (⇔)

        ↳52 PiDP

        ↳53 PiDPToQDPProof (⇐)

        ↳54 QDP

        ↳55 QDPSizeChangeProof (⇔)

        ↳56 TRUE

      ↳57 PiDP

        ↳58 UsableRulesProof (⇔)

        ↳59 PiDP

        ↳60 PiDPToQDPProof (⇔)

        ↳61 QDP

        ↳62 QDPSizeChangeProof (⇔)

        ↳63 TRUE

      ↳64 PiDP

        ↳65 UsableRulesProof (⇔)

        ↳66 PiDP

        ↳67 PiDPToQDPProof (⇐)

        ↳68 QDP

        ↳69 QDPOrderProof (⇔)

        ↳70 QDP

        ↳71 DependencyGraphProof (⇔)

        ↳72 QDP

        ↳73 UsableRulesProof (⇔)

        ↳74 QDP

        ↳75 QReductionProof (⇔)

        ↳76 QDP

(0) Obligation:

Clauses:

gcd(X, Y, D) :- ','(le(X, Y), gcd_le(X, Y, D)).
gcd(X, Y, D) :- ','(gt(X, Y), gcd_le(Y, X, D)).
gcd_le(0, Y, Y).
gcd_le(s(X), Y, D) :- ','(add(s(X), Z, Y), gcd(s(X), Z, D)).
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).
add(s(X), Y, s(Z)) :- add(X, Y, Z).
add(0, X, X).

Queries:

gcd(g,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:
gcd_in: (b,b,f)
le_in: (b,b)
gcd_le_in: (b,b,f)
add_in: (b,f,b)
gt_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

gcd_in_gga(X, Y, D) → U1_gga(X, Y, D, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U1_gga(X, Y, D, le_out_gg(X, Y)) → U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D))
gcd_le_in_gga(0, Y, Y) → gcd_le_out_gga(0, Y, Y)
gcd_le_in_gga(s(X), Y, D) → U5_gga(X, Y, D, add_in_gag(s(X), Z, Y))
add_in_gag(s(X), Y, s(Z)) → U9_gag(X, Y, Z, add_in_gag(X, Y, Z))
add_in_gag(0, X, X) → add_out_gag(0, X, X)
U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) → U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D))
gcd_in_gga(X, Y, D) → U3_gga(X, Y, D, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U3_gga(X, Y, D, gt_out_gg(X, Y)) → U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D))
U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) → gcd_out_gga(X, Y, D)
U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) → gcd_le_out_gga(s(X), Y, D)
U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) → gcd_out_gga(X, Y, D)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

gcd_in_gga(X, Y, D) → U1_gga(X, Y, D, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U1_gga(X, Y, D, le_out_gg(X, Y)) → U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D))
gcd_le_in_gga(0, Y, Y) → gcd_le_out_gga(0, Y, Y)
gcd_le_in_gga(s(X), Y, D) → U5_gga(X, Y, D, add_in_gag(s(X), Z, Y))
add_in_gag(s(X), Y, s(Z)) → U9_gag(X, Y, Z, add_in_gag(X, Y, Z))
add_in_gag(0, X, X) → add_out_gag(0, X, X)
U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) → U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D))
gcd_in_gga(X, Y, D) → U3_gga(X, Y, D, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U3_gga(X, Y, D, gt_out_gg(X, Y)) → U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D))
U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) → gcd_out_gga(X, Y, D)
U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) → gcd_le_out_gga(s(X), Y, D)
U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) → gcd_out_gga(X, Y, D)

(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:

GCD_IN_GGA(X, Y, D) → U1_GGA(X, Y, D, le_in_gg(X, Y))
GCD_IN_GGA(X, Y, D) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U8_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U1_GGA(X, Y, D, le_out_gg(X, Y)) → U2_GGA(X, Y, D, gcd_le_in_gga(X, Y, D))
U1_GGA(X, Y, D, le_out_gg(X, Y)) → GCD_LE_IN_GGA(X, Y, D)
GCD_LE_IN_GGA(s(X), Y, D) → U5_GGA(X, Y, D, add_in_gag(s(X), Z, Y))
GCD_LE_IN_GGA(s(X), Y, D) → ADD_IN_GAG(s(X), Z, Y)
ADD_IN_GAG(s(X), Y, s(Z)) → U9_GAG(X, Y, Z, add_in_gag(X, Y, Z))
ADD_IN_GAG(s(X), Y, s(Z)) → ADD_IN_GAG(X, Y, Z)
U5_GGA(X, Y, D, add_out_gag(s(X), Z, Y)) → U6_GGA(X, Y, D, gcd_in_gga(s(X), Z, D))
U5_GGA(X, Y, D, add_out_gag(s(X), Z, Y)) → GCD_IN_GGA(s(X), Z, D)
GCD_IN_GGA(X, Y, D) → U3_GGA(X, Y, D, gt_in_gg(X, Y))
GCD_IN_GGA(X, Y, D) → GT_IN_GG(X, Y)
GT_IN_GG(s(X), s(Y)) → U7_GG(X, Y, gt_in_gg(X, Y))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
U3_GGA(X, Y, D, gt_out_gg(X, Y)) → U4_GGA(X, Y, D, gcd_le_in_gga(Y, X, D))
U3_GGA(X, Y, D, gt_out_gg(X, Y)) → GCD_LE_IN_GGA(Y, X, D)

The TRS R consists of the following rules:

gcd_in_gga(X, Y, D) → U1_gga(X, Y, D, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U1_gga(X, Y, D, le_out_gg(X, Y)) → U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D))
gcd_le_in_gga(0, Y, Y) → gcd_le_out_gga(0, Y, Y)
gcd_le_in_gga(s(X), Y, D) → U5_gga(X, Y, D, add_in_gag(s(X), Z, Y))
add_in_gag(s(X), Y, s(Z)) → U9_gag(X, Y, Z, add_in_gag(X, Y, Z))
add_in_gag(0, X, X) → add_out_gag(0, X, X)
U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) → U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D))
gcd_in_gga(X, Y, D) → U3_gga(X, Y, D, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U3_gga(X, Y, D, gt_out_gg(X, Y)) → U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D))
U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) → gcd_out_gga(X, Y, D)
U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) → gcd_le_out_gga(s(X), Y, D)
U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) → gcd_out_gga(X, Y, D)

The argument filtering Pi contains the following mapping:
gcd_in_gga(x1, x2, x3) = gcd_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4)
le_in_gg(x1, x2) = le_in_gg(x1, x2)
s(x1) = s(x1)
U8_gg(x1, x2, x3) = U8_gg(x3)
0 = 0
le_out_gg(x1, x2) = le_out_gg
U2_gga(x1, x2, x3, x4) = U2_gga(x4)
gcd_le_in_gga(x1, x2, x3) = gcd_le_in_gga(x1, x2)
gcd_le_out_gga(x1, x2, x3) = gcd_le_out_gga(x3)
U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4)
add_in_gag(x1, x2, x3) = add_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4) = U9_gag(x4)
add_out_gag(x1, x2, x3) = add_out_gag(x2)
U6_gga(x1, x2, x3, x4) = U6_gga(x4)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
gt_in_gg(x1, x2) = gt_in_gg(x1, x2)
U7_gg(x1, x2, x3) = U7_gg(x3)
gt_out_gg(x1, x2) = gt_out_gg
U4_gga(x1, x2, x3, x4) = U4_gga(x4)
gcd_out_gga(x1, x2, x3) = gcd_out_gga(x3)
GCD_IN_GGA(x1, x2, x3) = GCD_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4)
LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2)
U8_GG(x1, x2, x3) = U8_GG(x3)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x4)
GCD_LE_IN_GGA(x1, x2, x3) = GCD_LE_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x4)
ADD_IN_GAG(x1, x2, x3) = ADD_IN_GAG(x1, x3)
U9_GAG(x1, x2, x3, x4) = U9_GAG(x4)
U6_GGA(x1, x2, x3, x4) = U6_GGA(x4)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2)
U7_GG(x1, x2, x3) = U7_GG(x3)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x4)

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

(4) Obligation:

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

GCD_IN_GGA(X, Y, D) → U1_GGA(X, Y, D, le_in_gg(X, Y))
GCD_IN_GGA(X, Y, D) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U8_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U1_GGA(X, Y, D, le_out_gg(X, Y)) → U2_GGA(X, Y, D, gcd_le_in_gga(X, Y, D))
U1_GGA(X, Y, D, le_out_gg(X, Y)) → GCD_LE_IN_GGA(X, Y, D)
GCD_LE_IN_GGA(s(X), Y, D) → U5_GGA(X, Y, D, add_in_gag(s(X), Z, Y))
GCD_LE_IN_GGA(s(X), Y, D) → ADD_IN_GAG(s(X), Z, Y)
ADD_IN_GAG(s(X), Y, s(Z)) → U9_GAG(X, Y, Z, add_in_gag(X, Y, Z))
ADD_IN_GAG(s(X), Y, s(Z)) → ADD_IN_GAG(X, Y, Z)
U5_GGA(X, Y, D, add_out_gag(s(X), Z, Y)) → U6_GGA(X, Y, D, gcd_in_gga(s(X), Z, D))
U5_GGA(X, Y, D, add_out_gag(s(X), Z, Y)) → GCD_IN_GGA(s(X), Z, D)
GCD_IN_GGA(X, Y, D) → U3_GGA(X, Y, D, gt_in_gg(X, Y))
GCD_IN_GGA(X, Y, D) → GT_IN_GG(X, Y)
GT_IN_GG(s(X), s(Y)) → U7_GG(X, Y, gt_in_gg(X, Y))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
U3_GGA(X, Y, D, gt_out_gg(X, Y)) → U4_GGA(X, Y, D, gcd_le_in_gga(Y, X, D))
U3_GGA(X, Y, D, gt_out_gg(X, Y)) → GCD_LE_IN_GGA(Y, X, D)

The TRS R consists of the following rules:

gcd_in_gga(X, Y, D) → U1_gga(X, Y, D, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U1_gga(X, Y, D, le_out_gg(X, Y)) → U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D))
gcd_le_in_gga(0, Y, Y) → gcd_le_out_gga(0, Y, Y)
gcd_le_in_gga(s(X), Y, D) → U5_gga(X, Y, D, add_in_gag(s(X), Z, Y))
add_in_gag(s(X), Y, s(Z)) → U9_gag(X, Y, Z, add_in_gag(X, Y, Z))
add_in_gag(0, X, X) → add_out_gag(0, X, X)
U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) → U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D))
gcd_in_gga(X, Y, D) → U3_gga(X, Y, D, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U3_gga(X, Y, D, gt_out_gg(X, Y)) → U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D))
U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) → gcd_out_gga(X, Y, D)
U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) → gcd_le_out_gga(s(X), Y, D)
U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) → gcd_out_gga(X, Y, D)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 9 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)

The TRS R consists of the following rules:

gcd_in_gga(X, Y, D) → U1_gga(X, Y, D, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U1_gga(X, Y, D, le_out_gg(X, Y)) → U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D))
gcd_le_in_gga(0, Y, Y) → gcd_le_out_gga(0, Y, Y)
gcd_le_in_gga(s(X), Y, D) → U5_gga(X, Y, D, add_in_gag(s(X), Z, Y))
add_in_gag(s(X), Y, s(Z)) → U9_gag(X, Y, Z, add_in_gag(X, Y, Z))
add_in_gag(0, X, X) → add_out_gag(0, X, X)
U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) → U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D))
gcd_in_gga(X, Y, D) → U3_gga(X, Y, D, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U3_gga(X, Y, D, gt_out_gg(X, Y)) → U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D))
U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) → gcd_out_gga(X, Y, D)
U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) → gcd_le_out_gga(s(X), Y, D)
U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) → gcd_out_gga(X, Y, D)

(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:

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)

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

(10) PiDPToQDPProof (EQUIVALENT 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:

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(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:

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(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_GAG(s(X), Y, s(Z)) → ADD_IN_GAG(X, Y, Z)

The TRS R consists of the following rules:

gcd_in_gga(X, Y, D) → U1_gga(X, Y, D, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U1_gga(X, Y, D, le_out_gg(X, Y)) → U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D))
gcd_le_in_gga(0, Y, Y) → gcd_le_out_gga(0, Y, Y)
gcd_le_in_gga(s(X), Y, D) → U5_gga(X, Y, D, add_in_gag(s(X), Z, Y))
add_in_gag(s(X), Y, s(Z)) → U9_gag(X, Y, Z, add_in_gag(X, Y, Z))
add_in_gag(0, X, X) → add_out_gag(0, X, X)
U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) → U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D))
gcd_in_gga(X, Y, D) → U3_gga(X, Y, D, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U3_gga(X, Y, D, gt_out_gg(X, Y)) → U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D))
U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) → gcd_out_gga(X, Y, D)
U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) → gcd_le_out_gga(s(X), Y, D)
U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) → gcd_out_gga(X, Y, D)

(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_GAG(s(X), Y, s(Z)) → ADD_IN_GAG(X, Y, Z)

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

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_GAG(s(X), s(Z)) → ADD_IN_GAG(X, Z)

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_GAG(s(X), s(Z)) → ADD_IN_GAG(X, Z)
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:

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

The TRS R consists of the following rules:

gcd_in_gga(X, Y, D) → U1_gga(X, Y, D, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U1_gga(X, Y, D, le_out_gg(X, Y)) → U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D))
gcd_le_in_gga(0, Y, Y) → gcd_le_out_gga(0, Y, Y)
gcd_le_in_gga(s(X), Y, D) → U5_gga(X, Y, D, add_in_gag(s(X), Z, Y))
add_in_gag(s(X), Y, s(Z)) → U9_gag(X, Y, Z, add_in_gag(X, Y, Z))
add_in_gag(0, X, X) → add_out_gag(0, X, X)
U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) → U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D))
gcd_in_gga(X, Y, D) → U3_gga(X, Y, D, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U3_gga(X, Y, D, gt_out_gg(X, Y)) → U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D))
U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) → gcd_out_gga(X, Y, D)
U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) → gcd_le_out_gga(s(X), Y, D)
U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) → gcd_out_gga(X, Y, D)

(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:

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

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

(24) PiDPToQDPProof (EQUIVALENT 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:

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(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:

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(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:

U1_GGA(X, Y, D, le_out_gg(X, Y)) → GCD_LE_IN_GGA(X, Y, D)
GCD_LE_IN_GGA(s(X), Y, D) → U5_GGA(X, Y, D, add_in_gag(s(X), Z, Y))
U5_GGA(X, Y, D, add_out_gag(s(X), Z, Y)) → GCD_IN_GGA(s(X), Z, D)
GCD_IN_GGA(X, Y, D) → U1_GGA(X, Y, D, le_in_gg(X, Y))
GCD_IN_GGA(X, Y, D) → U3_GGA(X, Y, D, gt_in_gg(X, Y))
U3_GGA(X, Y, D, gt_out_gg(X, Y)) → GCD_LE_IN_GGA(Y, X, D)

The TRS R consists of the following rules:

gcd_in_gga(X, Y, D) → U1_gga(X, Y, D, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U1_gga(X, Y, D, le_out_gg(X, Y)) → U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D))
gcd_le_in_gga(0, Y, Y) → gcd_le_out_gga(0, Y, Y)
gcd_le_in_gga(s(X), Y, D) → U5_gga(X, Y, D, add_in_gag(s(X), Z, Y))
add_in_gag(s(X), Y, s(Z)) → U9_gag(X, Y, Z, add_in_gag(X, Y, Z))
add_in_gag(0, X, X) → add_out_gag(0, X, X)
U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) → U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D))
gcd_in_gga(X, Y, D) → U3_gga(X, Y, D, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U3_gga(X, Y, D, gt_out_gg(X, Y)) → U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D))
U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) → gcd_out_gga(X, Y, D)
U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) → gcd_le_out_gga(s(X), Y, D)
U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) → gcd_out_gga(X, Y, D)

(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:

U1_GGA(X, Y, D, le_out_gg(X, Y)) → GCD_LE_IN_GGA(X, Y, D)
GCD_LE_IN_GGA(s(X), Y, D) → U5_GGA(X, Y, D, add_in_gag(s(X), Z, Y))
U5_GGA(X, Y, D, add_out_gag(s(X), Z, Y)) → GCD_IN_GGA(s(X), Z, D)
GCD_IN_GGA(X, Y, D) → U1_GGA(X, Y, D, le_in_gg(X, Y))
GCD_IN_GGA(X, Y, D) → U3_GGA(X, Y, D, gt_in_gg(X, Y))
U3_GGA(X, Y, D, gt_out_gg(X, Y)) → GCD_LE_IN_GGA(Y, X, D)

The TRS R consists of the following rules:

add_in_gag(s(X), Y, s(Z)) → U9_gag(X, Y, Z, add_in_gag(X, Y, Z))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
add_in_gag(0, X, X) → add_out_gag(0, X, X)

The argument filtering Pi contains the following mapping:
le_in_gg(x1, x2) = le_in_gg(x1, x2)
s(x1) = s(x1)
U8_gg(x1, x2, x3) = U8_gg(x3)
0 = 0
le_out_gg(x1, x2) = le_out_gg
add_in_gag(x1, x2, x3) = add_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4) = U9_gag(x4)
add_out_gag(x1, x2, x3) = add_out_gag(x2)
gt_in_gg(x1, x2) = gt_in_gg(x1, x2)
U7_gg(x1, x2, x3) = U7_gg(x3)
gt_out_gg(x1, x2) = gt_out_gg
GCD_IN_GGA(x1, x2, x3) = GCD_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4)
GCD_LE_IN_GGA(x1, x2, x3) = GCD_LE_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x4)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)

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

(31) PiDPToQDPProof (SOUND 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:

U1_GGA(X, Y, le_out_gg) → GCD_LE_IN_GGA(X, Y)
GCD_LE_IN_GGA(s(X), Y) → U5_GGA(X, add_in_gag(s(X), Y))
U5_GGA(X, add_out_gag(Z)) → GCD_IN_GGA(s(X), Z)
GCD_IN_GGA(X, Y) → U1_GGA(X, Y, le_in_gg(X, Y))
GCD_IN_GGA(X, Y) → U3_GGA(X, Y, gt_in_gg(X, Y))
U3_GGA(X, Y, gt_out_gg) → GCD_LE_IN_GGA(Y, X)

The TRS R consists of the following rules:

add_in_gag(s(X), s(Z)) → U9_gag(add_in_gag(X, Z))
le_in_gg(s(X), s(Y)) → U8_gg(le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
gt_in_gg(s(X), s(Y)) → U7_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U9_gag(add_out_gag(Y)) → add_out_gag(Y)
U8_gg(le_out_gg) → le_out_gg
U7_gg(gt_out_gg) → gt_out_gg
add_in_gag(0, X) → add_out_gag(X)

The set Q consists of the following terms:

add_in_gag(x0, x1)
le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U9_gag(x0)
U8_gg(x0)
U7_gg(x0)

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

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

GCD_LE_IN_GGA(s(X), Y) → U5_GGA(X, add_in_gag(s(X), Y))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(U1_GGA(x₁, x₂, x₃)) = 0 +
[ 1, 0 ]

· x₁ +
[ 1, 0 ]

· x₂ +
[ 0, 0 ]

· x₃

POL(le_out_gg) =
/ 0 \

\ 0 /

POL(GCD_LE_IN_GGA(x₁, x₂)) = 0 +
[ 1, 0 ]

· x₁ +
[ 1, 0 ]

· x₂

POL(s(x₁)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁

POL(U5_GGA(x₁, x₂)) = 0 +
[ 1, 0 ]

· x₁ +
[ 0, 1 ]

· x₂

POL(add_in_gag(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 1 /

· x₁ +
/ 1 0 \

\ 1 0 /

· x₂

POL(add_out_gag(x₁)) =
/ 0 \

\ 1 /

+
/ 1 0 \

\ 1 0 /

· x₁

POL(GCD_IN_GGA(x₁, x₂)) = 0 +
[ 1, 0 ]

· x₁ +
[ 1, 0 ]

· x₂

POL(le_in_gg(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 1 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(U3_GGA(x₁, x₂, x₃)) = 0 +
[ 1, 0 ]

· x₁ +
[ 1, 0 ]

· x₂ +
[ 0, 0 ]

· x₃

POL(gt_in_gg(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(gt_out_gg) =
/ 1 \

\ 1 /

POL(U9_gag(x₁)) =
/ 0 \

\ 1 /

+
/ 1 0 \

\ 1 0 /

· x₁

POL(U8_gg(x₁)) =
/ 1 \

\ 1 /

+
/ 0 1 \

\ 0 1 /

· x₁

POL(0) =
/ 0 \

\ 1 /

POL(U7_gg(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 1 0 /

· x₁

The following usable rules [FROCOS05] were oriented:

add_in_gag(s(X), s(Z)) → U9_gag(add_in_gag(X, Z))
add_in_gag(0, X) → add_out_gag(X)
U9_gag(add_out_gag(Y)) → add_out_gag(Y)

(34) Obligation:

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

U1_GGA(X, Y, le_out_gg) → GCD_LE_IN_GGA(X, Y)
U5_GGA(X, add_out_gag(Z)) → GCD_IN_GGA(s(X), Z)
GCD_IN_GGA(X, Y) → U1_GGA(X, Y, le_in_gg(X, Y))
GCD_IN_GGA(X, Y) → U3_GGA(X, Y, gt_in_gg(X, Y))
U3_GGA(X, Y, gt_out_gg) → GCD_LE_IN_GGA(Y, X)

The TRS R consists of the following rules:

add_in_gag(s(X), s(Z)) → U9_gag(add_in_gag(X, Z))
le_in_gg(s(X), s(Y)) → U8_gg(le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
gt_in_gg(s(X), s(Y)) → U7_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U9_gag(add_out_gag(Y)) → add_out_gag(Y)
U8_gg(le_out_gg) → le_out_gg
U7_gg(gt_out_gg) → gt_out_gg
add_in_gag(0, X) → add_out_gag(X)

The set Q consists of the following terms:

add_in_gag(x0, x1)
le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U9_gag(x0)
U8_gg(x0)
U7_gg(x0)

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

(35) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 5 less nodes.

(36) TRUE

(37) PrologToPiTRSProof (SOUND transformation)

gcd_in_gga(X, Y, D) → U1_gga(X, Y, D, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U1_gga(X, Y, D, le_out_gg(X, Y)) → U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D))
gcd_le_in_gga(0, Y, Y) → gcd_le_out_gga(0, Y, Y)
gcd_le_in_gga(s(X), Y, D) → U5_gga(X, Y, D, add_in_gag(s(X), Z, Y))
add_in_gag(s(X), Y, s(Z)) → U9_gag(X, Y, Z, add_in_gag(X, Y, Z))
add_in_gag(0, X, X) → add_out_gag(0, X, X)
U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) → U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D))
gcd_in_gga(X, Y, D) → U3_gga(X, Y, D, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U3_gga(X, Y, D, gt_out_gg(X, Y)) → U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D))
U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) → gcd_out_gga(X, Y, D)
U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) → gcd_le_out_gga(s(X), Y, D)
U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) → gcd_out_gga(X, Y, D)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(38) Obligation:

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

gcd_in_gga(X, Y, D) → U1_gga(X, Y, D, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U1_gga(X, Y, D, le_out_gg(X, Y)) → U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D))
gcd_le_in_gga(0, Y, Y) → gcd_le_out_gga(0, Y, Y)
gcd_le_in_gga(s(X), Y, D) → U5_gga(X, Y, D, add_in_gag(s(X), Z, Y))
add_in_gag(s(X), Y, s(Z)) → U9_gag(X, Y, Z, add_in_gag(X, Y, Z))
add_in_gag(0, X, X) → add_out_gag(0, X, X)
U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) → U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D))
gcd_in_gga(X, Y, D) → U3_gga(X, Y, D, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U3_gga(X, Y, D, gt_out_gg(X, Y)) → U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D))
U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) → gcd_out_gga(X, Y, D)
U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) → gcd_le_out_gga(s(X), Y, D)
U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) → gcd_out_gga(X, Y, D)

(39) 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:

GCD_IN_GGA(X, Y, D) → U1_GGA(X, Y, D, le_in_gg(X, Y))
GCD_IN_GGA(X, Y, D) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U8_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U1_GGA(X, Y, D, le_out_gg(X, Y)) → U2_GGA(X, Y, D, gcd_le_in_gga(X, Y, D))
U1_GGA(X, Y, D, le_out_gg(X, Y)) → GCD_LE_IN_GGA(X, Y, D)
GCD_LE_IN_GGA(s(X), Y, D) → U5_GGA(X, Y, D, add_in_gag(s(X), Z, Y))
GCD_LE_IN_GGA(s(X), Y, D) → ADD_IN_GAG(s(X), Z, Y)
ADD_IN_GAG(s(X), Y, s(Z)) → U9_GAG(X, Y, Z, add_in_gag(X, Y, Z))
ADD_IN_GAG(s(X), Y, s(Z)) → ADD_IN_GAG(X, Y, Z)
U5_GGA(X, Y, D, add_out_gag(s(X), Z, Y)) → U6_GGA(X, Y, D, gcd_in_gga(s(X), Z, D))
U5_GGA(X, Y, D, add_out_gag(s(X), Z, Y)) → GCD_IN_GGA(s(X), Z, D)
GCD_IN_GGA(X, Y, D) → U3_GGA(X, Y, D, gt_in_gg(X, Y))
GCD_IN_GGA(X, Y, D) → GT_IN_GG(X, Y)
GT_IN_GG(s(X), s(Y)) → U7_GG(X, Y, gt_in_gg(X, Y))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
U3_GGA(X, Y, D, gt_out_gg(X, Y)) → U4_GGA(X, Y, D, gcd_le_in_gga(Y, X, D))
U3_GGA(X, Y, D, gt_out_gg(X, Y)) → GCD_LE_IN_GGA(Y, X, D)

The TRS R consists of the following rules:

gcd_in_gga(X, Y, D) → U1_gga(X, Y, D, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U1_gga(X, Y, D, le_out_gg(X, Y)) → U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D))
gcd_le_in_gga(0, Y, Y) → gcd_le_out_gga(0, Y, Y)
gcd_le_in_gga(s(X), Y, D) → U5_gga(X, Y, D, add_in_gag(s(X), Z, Y))
add_in_gag(s(X), Y, s(Z)) → U9_gag(X, Y, Z, add_in_gag(X, Y, Z))
add_in_gag(0, X, X) → add_out_gag(0, X, X)
U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) → U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D))
gcd_in_gga(X, Y, D) → U3_gga(X, Y, D, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U3_gga(X, Y, D, gt_out_gg(X, Y)) → U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D))
U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) → gcd_out_gga(X, Y, D)
U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) → gcd_le_out_gga(s(X), Y, D)
U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) → gcd_out_gga(X, Y, D)

The argument filtering Pi contains the following mapping:
gcd_in_gga(x1, x2, x3) = gcd_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4)
le_in_gg(x1, x2) = le_in_gg(x1, x2)
s(x1) = s(x1)
U8_gg(x1, x2, x3) = U8_gg(x1, x2, x3)
0 = 0
le_out_gg(x1, x2) = le_out_gg(x1, x2)
U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4)
gcd_le_in_gga(x1, x2, x3) = gcd_le_in_gga(x1, x2)
gcd_le_out_gga(x1, x2, x3) = gcd_le_out_gga(x1, x2, x3)
U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4)
add_in_gag(x1, x2, x3) = add_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4) = U9_gag(x1, x3, x4)
add_out_gag(x1, x2, x3) = add_out_gag(x1, x2, x3)
U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
gt_in_gg(x1, x2) = gt_in_gg(x1, x2)
U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3)
gt_out_gg(x1, x2) = gt_out_gg(x1, x2)
U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4)
gcd_out_gga(x1, x2, x3) = gcd_out_gga(x1, x2, x3)
GCD_IN_GGA(x1, x2, x3) = GCD_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4)
LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2)
U8_GG(x1, x2, x3) = U8_GG(x1, x2, x3)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4)
GCD_LE_IN_GGA(x1, x2, x3) = GCD_LE_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4)
ADD_IN_GAG(x1, x2, x3) = ADD_IN_GAG(x1, x3)
U9_GAG(x1, x2, x3, x4) = U9_GAG(x1, x3, x4)
U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2)
U7_GG(x1, x2, x3) = U7_GG(x1, x2, x3)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4)

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

(40) Obligation:

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

GCD_IN_GGA(X, Y, D) → U1_GGA(X, Y, D, le_in_gg(X, Y))
GCD_IN_GGA(X, Y, D) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U8_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U1_GGA(X, Y, D, le_out_gg(X, Y)) → U2_GGA(X, Y, D, gcd_le_in_gga(X, Y, D))
U1_GGA(X, Y, D, le_out_gg(X, Y)) → GCD_LE_IN_GGA(X, Y, D)
GCD_LE_IN_GGA(s(X), Y, D) → U5_GGA(X, Y, D, add_in_gag(s(X), Z, Y))
GCD_LE_IN_GGA(s(X), Y, D) → ADD_IN_GAG(s(X), Z, Y)
ADD_IN_GAG(s(X), Y, s(Z)) → U9_GAG(X, Y, Z, add_in_gag(X, Y, Z))
ADD_IN_GAG(s(X), Y, s(Z)) → ADD_IN_GAG(X, Y, Z)
U5_GGA(X, Y, D, add_out_gag(s(X), Z, Y)) → U6_GGA(X, Y, D, gcd_in_gga(s(X), Z, D))
U5_GGA(X, Y, D, add_out_gag(s(X), Z, Y)) → GCD_IN_GGA(s(X), Z, D)
GCD_IN_GGA(X, Y, D) → U3_GGA(X, Y, D, gt_in_gg(X, Y))
GCD_IN_GGA(X, Y, D) → GT_IN_GG(X, Y)
GT_IN_GG(s(X), s(Y)) → U7_GG(X, Y, gt_in_gg(X, Y))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
U3_GGA(X, Y, D, gt_out_gg(X, Y)) → U4_GGA(X, Y, D, gcd_le_in_gga(Y, X, D))
U3_GGA(X, Y, D, gt_out_gg(X, Y)) → GCD_LE_IN_GGA(Y, X, D)

The TRS R consists of the following rules:

gcd_in_gga(X, Y, D) → U1_gga(X, Y, D, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U1_gga(X, Y, D, le_out_gg(X, Y)) → U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D))
gcd_le_in_gga(0, Y, Y) → gcd_le_out_gga(0, Y, Y)
gcd_le_in_gga(s(X), Y, D) → U5_gga(X, Y, D, add_in_gag(s(X), Z, Y))
add_in_gag(s(X), Y, s(Z)) → U9_gag(X, Y, Z, add_in_gag(X, Y, Z))
add_in_gag(0, X, X) → add_out_gag(0, X, X)
U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) → U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D))
gcd_in_gga(X, Y, D) → U3_gga(X, Y, D, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U3_gga(X, Y, D, gt_out_gg(X, Y)) → U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D))
U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) → gcd_out_gga(X, Y, D)
U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) → gcd_le_out_gga(s(X), Y, D)
U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) → gcd_out_gga(X, Y, D)

The argument filtering Pi contains the following mapping:
gcd_in_gga(x1, x2, x3) = gcd_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4)
le_in_gg(x1, x2) = le_in_gg(x1, x2)
s(x1) = s(x1)
U8_gg(x1, x2, x3) = U8_gg(x1, x2, x3)
0 = 0
le_out_gg(x1, x2) = le_out_gg(x1, x2)
U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4)
gcd_le_in_gga(x1, x2, x3) = gcd_le_in_gga(x1, x2)
gcd_le_out_gga(x1, x2, x3) = gcd_le_out_gga(x1, x2, x3)
U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4)
add_in_gag(x1, x2, x3) = add_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4) = U9_gag(x1, x3, x4)
add_out_gag(x1, x2, x3) = add_out_gag(x1, x2, x3)
U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
gt_in_gg(x1, x2) = gt_in_gg(x1, x2)
U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3)
gt_out_gg(x1, x2) = gt_out_gg(x1, x2)
U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4)
gcd_out_gga(x1, x2, x3) = gcd_out_gga(x1, x2, x3)
GCD_IN_GGA(x1, x2, x3) = GCD_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4)
LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2)
U8_GG(x1, x2, x3) = U8_GG(x1, x2, x3)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4)
GCD_LE_IN_GGA(x1, x2, x3) = GCD_LE_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4)
ADD_IN_GAG(x1, x2, x3) = ADD_IN_GAG(x1, x3)
U9_GAG(x1, x2, x3, x4) = U9_GAG(x1, x3, x4)
U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2)
U7_GG(x1, x2, x3) = U7_GG(x1, x2, x3)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4)

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

(41) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 9 less nodes.

(42) Complex Obligation (AND)

(43) Obligation:

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

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)

The TRS R consists of the following rules:

gcd_in_gga(X, Y, D) → U1_gga(X, Y, D, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U1_gga(X, Y, D, le_out_gg(X, Y)) → U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D))
gcd_le_in_gga(0, Y, Y) → gcd_le_out_gga(0, Y, Y)
gcd_le_in_gga(s(X), Y, D) → U5_gga(X, Y, D, add_in_gag(s(X), Z, Y))
add_in_gag(s(X), Y, s(Z)) → U9_gag(X, Y, Z, add_in_gag(X, Y, Z))
add_in_gag(0, X, X) → add_out_gag(0, X, X)
U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) → U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D))
gcd_in_gga(X, Y, D) → U3_gga(X, Y, D, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U3_gga(X, Y, D, gt_out_gg(X, Y)) → U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D))
U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) → gcd_out_gga(X, Y, D)
U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) → gcd_le_out_gga(s(X), Y, D)
U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) → gcd_out_gga(X, Y, D)

(44) UsableRulesProof (EQUIVALENT transformation)

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

(45) Obligation:

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

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)

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

(46) PiDPToQDPProof (EQUIVALENT transformation)

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

(47) Obligation:

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

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)

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

(48) 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:

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

(49) TRUE

(50) Obligation:

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

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

The TRS R consists of the following rules:

gcd_in_gga(X, Y, D) → U1_gga(X, Y, D, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U1_gga(X, Y, D, le_out_gg(X, Y)) → U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D))
gcd_le_in_gga(0, Y, Y) → gcd_le_out_gga(0, Y, Y)
gcd_le_in_gga(s(X), Y, D) → U5_gga(X, Y, D, add_in_gag(s(X), Z, Y))
add_in_gag(s(X), Y, s(Z)) → U9_gag(X, Y, Z, add_in_gag(X, Y, Z))
add_in_gag(0, X, X) → add_out_gag(0, X, X)
U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) → U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D))
gcd_in_gga(X, Y, D) → U3_gga(X, Y, D, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U3_gga(X, Y, D, gt_out_gg(X, Y)) → U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D))
U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) → gcd_out_gga(X, Y, D)
U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) → gcd_le_out_gga(s(X), Y, D)
U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) → gcd_out_gga(X, Y, D)

(51) UsableRulesProof (EQUIVALENT transformation)

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

(52) Obligation:

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

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

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

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

(53) PiDPToQDPProof (SOUND transformation)

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

(54) Obligation:

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

ADD_IN_GAG(s(X), s(Z)) → ADD_IN_GAG(X, Z)

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

(55) 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_GAG(s(X), s(Z)) → ADD_IN_GAG(X, Z)
The graph contains the following edges 1 > 1, 2 > 2

(56) TRUE

(57) Obligation:

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

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

The TRS R consists of the following rules:

gcd_in_gga(X, Y, D) → U1_gga(X, Y, D, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U1_gga(X, Y, D, le_out_gg(X, Y)) → U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D))
gcd_le_in_gga(0, Y, Y) → gcd_le_out_gga(0, Y, Y)
gcd_le_in_gga(s(X), Y, D) → U5_gga(X, Y, D, add_in_gag(s(X), Z, Y))
add_in_gag(s(X), Y, s(Z)) → U9_gag(X, Y, Z, add_in_gag(X, Y, Z))
add_in_gag(0, X, X) → add_out_gag(0, X, X)
U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) → U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D))
gcd_in_gga(X, Y, D) → U3_gga(X, Y, D, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U3_gga(X, Y, D, gt_out_gg(X, Y)) → U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D))
U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) → gcd_out_gga(X, Y, D)
U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) → gcd_le_out_gga(s(X), Y, D)
U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) → gcd_out_gga(X, Y, D)

(58) UsableRulesProof (EQUIVALENT transformation)

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

(59) Obligation:

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

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

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

(60) PiDPToQDPProof (EQUIVALENT transformation)

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

(61) Obligation:

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

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

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

(62) 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:

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

(63) TRUE

(64) Obligation:

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

U1_GGA(X, Y, D, le_out_gg(X, Y)) → GCD_LE_IN_GGA(X, Y, D)
GCD_LE_IN_GGA(s(X), Y, D) → U5_GGA(X, Y, D, add_in_gag(s(X), Z, Y))
U5_GGA(X, Y, D, add_out_gag(s(X), Z, Y)) → GCD_IN_GGA(s(X), Z, D)
GCD_IN_GGA(X, Y, D) → U1_GGA(X, Y, D, le_in_gg(X, Y))
GCD_IN_GGA(X, Y, D) → U3_GGA(X, Y, D, gt_in_gg(X, Y))
U3_GGA(X, Y, D, gt_out_gg(X, Y)) → GCD_LE_IN_GGA(Y, X, D)

The TRS R consists of the following rules:

gcd_in_gga(X, Y, D) → U1_gga(X, Y, D, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U1_gga(X, Y, D, le_out_gg(X, Y)) → U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D))
gcd_le_in_gga(0, Y, Y) → gcd_le_out_gga(0, Y, Y)
gcd_le_in_gga(s(X), Y, D) → U5_gga(X, Y, D, add_in_gag(s(X), Z, Y))
add_in_gag(s(X), Y, s(Z)) → U9_gag(X, Y, Z, add_in_gag(X, Y, Z))
add_in_gag(0, X, X) → add_out_gag(0, X, X)
U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) → U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D))
gcd_in_gga(X, Y, D) → U3_gga(X, Y, D, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U3_gga(X, Y, D, gt_out_gg(X, Y)) → U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D))
U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) → gcd_out_gga(X, Y, D)
U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) → gcd_le_out_gga(s(X), Y, D)
U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) → gcd_out_gga(X, Y, D)

The argument filtering Pi contains the following mapping:
gcd_in_gga(x1, x2, x3) = gcd_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4)
le_in_gg(x1, x2) = le_in_gg(x1, x2)
s(x1) = s(x1)
U8_gg(x1, x2, x3) = U8_gg(x1, x2, x3)
0 = 0
le_out_gg(x1, x2) = le_out_gg(x1, x2)
U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4)
gcd_le_in_gga(x1, x2, x3) = gcd_le_in_gga(x1, x2)
gcd_le_out_gga(x1, x2, x3) = gcd_le_out_gga(x1, x2, x3)
U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4)
add_in_gag(x1, x2, x3) = add_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4) = U9_gag(x1, x3, x4)
add_out_gag(x1, x2, x3) = add_out_gag(x1, x2, x3)
U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
gt_in_gg(x1, x2) = gt_in_gg(x1, x2)
U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3)
gt_out_gg(x1, x2) = gt_out_gg(x1, x2)
U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4)
gcd_out_gga(x1, x2, x3) = gcd_out_gga(x1, x2, x3)
GCD_IN_GGA(x1, x2, x3) = GCD_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4)
GCD_LE_IN_GGA(x1, x2, x3) = GCD_LE_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)

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

(65) UsableRulesProof (EQUIVALENT transformation)

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

(66) Obligation:

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

U1_GGA(X, Y, D, le_out_gg(X, Y)) → GCD_LE_IN_GGA(X, Y, D)
GCD_LE_IN_GGA(s(X), Y, D) → U5_GGA(X, Y, D, add_in_gag(s(X), Z, Y))
U5_GGA(X, Y, D, add_out_gag(s(X), Z, Y)) → GCD_IN_GGA(s(X), Z, D)
GCD_IN_GGA(X, Y, D) → U1_GGA(X, Y, D, le_in_gg(X, Y))
GCD_IN_GGA(X, Y, D) → U3_GGA(X, Y, D, gt_in_gg(X, Y))
U3_GGA(X, Y, D, gt_out_gg(X, Y)) → GCD_LE_IN_GGA(Y, X, D)

The TRS R consists of the following rules:

add_in_gag(s(X), Y, s(Z)) → U9_gag(X, Y, Z, add_in_gag(X, Y, Z))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
add_in_gag(0, X, X) → add_out_gag(0, X, X)

The argument filtering Pi contains the following mapping:
le_in_gg(x1, x2) = le_in_gg(x1, x2)
s(x1) = s(x1)
U8_gg(x1, x2, x3) = U8_gg(x1, x2, x3)
0 = 0
le_out_gg(x1, x2) = le_out_gg(x1, x2)
add_in_gag(x1, x2, x3) = add_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4) = U9_gag(x1, x3, x4)
add_out_gag(x1, x2, x3) = add_out_gag(x1, x2, x3)
gt_in_gg(x1, x2) = gt_in_gg(x1, x2)
U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3)
gt_out_gg(x1, x2) = gt_out_gg(x1, x2)
GCD_IN_GGA(x1, x2, x3) = GCD_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4)
GCD_LE_IN_GGA(x1, x2, x3) = GCD_LE_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)

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

(67) PiDPToQDPProof (SOUND transformation)

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

(68) Obligation:

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

U1_GGA(X, Y, le_out_gg(X, Y)) → GCD_LE_IN_GGA(X, Y)
GCD_LE_IN_GGA(s(X), Y) → U5_GGA(X, Y, add_in_gag(s(X), Y))
U5_GGA(X, Y, add_out_gag(s(X), Z, Y)) → GCD_IN_GGA(s(X), Z)
GCD_IN_GGA(X, Y) → U1_GGA(X, Y, le_in_gg(X, Y))
GCD_IN_GGA(X, Y) → U3_GGA(X, Y, gt_in_gg(X, Y))
U3_GGA(X, Y, gt_out_gg(X, Y)) → GCD_LE_IN_GGA(Y, X)

The TRS R consists of the following rules:

add_in_gag(s(X), s(Z)) → U9_gag(X, Z, add_in_gag(X, Z))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U9_gag(X, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
add_in_gag(0, X) → add_out_gag(0, X, X)

The set Q consists of the following terms:

add_in_gag(x0, x1)
le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U9_gag(x0, x1, x2)
U8_gg(x0, x1, x2)
U7_gg(x0, x1, x2)

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

(69) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U3_GGA(X, Y, gt_out_gg(X, Y)) → GCD_LE_IN_GGA(Y, X)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(GCD_IN_GGA(x₁, x₂)) = x₁
POL(GCD_LE_IN_GGA(x₁, x₂)) = x₁
POL(U1_GGA(x₁, x₂, x₃)) = x₁
POL(U3_GGA(x₁, x₂, x₃)) = x₃
POL(U5_GGA(x₁, x₂, x₃)) = 1 + x₁
POL(U7_gg(x₁, x₂, x₃)) = 1 + x₃
POL(U8_gg(x₁, x₂, x₃)) = 0
POL(U9_gag(x₁, x₂, x₃)) = 0
POL(add_in_gag(x₁, x₂)) = 1
POL(add_out_gag(x₁, x₂, x₃)) = 0
POL(gt_in_gg(x₁, x₂)) = x₁
POL(gt_out_gg(x₁, x₂)) = 1 + x₂
POL(le_in_gg(x₁, x₂)) = 0
POL(le_out_gg(x₁, x₂)) = 0
POL(s(x₁)) = 1 + x₁

The following usable rules [FROCOS05] were oriented:

gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))

(70) Obligation:

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

U1_GGA(X, Y, le_out_gg(X, Y)) → GCD_LE_IN_GGA(X, Y)
GCD_LE_IN_GGA(s(X), Y) → U5_GGA(X, Y, add_in_gag(s(X), Y))
U5_GGA(X, Y, add_out_gag(s(X), Z, Y)) → GCD_IN_GGA(s(X), Z)
GCD_IN_GGA(X, Y) → U1_GGA(X, Y, le_in_gg(X, Y))
GCD_IN_GGA(X, Y) → U3_GGA(X, Y, gt_in_gg(X, Y))

The TRS R consists of the following rules:

add_in_gag(s(X), s(Z)) → U9_gag(X, Z, add_in_gag(X, Z))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U9_gag(X, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
add_in_gag(0, X) → add_out_gag(0, X, X)

The set Q consists of the following terms:

add_in_gag(x0, x1)
le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U9_gag(x0, x1, x2)
U8_gg(x0, x1, x2)
U7_gg(x0, x1, x2)

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

(71) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(72) Obligation:

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

GCD_LE_IN_GGA(s(X), Y) → U5_GGA(X, Y, add_in_gag(s(X), Y))
U5_GGA(X, Y, add_out_gag(s(X), Z, Y)) → GCD_IN_GGA(s(X), Z)
GCD_IN_GGA(X, Y) → U1_GGA(X, Y, le_in_gg(X, Y))
U1_GGA(X, Y, le_out_gg(X, Y)) → GCD_LE_IN_GGA(X, Y)

The TRS R consists of the following rules:

add_in_gag(s(X), s(Z)) → U9_gag(X, Z, add_in_gag(X, Z))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
gt_in_gg(s(X), s(Y)) → U7_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U9_gag(X, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
add_in_gag(0, X) → add_out_gag(0, X, X)

The set Q consists of the following terms:

add_in_gag(x0, x1)
le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U9_gag(x0, x1, x2)
U8_gg(x0, x1, x2)
U7_gg(x0, x1, x2)

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

(73) UsableRulesProof (EQUIVALENT transformation)

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

(74) Obligation:

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

GCD_LE_IN_GGA(s(X), Y) → U5_GGA(X, Y, add_in_gag(s(X), Y))
U5_GGA(X, Y, add_out_gag(s(X), Z, Y)) → GCD_IN_GGA(s(X), Z)
GCD_IN_GGA(X, Y) → U1_GGA(X, Y, le_in_gg(X, Y))
U1_GGA(X, Y, le_out_gg(X, Y)) → GCD_LE_IN_GGA(X, Y)

The TRS R consists of the following rules:

le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
add_in_gag(s(X), s(Z)) → U9_gag(X, Z, add_in_gag(X, Z))
add_in_gag(0, X) → add_out_gag(0, X, X)
U9_gag(X, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))

The set Q consists of the following terms:

add_in_gag(x0, x1)
le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U9_gag(x0, x1, x2)
U8_gg(x0, x1, x2)
U7_gg(x0, x1, x2)

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

(75) QReductionProof (EQUIVALENT transformation)

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

gt_in_gg(x0, x1)
U7_gg(x0, x1, x2)

(76) Obligation:

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

GCD_LE_IN_GGA(s(X), Y) → U5_GGA(X, Y, add_in_gag(s(X), Y))
U5_GGA(X, Y, add_out_gag(s(X), Z, Y)) → GCD_IN_GGA(s(X), Z)
GCD_IN_GGA(X, Y) → U1_GGA(X, Y, le_in_gg(X, Y))
U1_GGA(X, Y, le_out_gg(X, Y)) → GCD_LE_IN_GGA(X, Y)

The TRS R consists of the following rules:

le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
add_in_gag(s(X), s(Z)) → U9_gag(X, Z, add_in_gag(X, Z))
add_in_gag(0, X) → add_out_gag(0, X, X)
U9_gag(X, Z, add_out_gag(X, Y, Z)) → add_out_gag(s(X), Y, s(Z))

The set Q consists of the following terms:

add_in_gag(x0, x1)
le_in_gg(x0, x1)
U9_gag(x0, x1, x2)
U8_gg(x0, x1, x2)

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