0 Prolog
↳1 PrologToPiTRSProof (⇒, 55 ms)
↳2 PiTRS
↳3 DependencyPairsProof (⇔, 89 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 20 ms)
↳6 AND
↳7 PiDP
↳8 UsableRulesProof (⇔, 0 ms)
↳9 PiDP
↳10 PiDPToQDPProof (⇒, 26 ms)
↳11 QDP
↳12 QDPSizeChangeProof (⇔, 0 ms)
↳13 YES
↳14 PiDP
↳15 UsableRulesProof (⇔, 0 ms)
↳16 PiDP
↳17 PiDPToQDPProof (⇔, 0 ms)
↳18 QDP
↳19 QDPSizeChangeProof (⇔, 0 ms)
↳20 YES
↳21 PiDP
↳22 UsableRulesProof (⇔, 0 ms)
↳23 PiDP
↳24 PiDPToQDPProof (⇔, 0 ms)
↳25 QDP
↳26 QDPSizeChangeProof (⇔, 0 ms)
↳27 YES
↳28 PiDP
↳29 UsableRulesProof (⇔, 0 ms)
↳30 PiDP
↳31 PiDPToQDPProof (⇒, 0 ms)
↳32 QDP
↳33 QDPSizeChangeProof (⇔, 0 ms)
↳34 YES
↳35 PiDP
↳36 UsableRulesProof (⇔, 0 ms)
↳37 PiDP
↳38 PiDPToQDPProof (⇒, 0 ms)
↳39 QDP
↳40 QDPSizeChangeProof (⇔, 0 ms)
↳41 YES
in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))
IN_IN_AG(X, tree(Y, Left, X3)) → U1_AG(X, Y, Left, X3, less_in_ag(X, Y))
IN_IN_AG(X, tree(Y, Left, X3)) → LESS_IN_AG(X, Y)
LESS_IN_AG(s(X), s(Y)) → U5_AG(X, Y, less_in_ag(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
U1_AG(X, Y, Left, X3, less_out_ag(X, Y)) → U2_AG(X, Y, Left, X3, in_in_gg(X, Left))
U1_AG(X, Y, Left, X3, less_out_ag(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X3)) → U1_GG(X, Y, Left, X3, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, Left, X3)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U5_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → U2_GG(X, Y, Left, X3, in_in_gg(X, Left))
U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, X4, Right)) → U3_GG(X, Y, X4, Right, less_in_gg(Y, X))
IN_IN_GG(X, tree(Y, X4, Right)) → LESS_IN_GG(Y, X)
U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → U4_GG(X, Y, X4, Right, in_in_gg(X, Right))
U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
IN_IN_AG(X, tree(Y, X4, Right)) → U3_AG(X, Y, X4, Right, less_in_ga(Y, X))
IN_IN_AG(X, tree(Y, X4, Right)) → LESS_IN_GA(Y, X)
LESS_IN_GA(s(X), s(Y)) → U5_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) → U4_AG(X, Y, X4, Right, in_in_ag(X, Right))
U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) → IN_IN_AG(X, Right)
in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))
IN_IN_AG(X, tree(Y, Left, X3)) → U1_AG(X, Y, Left, X3, less_in_ag(X, Y))
IN_IN_AG(X, tree(Y, Left, X3)) → LESS_IN_AG(X, Y)
LESS_IN_AG(s(X), s(Y)) → U5_AG(X, Y, less_in_ag(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
U1_AG(X, Y, Left, X3, less_out_ag(X, Y)) → U2_AG(X, Y, Left, X3, in_in_gg(X, Left))
U1_AG(X, Y, Left, X3, less_out_ag(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X3)) → U1_GG(X, Y, Left, X3, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, Left, X3)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U5_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → U2_GG(X, Y, Left, X3, in_in_gg(X, Left))
U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, X4, Right)) → U3_GG(X, Y, X4, Right, less_in_gg(Y, X))
IN_IN_GG(X, tree(Y, X4, Right)) → LESS_IN_GG(Y, X)
U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → U4_GG(X, Y, X4, Right, in_in_gg(X, Right))
U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
IN_IN_AG(X, tree(Y, X4, Right)) → U3_AG(X, Y, X4, Right, less_in_ga(Y, X))
IN_IN_AG(X, tree(Y, X4, Right)) → LESS_IN_GA(Y, X)
LESS_IN_GA(s(X), s(Y)) → U5_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) → U4_AG(X, Y, X4, Right, in_in_ag(X, Right))
U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) → IN_IN_AG(X, Right)
in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
LESS_IN_GA(s(X)) → LESS_IN_GA(X)
From the DPs we obtained the following set of size-change graphs:
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
From the DPs we obtained the following set of size-change graphs:
U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X3)) → U1_GG(X, Y, Left, X3, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, X4, Right)) → U3_GG(X, Y, X4, Right, less_in_gg(Y, X))
U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))
U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X3)) → U1_GG(X, Y, Left, X3, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, X4, Right)) → U3_GG(X, Y, X4, Right, less_in_gg(Y, X))
U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X3)) → U1_GG(X, Y, Left, X3, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, X4, Right)) → U3_GG(X, Y, X4, Right, less_in_gg(Y, X))
U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
less_in_gg(x0, x1)
U5_gg(x0, x1, x2)
From the DPs we obtained the following set of size-change graphs:
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
LESS_IN_AG(s(Y)) → LESS_IN_AG(Y)
From the DPs we obtained the following set of size-change graphs:
IN_IN_AG(X, tree(Y, X4, Right)) → U3_AG(X, Y, X4, Right, less_in_ga(Y, X))
U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) → IN_IN_AG(X, Right)
in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))
IN_IN_AG(X, tree(Y, X4, Right)) → U3_AG(X, Y, X4, Right, less_in_ga(Y, X))
U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) → IN_IN_AG(X, Right)
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
IN_IN_AG(tree(Y, X4, Right)) → U3_AG(Y, X4, Right, less_in_ga(Y))
U3_AG(Y, X4, Right, less_out_ga(Y)) → IN_IN_AG(Right)
less_in_ga(0) → less_out_ga(0)
less_in_ga(s(X)) → U5_ga(X, less_in_ga(X))
U5_ga(X, less_out_ga(X)) → less_out_ga(s(X))
less_in_ga(x0)
U5_ga(x0, x1)
From the DPs we obtained the following set of size-change graphs: