0 Prolog
↳1 PrologToPiTRSProof (⇒, 61 ms)
↳2 PiTRS
↳3 DependencyPairsProof (⇔, 185 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 AND
↳7 PiDP
↳8 UsableRulesProof (⇔, 0 ms)
↳9 PiDP
↳10 PiDPToQDPProof (⇒, 0 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 (⇔, 2 ms)
↳37 PiDP
↳38 PiDPToQDPProof (⇒, 0 ms)
↳39 QDP
↳40 QDPSizeChangeProof (⇔, 0 ms)
↳41 YES
↳42 PiDP
↳43 UsableRulesProof (⇔, 0 ms)
↳44 PiDP
↳45 PiDPToQDPProof (⇒, 3 ms)
↳46 QDP
↳47 QDPSizeChangeProof (⇔, 0 ms)
↳48 YES
delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
DELETE_IN_AGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_AGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
DELETE_IN_AGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_GAA(Right, Y, Right1)
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_GAA(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_GAA(Left, Y, Left1)
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_AGA(X, Y, Left, Right, Left1, less_in_ag(X, Y))
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_AG(X, Y)
LESS_IN_AG(s(X), s(Y)) → U7_AG(X, Y, less_in_ag(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
U2_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_AGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
U2_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_GAA(Right, Y, Right1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U7_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_GGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_GGA(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right, Right1)
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X))
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GA(Y, X)
LESS_IN_GA(s(X), s(Y)) → U7_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_AGA(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AGA(X, Right, Right1)
delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
DELETE_IN_AGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_AGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
DELETE_IN_AGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_GAA(Right, Y, Right1)
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_GAA(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_GAA(Left, Y, Left1)
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_AGA(X, Y, Left, Right, Left1, less_in_ag(X, Y))
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_AG(X, Y)
LESS_IN_AG(s(X), s(Y)) → U7_AG(X, Y, less_in_ag(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
U2_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_AGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
U2_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_GAA(Right, Y, Right1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U7_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_GGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_GGA(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right, Right1)
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X))
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GA(Y, X)
LESS_IN_GA(s(X), s(Y)) → U7_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_AGA(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AGA(X, Right, Right1)
delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
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)
delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
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:
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
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:
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_GAA(Left, Y, Left1)
delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_GAA(Left, Y, Left1)
DELMIN_IN_GAA(tree(X, Left, X1)) → DELMIN_IN_GAA(Left)
From the DPs we obtained the following set of size-change graphs:
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right, Right1)
delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right, Right1)
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
DELETE_IN_GGA(X, tree(Y, Left, Right)) → U2_GGA(X, Left, less_in_gg(X, Y))
U2_GGA(X, Left, less_out_gg) → DELETE_IN_GGA(X, Left)
DELETE_IN_GGA(X, tree(Y, Left, Right)) → U4_GGA(X, Right, less_in_gg(Y, X))
U4_GGA(X, Right, less_out_gg) → DELETE_IN_GGA(X, Right)
less_in_gg(0, s(X3)) → less_out_gg
less_in_gg(s(X), s(Y)) → U7_gg(less_in_gg(X, Y))
U7_gg(less_out_gg) → less_out_gg
less_in_gg(x0, x1)
U7_gg(x0)
From the DPs we obtained the following set of size-change graphs:
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X))
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AGA(X, Right, Right1)
delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X))
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AGA(X, Right, Right1)
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
DELETE_IN_AGA(tree(Y, Left, Right)) → U4_AGA(Right, less_in_ga(Y))
U4_AGA(Right, less_out_ga) → DELETE_IN_AGA(Right)
less_in_ga(0) → less_out_ga
less_in_ga(s(X)) → U7_ga(less_in_ga(X))
U7_ga(less_out_ga) → less_out_ga
less_in_ga(x0)
U7_ga(x0)
From the DPs we obtained the following set of size-change graphs: