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 PrologToPiTRSProof (⇐)
↳29 PiTRS
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(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_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(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_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GAG(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_AGG(Right, Y, Right1)
DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_AGG(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_AGG(Left, Y, Left1)
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GAG(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_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_GAG(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GAG(X, Left, Left1)
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_GAG(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GAG(X, Right, Right1)
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(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_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GAG(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_AGG(Right, Y, Right1)
DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_AGG(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_AGG(Left, Y, Left1)
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GAG(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_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_GAG(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GAG(X, Left, Left1)
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_GAG(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GAG(X, Right, Right1)
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(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_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(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_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, 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:
DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_AGG(Left, Y, Left1)
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(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_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_AGG(Left, Y, Left1)
DELMIN_IN_AGG(Y, tree(X, Left1, X2)) → DELMIN_IN_AGG(Y, Left1)
From the DPs we obtained the following set of size-change graphs:
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GAG(X, Left, Left1)
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GAG(X, Right, Right1)
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(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_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GAG(X, Left, Left1)
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GAG(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_GAG(X, tree(Y, Left1, Right)) → U2_GAG(X, Y, Right, Left1, less_in_gg(X, Y))
U2_GAG(X, Y, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GAG(X, Left1)
DELETE_IN_GAG(X, tree(Y, Left, Right1)) → U4_GAG(X, Y, Left, Right1, less_in_gg(Y, X))
U4_GAG(X, Y, Left, Right1, less_out_gg(Y, X)) → DELETE_IN_GAG(X, 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))
less_in_gg(x0, x1)
U7_gg(x0, x1, x2)
From the DPs we obtained the following set of size-change graphs:
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(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_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(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_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))