0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇒, 89 ms)
↳2 Prolog
↳3 PrologToPiTRSProof (⇒, 0 ms)
↳4 PiTRS
↳5 DependencyPairsProof (⇔, 5 ms)
↳6 PiDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 PiDP
↳9 UsableRulesProof (⇔, 0 ms)
↳10 PiDP
↳11 PiDPToQDPProof (⇒, 5 ms)
↳12 QDP
↳13 UsableRulesReductionPairsProof (⇔, 121 ms)
↳14 QDP
↳15 PisEmptyProof (⇔, 0 ms)
↳16 YES
flatA_in_ga(niltree, nil) → flatA_out_ga(niltree, nil)
flatA_in_ga(tree(T9, niltree, niltree), cons(T9, nil)) → flatA_out_ga(tree(T9, niltree, niltree), cons(T9, nil))
flatA_in_ga(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → U1_ga(T9, T25, T26, T28, flatA_in_ga(T26, T28))
flatA_in_ga(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → U2_ga(T9, T53, T54, T55, T56, T57, T59, flatA_in_ga(tree(T54, T55, tree(T53, T56, T57)), T59))
flatA_in_ga(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → U3_ga(T95, T94, T96, T97, T99, flatA_in_ga(tree(T95, T96, T97), T99))
flatA_in_ga(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → U4_ga(T120, T116, T117, T118, T119, T121, T122, T124, flatA_in_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124))
U4_ga(T120, T116, T117, T118, T119, T121, T122, T124, flatA_out_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124)) → flatA_out_ga(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124)
U3_ga(T95, T94, T96, T97, T99, flatA_out_ga(tree(T95, T96, T97), T99)) → flatA_out_ga(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99))
U2_ga(T9, T53, T54, T55, T56, T57, T59, flatA_out_ga(tree(T54, T55, tree(T53, T56, T57)), T59)) → flatA_out_ga(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59))
U1_ga(T9, T25, T26, T28, flatA_out_ga(T26, T28)) → flatA_out_ga(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28)))
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
flatA_in_ga(niltree, nil) → flatA_out_ga(niltree, nil)
flatA_in_ga(tree(T9, niltree, niltree), cons(T9, nil)) → flatA_out_ga(tree(T9, niltree, niltree), cons(T9, nil))
flatA_in_ga(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → U1_ga(T9, T25, T26, T28, flatA_in_ga(T26, T28))
flatA_in_ga(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → U2_ga(T9, T53, T54, T55, T56, T57, T59, flatA_in_ga(tree(T54, T55, tree(T53, T56, T57)), T59))
flatA_in_ga(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → U3_ga(T95, T94, T96, T97, T99, flatA_in_ga(tree(T95, T96, T97), T99))
flatA_in_ga(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → U4_ga(T120, T116, T117, T118, T119, T121, T122, T124, flatA_in_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124))
U4_ga(T120, T116, T117, T118, T119, T121, T122, T124, flatA_out_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124)) → flatA_out_ga(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124)
U3_ga(T95, T94, T96, T97, T99, flatA_out_ga(tree(T95, T96, T97), T99)) → flatA_out_ga(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99))
U2_ga(T9, T53, T54, T55, T56, T57, T59, flatA_out_ga(tree(T54, T55, tree(T53, T56, T57)), T59)) → flatA_out_ga(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59))
U1_ga(T9, T25, T26, T28, flatA_out_ga(T26, T28)) → flatA_out_ga(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28)))
FLATA_IN_GA(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → U1_GA(T9, T25, T26, T28, flatA_in_ga(T26, T28))
FLATA_IN_GA(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → FLATA_IN_GA(T26, T28)
FLATA_IN_GA(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → U2_GA(T9, T53, T54, T55, T56, T57, T59, flatA_in_ga(tree(T54, T55, tree(T53, T56, T57)), T59))
FLATA_IN_GA(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → FLATA_IN_GA(tree(T54, T55, tree(T53, T56, T57)), T59)
FLATA_IN_GA(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → U3_GA(T95, T94, T96, T97, T99, flatA_in_ga(tree(T95, T96, T97), T99))
FLATA_IN_GA(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → FLATA_IN_GA(tree(T95, T96, T97), T99)
FLATA_IN_GA(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → U4_GA(T120, T116, T117, T118, T119, T121, T122, T124, flatA_in_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124))
FLATA_IN_GA(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → FLATA_IN_GA(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124)
flatA_in_ga(niltree, nil) → flatA_out_ga(niltree, nil)
flatA_in_ga(tree(T9, niltree, niltree), cons(T9, nil)) → flatA_out_ga(tree(T9, niltree, niltree), cons(T9, nil))
flatA_in_ga(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → U1_ga(T9, T25, T26, T28, flatA_in_ga(T26, T28))
flatA_in_ga(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → U2_ga(T9, T53, T54, T55, T56, T57, T59, flatA_in_ga(tree(T54, T55, tree(T53, T56, T57)), T59))
flatA_in_ga(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → U3_ga(T95, T94, T96, T97, T99, flatA_in_ga(tree(T95, T96, T97), T99))
flatA_in_ga(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → U4_ga(T120, T116, T117, T118, T119, T121, T122, T124, flatA_in_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124))
U4_ga(T120, T116, T117, T118, T119, T121, T122, T124, flatA_out_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124)) → flatA_out_ga(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124)
U3_ga(T95, T94, T96, T97, T99, flatA_out_ga(tree(T95, T96, T97), T99)) → flatA_out_ga(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99))
U2_ga(T9, T53, T54, T55, T56, T57, T59, flatA_out_ga(tree(T54, T55, tree(T53, T56, T57)), T59)) → flatA_out_ga(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59))
U1_ga(T9, T25, T26, T28, flatA_out_ga(T26, T28)) → flatA_out_ga(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28)))
FLATA_IN_GA(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → U1_GA(T9, T25, T26, T28, flatA_in_ga(T26, T28))
FLATA_IN_GA(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → FLATA_IN_GA(T26, T28)
FLATA_IN_GA(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → U2_GA(T9, T53, T54, T55, T56, T57, T59, flatA_in_ga(tree(T54, T55, tree(T53, T56, T57)), T59))
FLATA_IN_GA(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → FLATA_IN_GA(tree(T54, T55, tree(T53, T56, T57)), T59)
FLATA_IN_GA(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → U3_GA(T95, T94, T96, T97, T99, flatA_in_ga(tree(T95, T96, T97), T99))
FLATA_IN_GA(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → FLATA_IN_GA(tree(T95, T96, T97), T99)
FLATA_IN_GA(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → U4_GA(T120, T116, T117, T118, T119, T121, T122, T124, flatA_in_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124))
FLATA_IN_GA(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → FLATA_IN_GA(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124)
flatA_in_ga(niltree, nil) → flatA_out_ga(niltree, nil)
flatA_in_ga(tree(T9, niltree, niltree), cons(T9, nil)) → flatA_out_ga(tree(T9, niltree, niltree), cons(T9, nil))
flatA_in_ga(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → U1_ga(T9, T25, T26, T28, flatA_in_ga(T26, T28))
flatA_in_ga(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → U2_ga(T9, T53, T54, T55, T56, T57, T59, flatA_in_ga(tree(T54, T55, tree(T53, T56, T57)), T59))
flatA_in_ga(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → U3_ga(T95, T94, T96, T97, T99, flatA_in_ga(tree(T95, T96, T97), T99))
flatA_in_ga(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → U4_ga(T120, T116, T117, T118, T119, T121, T122, T124, flatA_in_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124))
U4_ga(T120, T116, T117, T118, T119, T121, T122, T124, flatA_out_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124)) → flatA_out_ga(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124)
U3_ga(T95, T94, T96, T97, T99, flatA_out_ga(tree(T95, T96, T97), T99)) → flatA_out_ga(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99))
U2_ga(T9, T53, T54, T55, T56, T57, T59, flatA_out_ga(tree(T54, T55, tree(T53, T56, T57)), T59)) → flatA_out_ga(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59))
U1_ga(T9, T25, T26, T28, flatA_out_ga(T26, T28)) → flatA_out_ga(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28)))
FLATA_IN_GA(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → FLATA_IN_GA(tree(T54, T55, tree(T53, T56, T57)), T59)
FLATA_IN_GA(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → FLATA_IN_GA(T26, T28)
FLATA_IN_GA(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → FLATA_IN_GA(tree(T95, T96, T97), T99)
FLATA_IN_GA(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → FLATA_IN_GA(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124)
flatA_in_ga(niltree, nil) → flatA_out_ga(niltree, nil)
flatA_in_ga(tree(T9, niltree, niltree), cons(T9, nil)) → flatA_out_ga(tree(T9, niltree, niltree), cons(T9, nil))
flatA_in_ga(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → U1_ga(T9, T25, T26, T28, flatA_in_ga(T26, T28))
flatA_in_ga(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → U2_ga(T9, T53, T54, T55, T56, T57, T59, flatA_in_ga(tree(T54, T55, tree(T53, T56, T57)), T59))
flatA_in_ga(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → U3_ga(T95, T94, T96, T97, T99, flatA_in_ga(tree(T95, T96, T97), T99))
flatA_in_ga(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → U4_ga(T120, T116, T117, T118, T119, T121, T122, T124, flatA_in_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124))
U4_ga(T120, T116, T117, T118, T119, T121, T122, T124, flatA_out_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124)) → flatA_out_ga(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124)
U3_ga(T95, T94, T96, T97, T99, flatA_out_ga(tree(T95, T96, T97), T99)) → flatA_out_ga(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99))
U2_ga(T9, T53, T54, T55, T56, T57, T59, flatA_out_ga(tree(T54, T55, tree(T53, T56, T57)), T59)) → flatA_out_ga(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59))
U1_ga(T9, T25, T26, T28, flatA_out_ga(T26, T28)) → flatA_out_ga(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28)))
FLATA_IN_GA(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → FLATA_IN_GA(tree(T54, T55, tree(T53, T56, T57)), T59)
FLATA_IN_GA(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → FLATA_IN_GA(T26, T28)
FLATA_IN_GA(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → FLATA_IN_GA(tree(T95, T96, T97), T99)
FLATA_IN_GA(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → FLATA_IN_GA(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124)
FLATA_IN_GA(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57))) → FLATA_IN_GA(tree(T54, T55, tree(T53, T56, T57)))
FLATA_IN_GA(tree(T9, niltree, tree(T25, niltree, T26))) → FLATA_IN_GA(T26)
FLATA_IN_GA(tree(T95, tree(T94, niltree, T96), T97)) → FLATA_IN_GA(tree(T95, T96, T97))
FLATA_IN_GA(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122)) → FLATA_IN_GA(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))))
No rules are removed from R.
FLATA_IN_GA(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57))) → FLATA_IN_GA(tree(T54, T55, tree(T53, T56, T57)))
FLATA_IN_GA(tree(T9, niltree, tree(T25, niltree, T26))) → FLATA_IN_GA(T26)
FLATA_IN_GA(tree(T95, tree(T94, niltree, T96), T97)) → FLATA_IN_GA(tree(T95, T96, T97))
FLATA_IN_GA(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122)) → FLATA_IN_GA(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))))
POL(FLATA_IN_GA(x1)) = 2·x1
POL(niltree) = 0
POL(tree(x1, x2, x3)) = 2 + 2·x1 + 2·x2 + x3