/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 164 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
    (7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (8) CdtProblem
    (9) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CdtProblem
    (13) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (14) CdtProblem
    (15) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 7 ms]
    (16) CdtProblem
    (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 128 ms]
    (18) CdtProblem
    (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 50 ms]
    (20) CdtProblem
    (21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 43 ms]
    (22) CdtProblem
    (23) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 271 ms]
    (24) CdtProblem
    (25) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) BOUNDS(1, 1)
(27) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(28) CpxRelTRS
    (29) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) typed CpxTrs
    (31) OrderProof [LOWER BOUND(ID), 0 ms]
    (32) typed CpxTrs
    (33) RewriteLemmaProof [LOWER BOUND(ID), 440 ms]
    (34) BEST
        (35) proven lower bound
            (36) LowerBoundPropagationProof [FINISHED, 0 ms]
            (37) BOUNDS(n^1, INF)
        (38) typed CpxTrs
            (39) RewriteLemmaProof [LOWER BOUND(ID), 87 ms]
            (40) BOUNDS(1, INF)


----------------------------------------

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   h(X, Z) -> f(X, s(X), Z)
   f(X, Y, g(X, Y)) -> h(0, g(X, Y))
   g(0, Y) -> 0
   g(X, s(Y)) -> g(X, Y)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_s(x_1) -> s(encArg(x_1))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0


----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   h(X, Z) -> f(X, s(X), Z)
   f(X, Y, g(X, Y)) -> h(0, g(X, Y))
   g(0, Y) -> 0
   g(X, s(Y)) -> g(X, Y)

The (relative) TRS S consists of the following rules:

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_s(x_1) -> s(encArg(x_1))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0

Rewrite Strategy: INNERMOST
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   h(X, Z) -> f(X, s(X), Z)
   f(X, Y, g(X, Y)) -> h(0, g(X, Y))
   g(0, Y) -> 0
   g(X, s(Y)) -> g(X, Y)

The (relative) TRS S consists of the following rules:

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_s(x_1) -> s(encArg(x_1))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0

Rewrite Strategy: INNERMOST
----------------------------------------

(5) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_h(z0, z1)) -> h(encArg(z0), encArg(z1))
   encArg(cons_f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encode_h(z0, z1) -> h(encArg(z0), encArg(z1))
   encode_f(z0, z1, z2) -> f(encArg(z0), encArg(z1), encArg(z2))
   encode_s(z0) -> s(encArg(z0))
   encode_g(z0, z1) -> g(encArg(z0), encArg(z1))
   encode_0 -> 0
   h(z0, z1) -> f(z0, s(z0), z1)
   f(z0, z1, g(z0, z1)) -> h(0, g(z0, z1))
   g(0, z0) -> 0
   g(z0, s(z1)) -> g(z0, z1)
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(0) -> c1
   ENCARG(cons_h(z0, z1)) -> c2(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_f(z0, z1, z2)) -> c3(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_g(z0, z1)) -> c4(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_H(z0, z1) -> c5(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_F(z0, z1, z2) -> c6(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCODE_S(z0) -> c7(ENCARG(z0))
   ENCODE_G(z0, z1) -> c8(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_0 -> c9
   H(z0, z1) -> c10(F(z0, s(z0), z1))
   F(z0, z1, g(z0, z1)) -> c11(H(0, g(z0, z1)), G(z0, z1))
   G(0, z0) -> c12
   G(z0, s(z1)) -> c13(G(z0, z1))
S tuples:
   H(z0, z1) -> c10(F(z0, s(z0), z1))
   F(z0, z1, g(z0, z1)) -> c11(H(0, g(z0, z1)), G(z0, z1))
   G(0, z0) -> c12
   G(z0, s(z1)) -> c13(G(z0, z1))
K tuples:none
Defined Rule Symbols:   h_2, f_3, g_2, encArg_1, encode_h_2, encode_f_3, encode_s_1, encode_g_2, encode_0

Defined Pair Symbols:   ENCARG_1, ENCODE_H_2, ENCODE_F_3, ENCODE_S_1, ENCODE_G_2, ENCODE_0, H_2, F_3, G_2

Compound Symbols:   c_1, c1, c2_3, c3_4, c4_3, c5_3, c6_4, c7_1, c8_3, c9, c10_1, c11_2, c12, c13_1


----------------------------------------

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   ENCODE_S(z0) -> c7(ENCARG(z0))
Removed 2 trailing nodes:
   ENCODE_0 -> c9
   ENCARG(0) -> c1

----------------------------------------

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_h(z0, z1)) -> h(encArg(z0), encArg(z1))
   encArg(cons_f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encode_h(z0, z1) -> h(encArg(z0), encArg(z1))
   encode_f(z0, z1, z2) -> f(encArg(z0), encArg(z1), encArg(z2))
   encode_s(z0) -> s(encArg(z0))
   encode_g(z0, z1) -> g(encArg(z0), encArg(z1))
   encode_0 -> 0
   h(z0, z1) -> f(z0, s(z0), z1)
   f(z0, z1, g(z0, z1)) -> h(0, g(z0, z1))
   g(0, z0) -> 0
   g(z0, s(z1)) -> g(z0, z1)
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_h(z0, z1)) -> c2(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_f(z0, z1, z2)) -> c3(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_g(z0, z1)) -> c4(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_H(z0, z1) -> c5(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_F(z0, z1, z2) -> c6(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCODE_G(z0, z1) -> c8(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   H(z0, z1) -> c10(F(z0, s(z0), z1))
   F(z0, z1, g(z0, z1)) -> c11(H(0, g(z0, z1)), G(z0, z1))
   G(0, z0) -> c12
   G(z0, s(z1)) -> c13(G(z0, z1))
S tuples:
   H(z0, z1) -> c10(F(z0, s(z0), z1))
   F(z0, z1, g(z0, z1)) -> c11(H(0, g(z0, z1)), G(z0, z1))
   G(0, z0) -> c12
   G(z0, s(z1)) -> c13(G(z0, z1))
K tuples:none
Defined Rule Symbols:   h_2, f_3, g_2, encArg_1, encode_h_2, encode_f_3, encode_s_1, encode_g_2, encode_0

Defined Pair Symbols:   ENCARG_1, ENCODE_H_2, ENCODE_F_3, ENCODE_G_2, H_2, F_3, G_2

Compound Symbols:   c_1, c2_3, c3_4, c4_3, c5_3, c6_4, c8_3, c10_1, c11_2, c12, c13_1


----------------------------------------

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_h(z0, z1)) -> h(encArg(z0), encArg(z1))
   encArg(cons_f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encode_h(z0, z1) -> h(encArg(z0), encArg(z1))
   encode_f(z0, z1, z2) -> f(encArg(z0), encArg(z1), encArg(z2))
   encode_s(z0) -> s(encArg(z0))
   encode_g(z0, z1) -> g(encArg(z0), encArg(z1))
   encode_0 -> 0
   h(z0, z1) -> f(z0, s(z0), z1)
   f(z0, z1, g(z0, z1)) -> h(0, g(z0, z1))
   g(0, z0) -> 0
   g(z0, s(z1)) -> g(z0, z1)
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_h(z0, z1)) -> c2(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_f(z0, z1, z2)) -> c3(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_g(z0, z1)) -> c4(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_H(z0, z1) -> c5(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_F(z0, z1, z2) -> c6(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCODE_G(z0, z1) -> c8(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(z0, z1, g(z0, z1)) -> c11(H(0, g(z0, z1)), G(z0, z1))
   G(0, z0) -> c12
   G(z0, s(z1)) -> c13(G(z0, z1))
   H(z0, z1) -> c10
S tuples:
   F(z0, z1, g(z0, z1)) -> c11(H(0, g(z0, z1)), G(z0, z1))
   G(0, z0) -> c12
   G(z0, s(z1)) -> c13(G(z0, z1))
   H(z0, z1) -> c10
K tuples:none
Defined Rule Symbols:   h_2, f_3, g_2, encArg_1, encode_h_2, encode_f_3, encode_s_1, encode_g_2, encode_0

Defined Pair Symbols:   ENCARG_1, ENCODE_H_2, ENCODE_F_3, ENCODE_G_2, F_3, G_2, H_2

Compound Symbols:   c_1, c2_3, c3_4, c4_3, c5_3, c6_4, c8_3, c11_2, c12, c13_1, c10


----------------------------------------

(11) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_h(z0, z1)) -> h(encArg(z0), encArg(z1))
   encArg(cons_f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encode_h(z0, z1) -> h(encArg(z0), encArg(z1))
   encode_f(z0, z1, z2) -> f(encArg(z0), encArg(z1), encArg(z2))
   encode_s(z0) -> s(encArg(z0))
   encode_g(z0, z1) -> g(encArg(z0), encArg(z1))
   encode_0 -> 0
   h(z0, z1) -> f(z0, s(z0), z1)
   f(z0, z1, g(z0, z1)) -> h(0, g(z0, z1))
   g(0, z0) -> 0
   g(z0, s(z1)) -> g(z0, z1)
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_h(z0, z1)) -> c2(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_f(z0, z1, z2)) -> c3(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_g(z0, z1)) -> c4(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, z0) -> c12
   G(z0, s(z1)) -> c13(G(z0, z1))
   H(z0, z1) -> c10
   ENCODE_H(z0, z1) -> c1(H(encArg(z0), encArg(z1)))
   ENCODE_H(z0, z1) -> c1(ENCARG(z0))
   ENCODE_H(z0, z1) -> c1(ENCARG(z1))
   ENCODE_F(z0, z1, z2) -> c1(F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_F(z0, z1, z2) -> c1(ENCARG(z0))
   ENCODE_F(z0, z1, z2) -> c1(ENCARG(z1))
   ENCODE_F(z0, z1, z2) -> c1(ENCARG(z2))
   ENCODE_G(z0, z1) -> c1(G(encArg(z0), encArg(z1)))
   ENCODE_G(z0, z1) -> c1(ENCARG(z0))
   ENCODE_G(z0, z1) -> c1(ENCARG(z1))
   F(z0, z1, g(z0, z1)) -> c1(H(0, g(z0, z1)))
   F(z0, z1, g(z0, z1)) -> c1(G(z0, z1))
S tuples:
   G(0, z0) -> c12
   G(z0, s(z1)) -> c13(G(z0, z1))
   H(z0, z1) -> c10
   F(z0, z1, g(z0, z1)) -> c1(H(0, g(z0, z1)))
   F(z0, z1, g(z0, z1)) -> c1(G(z0, z1))
K tuples:none
Defined Rule Symbols:   h_2, f_3, g_2, encArg_1, encode_h_2, encode_f_3, encode_s_1, encode_g_2, encode_0

Defined Pair Symbols:   ENCARG_1, G_2, H_2, ENCODE_H_2, ENCODE_F_3, ENCODE_G_2, F_3

Compound Symbols:   c_1, c2_3, c3_4, c4_3, c12, c13_1, c10, c1_1


----------------------------------------

(13) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 7 leading nodes:
   ENCODE_H(z0, z1) -> c1(ENCARG(z0))
   ENCODE_H(z0, z1) -> c1(ENCARG(z1))
   ENCODE_F(z0, z1, z2) -> c1(ENCARG(z0))
   ENCODE_F(z0, z1, z2) -> c1(ENCARG(z1))
   ENCODE_F(z0, z1, z2) -> c1(ENCARG(z2))
   ENCODE_G(z0, z1) -> c1(ENCARG(z0))
   ENCODE_G(z0, z1) -> c1(ENCARG(z1))

----------------------------------------

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_h(z0, z1)) -> h(encArg(z0), encArg(z1))
   encArg(cons_f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encode_h(z0, z1) -> h(encArg(z0), encArg(z1))
   encode_f(z0, z1, z2) -> f(encArg(z0), encArg(z1), encArg(z2))
   encode_s(z0) -> s(encArg(z0))
   encode_g(z0, z1) -> g(encArg(z0), encArg(z1))
   encode_0 -> 0
   h(z0, z1) -> f(z0, s(z0), z1)
   f(z0, z1, g(z0, z1)) -> h(0, g(z0, z1))
   g(0, z0) -> 0
   g(z0, s(z1)) -> g(z0, z1)
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_h(z0, z1)) -> c2(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_f(z0, z1, z2)) -> c3(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_g(z0, z1)) -> c4(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, z0) -> c12
   G(z0, s(z1)) -> c13(G(z0, z1))
   H(z0, z1) -> c10
   ENCODE_H(z0, z1) -> c1(H(encArg(z0), encArg(z1)))
   ENCODE_F(z0, z1, z2) -> c1(F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_G(z0, z1) -> c1(G(encArg(z0), encArg(z1)))
   F(z0, z1, g(z0, z1)) -> c1(H(0, g(z0, z1)))
   F(z0, z1, g(z0, z1)) -> c1(G(z0, z1))
S tuples:
   G(0, z0) -> c12
   G(z0, s(z1)) -> c13(G(z0, z1))
   H(z0, z1) -> c10
   F(z0, z1, g(z0, z1)) -> c1(H(0, g(z0, z1)))
   F(z0, z1, g(z0, z1)) -> c1(G(z0, z1))
K tuples:none
Defined Rule Symbols:   h_2, f_3, g_2, encArg_1, encode_h_2, encode_f_3, encode_s_1, encode_g_2, encode_0

Defined Pair Symbols:   ENCARG_1, G_2, H_2, ENCODE_H_2, ENCODE_F_3, ENCODE_G_2, F_3

Compound Symbols:   c_1, c2_3, c3_4, c4_3, c12, c13_1, c10, c1_1


----------------------------------------

(15) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_h(z0, z1) -> h(encArg(z0), encArg(z1))
   encode_f(z0, z1, z2) -> f(encArg(z0), encArg(z1), encArg(z2))
   encode_s(z0) -> s(encArg(z0))
   encode_g(z0, z1) -> g(encArg(z0), encArg(z1))
   encode_0 -> 0

----------------------------------------

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_h(z0, z1)) -> h(encArg(z0), encArg(z1))
   encArg(cons_f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   h(z0, z1) -> f(z0, s(z0), z1)
   f(z0, z1, g(z0, z1)) -> h(0, g(z0, z1))
   g(0, z0) -> 0
   g(z0, s(z1)) -> g(z0, z1)
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_h(z0, z1)) -> c2(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_f(z0, z1, z2)) -> c3(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_g(z0, z1)) -> c4(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, z0) -> c12
   G(z0, s(z1)) -> c13(G(z0, z1))
   H(z0, z1) -> c10
   ENCODE_H(z0, z1) -> c1(H(encArg(z0), encArg(z1)))
   ENCODE_F(z0, z1, z2) -> c1(F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_G(z0, z1) -> c1(G(encArg(z0), encArg(z1)))
   F(z0, z1, g(z0, z1)) -> c1(H(0, g(z0, z1)))
   F(z0, z1, g(z0, z1)) -> c1(G(z0, z1))
S tuples:
   G(0, z0) -> c12
   G(z0, s(z1)) -> c13(G(z0, z1))
   H(z0, z1) -> c10
   F(z0, z1, g(z0, z1)) -> c1(H(0, g(z0, z1)))
   F(z0, z1, g(z0, z1)) -> c1(G(z0, z1))
K tuples:none
Defined Rule Symbols:   encArg_1, h_2, f_3, g_2

Defined Pair Symbols:   ENCARG_1, G_2, H_2, ENCODE_H_2, ENCODE_F_3, ENCODE_G_2, F_3

Compound Symbols:   c_1, c2_3, c3_4, c4_3, c12, c13_1, c10, c1_1


----------------------------------------

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(0, z0) -> c12
   F(z0, z1, g(z0, z1)) -> c1(H(0, g(z0, z1)))
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_h(z0, z1)) -> c2(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_f(z0, z1, z2)) -> c3(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_g(z0, z1)) -> c4(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, z0) -> c12
   G(z0, s(z1)) -> c13(G(z0, z1))
   H(z0, z1) -> c10
   ENCODE_H(z0, z1) -> c1(H(encArg(z0), encArg(z1)))
   ENCODE_F(z0, z1, z2) -> c1(F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_G(z0, z1) -> c1(G(encArg(z0), encArg(z1)))
   F(z0, z1, g(z0, z1)) -> c1(H(0, g(z0, z1)))
   F(z0, z1, g(z0, z1)) -> c1(G(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_F(x_1, x_2, x_3)) = [1]
   POL(ENCODE_G(x_1, x_2)) = [1]
   POL(ENCODE_H(x_1, x_2)) = 0
   POL(F(x_1, x_2, x_3)) = [1]
   POL(G(x_1, x_2)) = [1]
   POL(H(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10) = 0
   POL(c12) = 0
   POL(c13(x_1)) = x_1
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c4(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(cons_f(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(cons_g(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_h(x_1, x_2)) = [1] + x_1 + x_2
   POL(encArg(x_1)) = [1] + x_1
   POL(f(x_1, x_2, x_3)) = [1] + x_1
   POL(g(x_1, x_2)) = [1] + x_2
   POL(h(x_1, x_2)) = [1] + x_1
   POL(s(x_1)) = [1] + x_1

----------------------------------------

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_h(z0, z1)) -> h(encArg(z0), encArg(z1))
   encArg(cons_f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   h(z0, z1) -> f(z0, s(z0), z1)
   f(z0, z1, g(z0, z1)) -> h(0, g(z0, z1))
   g(0, z0) -> 0
   g(z0, s(z1)) -> g(z0, z1)
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_h(z0, z1)) -> c2(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_f(z0, z1, z2)) -> c3(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_g(z0, z1)) -> c4(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, z0) -> c12
   G(z0, s(z1)) -> c13(G(z0, z1))
   H(z0, z1) -> c10
   ENCODE_H(z0, z1) -> c1(H(encArg(z0), encArg(z1)))
   ENCODE_F(z0, z1, z2) -> c1(F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_G(z0, z1) -> c1(G(encArg(z0), encArg(z1)))
   F(z0, z1, g(z0, z1)) -> c1(H(0, g(z0, z1)))
   F(z0, z1, g(z0, z1)) -> c1(G(z0, z1))
S tuples:
   G(z0, s(z1)) -> c13(G(z0, z1))
   H(z0, z1) -> c10
   F(z0, z1, g(z0, z1)) -> c1(G(z0, z1))
K tuples:
   G(0, z0) -> c12
   F(z0, z1, g(z0, z1)) -> c1(H(0, g(z0, z1)))
Defined Rule Symbols:   encArg_1, h_2, f_3, g_2

Defined Pair Symbols:   ENCARG_1, G_2, H_2, ENCODE_H_2, ENCODE_F_3, ENCODE_G_2, F_3

Compound Symbols:   c_1, c2_3, c3_4, c4_3, c12, c13_1, c10, c1_1


----------------------------------------

(19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(z0, z1, g(z0, z1)) -> c1(G(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_h(z0, z1)) -> c2(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_f(z0, z1, z2)) -> c3(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_g(z0, z1)) -> c4(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, z0) -> c12
   G(z0, s(z1)) -> c13(G(z0, z1))
   H(z0, z1) -> c10
   ENCODE_H(z0, z1) -> c1(H(encArg(z0), encArg(z1)))
   ENCODE_F(z0, z1, z2) -> c1(F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_G(z0, z1) -> c1(G(encArg(z0), encArg(z1)))
   F(z0, z1, g(z0, z1)) -> c1(H(0, g(z0, z1)))
   F(z0, z1, g(z0, z1)) -> c1(G(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_F(x_1, x_2, x_3)) = [1] + x_1 + x_2
   POL(ENCODE_G(x_1, x_2)) = 0
   POL(ENCODE_H(x_1, x_2)) = 0
   POL(F(x_1, x_2, x_3)) = [1]
   POL(G(x_1, x_2)) = 0
   POL(H(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10) = 0
   POL(c12) = 0
   POL(c13(x_1)) = x_1
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c4(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(cons_f(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(cons_g(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_h(x_1, x_2)) = [1] + x_1 + x_2
   POL(encArg(x_1)) = [1] + x_1
   POL(f(x_1, x_2, x_3)) = [1] + x_1
   POL(g(x_1, x_2)) = [1] + x_1 + x_2
   POL(h(x_1, x_2)) = [1] + x_1
   POL(s(x_1)) = [1] + x_1

----------------------------------------

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_h(z0, z1)) -> h(encArg(z0), encArg(z1))
   encArg(cons_f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   h(z0, z1) -> f(z0, s(z0), z1)
   f(z0, z1, g(z0, z1)) -> h(0, g(z0, z1))
   g(0, z0) -> 0
   g(z0, s(z1)) -> g(z0, z1)
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_h(z0, z1)) -> c2(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_f(z0, z1, z2)) -> c3(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_g(z0, z1)) -> c4(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, z0) -> c12
   G(z0, s(z1)) -> c13(G(z0, z1))
   H(z0, z1) -> c10
   ENCODE_H(z0, z1) -> c1(H(encArg(z0), encArg(z1)))
   ENCODE_F(z0, z1, z2) -> c1(F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_G(z0, z1) -> c1(G(encArg(z0), encArg(z1)))
   F(z0, z1, g(z0, z1)) -> c1(H(0, g(z0, z1)))
   F(z0, z1, g(z0, z1)) -> c1(G(z0, z1))
S tuples:
   G(z0, s(z1)) -> c13(G(z0, z1))
   H(z0, z1) -> c10
K tuples:
   G(0, z0) -> c12
   F(z0, z1, g(z0, z1)) -> c1(H(0, g(z0, z1)))
   F(z0, z1, g(z0, z1)) -> c1(G(z0, z1))
Defined Rule Symbols:   encArg_1, h_2, f_3, g_2

Defined Pair Symbols:   ENCARG_1, G_2, H_2, ENCODE_H_2, ENCODE_F_3, ENCODE_G_2, F_3

Compound Symbols:   c_1, c2_3, c3_4, c4_3, c12, c13_1, c10, c1_1


----------------------------------------

(21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   H(z0, z1) -> c10
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_h(z0, z1)) -> c2(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_f(z0, z1, z2)) -> c3(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_g(z0, z1)) -> c4(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, z0) -> c12
   G(z0, s(z1)) -> c13(G(z0, z1))
   H(z0, z1) -> c10
   ENCODE_H(z0, z1) -> c1(H(encArg(z0), encArg(z1)))
   ENCODE_F(z0, z1, z2) -> c1(F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_G(z0, z1) -> c1(G(encArg(z0), encArg(z1)))
   F(z0, z1, g(z0, z1)) -> c1(H(0, g(z0, z1)))
   F(z0, z1, g(z0, z1)) -> c1(G(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_F(x_1, x_2, x_3)) = [1] + x_1 + x_2
   POL(ENCODE_G(x_1, x_2)) = [1]
   POL(ENCODE_H(x_1, x_2)) = [1]
   POL(F(x_1, x_2, x_3)) = [1]
   POL(G(x_1, x_2)) = [1]
   POL(H(x_1, x_2)) = [1]
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10) = 0
   POL(c12) = 0
   POL(c13(x_1)) = x_1
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c4(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(cons_f(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(cons_g(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_h(x_1, x_2)) = [1] + x_1 + x_2
   POL(encArg(x_1)) = [1] + x_1
   POL(f(x_1, x_2, x_3)) = x_3
   POL(g(x_1, x_2)) = [1] + x_1
   POL(h(x_1, x_2)) = x_2
   POL(s(x_1)) = [1] + x_1

----------------------------------------

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_h(z0, z1)) -> h(encArg(z0), encArg(z1))
   encArg(cons_f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   h(z0, z1) -> f(z0, s(z0), z1)
   f(z0, z1, g(z0, z1)) -> h(0, g(z0, z1))
   g(0, z0) -> 0
   g(z0, s(z1)) -> g(z0, z1)
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_h(z0, z1)) -> c2(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_f(z0, z1, z2)) -> c3(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_g(z0, z1)) -> c4(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, z0) -> c12
   G(z0, s(z1)) -> c13(G(z0, z1))
   H(z0, z1) -> c10
   ENCODE_H(z0, z1) -> c1(H(encArg(z0), encArg(z1)))
   ENCODE_F(z0, z1, z2) -> c1(F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_G(z0, z1) -> c1(G(encArg(z0), encArg(z1)))
   F(z0, z1, g(z0, z1)) -> c1(H(0, g(z0, z1)))
   F(z0, z1, g(z0, z1)) -> c1(G(z0, z1))
S tuples:
   G(z0, s(z1)) -> c13(G(z0, z1))
K tuples:
   G(0, z0) -> c12
   F(z0, z1, g(z0, z1)) -> c1(H(0, g(z0, z1)))
   F(z0, z1, g(z0, z1)) -> c1(G(z0, z1))
   H(z0, z1) -> c10
Defined Rule Symbols:   encArg_1, h_2, f_3, g_2

Defined Pair Symbols:   ENCARG_1, G_2, H_2, ENCODE_H_2, ENCODE_F_3, ENCODE_G_2, F_3

Compound Symbols:   c_1, c2_3, c3_4, c4_3, c12, c13_1, c10, c1_1


----------------------------------------

(23) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(z0, s(z1)) -> c13(G(z0, z1))
We considered the (Usable) Rules:
   h(z0, z1) -> f(z0, s(z0), z1)
   encArg(s(z0)) -> s(encArg(z0))
   f(z0, z1, g(z0, z1)) -> h(0, g(z0, z1))
   g(z0, s(z1)) -> g(z0, z1)
   encArg(cons_h(z0, z1)) -> h(encArg(z0), encArg(z1))
   g(0, z0) -> 0
   encArg(cons_f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encArg(0) -> 0
And the Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_h(z0, z1)) -> c2(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_f(z0, z1, z2)) -> c3(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_g(z0, z1)) -> c4(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, z0) -> c12
   G(z0, s(z1)) -> c13(G(z0, z1))
   H(z0, z1) -> c10
   ENCODE_H(z0, z1) -> c1(H(encArg(z0), encArg(z1)))
   ENCODE_F(z0, z1, z2) -> c1(F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_G(z0, z1) -> c1(G(encArg(z0), encArg(z1)))
   F(z0, z1, g(z0, z1)) -> c1(H(0, g(z0, z1)))
   F(z0, z1, g(z0, z1)) -> c1(G(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_F(x_1, x_2, x_3)) = [2] + [2]x_1 + [2]x_2 + x_3 + [2]x_3^2 + [2]x_2*x_3 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + [2]x_2^2
   POL(ENCODE_G(x_1, x_2)) = [1] + [2]x_2 + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2
   POL(ENCODE_H(x_1, x_2)) = [2] + [2]x_1 + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2
   POL(F(x_1, x_2, x_3)) = x_3
   POL(G(x_1, x_2)) = x_2
   POL(H(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10) = 0
   POL(c12) = 0
   POL(c13(x_1)) = x_1
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c4(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(cons_f(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(cons_g(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_h(x_1, x_2)) = x_1 + x_2
   POL(encArg(x_1)) = x_1
   POL(f(x_1, x_2, x_3)) = 0
   POL(g(x_1, x_2)) = x_2
   POL(h(x_1, x_2)) = 0
   POL(s(x_1)) = [1] + x_1

----------------------------------------

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_h(z0, z1)) -> h(encArg(z0), encArg(z1))
   encArg(cons_f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   h(z0, z1) -> f(z0, s(z0), z1)
   f(z0, z1, g(z0, z1)) -> h(0, g(z0, z1))
   g(0, z0) -> 0
   g(z0, s(z1)) -> g(z0, z1)
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_h(z0, z1)) -> c2(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_f(z0, z1, z2)) -> c3(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_g(z0, z1)) -> c4(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, z0) -> c12
   G(z0, s(z1)) -> c13(G(z0, z1))
   H(z0, z1) -> c10
   ENCODE_H(z0, z1) -> c1(H(encArg(z0), encArg(z1)))
   ENCODE_F(z0, z1, z2) -> c1(F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_G(z0, z1) -> c1(G(encArg(z0), encArg(z1)))
   F(z0, z1, g(z0, z1)) -> c1(H(0, g(z0, z1)))
   F(z0, z1, g(z0, z1)) -> c1(G(z0, z1))
S tuples:none
K tuples:
   G(0, z0) -> c12
   F(z0, z1, g(z0, z1)) -> c1(H(0, g(z0, z1)))
   F(z0, z1, g(z0, z1)) -> c1(G(z0, z1))
   H(z0, z1) -> c10
   G(z0, s(z1)) -> c13(G(z0, z1))
Defined Rule Symbols:   encArg_1, h_2, f_3, g_2

Defined Pair Symbols:   ENCARG_1, G_2, H_2, ENCODE_H_2, ENCODE_F_3, ENCODE_G_2, F_3

Compound Symbols:   c_1, c2_3, c3_4, c4_3, c12, c13_1, c10, c1_1


----------------------------------------

(25) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
----------------------------------------

(26)
BOUNDS(1, 1)

----------------------------------------

(27) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(28)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   h(X, Z) -> f(X, s(X), Z)
   f(X, Y, g(X, Y)) -> h(0', g(X, Y))
   g(0', Y) -> 0'
   g(X, s(Y)) -> g(X, Y)

The (relative) TRS S consists of the following rules:

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0') -> 0'
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_s(x_1) -> s(encArg(x_1))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0'

Rewrite Strategy: INNERMOST
----------------------------------------

(29) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(30)
Obligation:
Innermost TRS:
Rules:
h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0', g(X, Y))
g(0', Y) -> 0'
g(X, s(Y)) -> g(X, Y)
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encode_s(x_1) -> s(encArg(x_1))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_0 -> 0'

Types:
h :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
f :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
s :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
g :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
0' :: s:0':cons_h:cons_f:cons_g
encArg :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
cons_h :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
cons_f :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
cons_g :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
encode_h :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
encode_f :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
encode_s :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
encode_g :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
encode_0 :: s:0':cons_h:cons_f:cons_g
hole_s:0':cons_h:cons_f:cons_g1_4 :: s:0':cons_h:cons_f:cons_g
gen_s:0':cons_h:cons_f:cons_g2_4 :: Nat -> s:0':cons_h:cons_f:cons_g

----------------------------------------

(31) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
h, f, g, encArg

They will be analysed ascendingly in the following order:
h = f
h < encArg
g < f
f < encArg
g < encArg

----------------------------------------

(32)
Obligation:
Innermost TRS:
Rules:
h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0', g(X, Y))
g(0', Y) -> 0'
g(X, s(Y)) -> g(X, Y)
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encode_s(x_1) -> s(encArg(x_1))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_0 -> 0'

Types:
h :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
f :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
s :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
g :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
0' :: s:0':cons_h:cons_f:cons_g
encArg :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
cons_h :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
cons_f :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
cons_g :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
encode_h :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
encode_f :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
encode_s :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
encode_g :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
encode_0 :: s:0':cons_h:cons_f:cons_g
hole_s:0':cons_h:cons_f:cons_g1_4 :: s:0':cons_h:cons_f:cons_g
gen_s:0':cons_h:cons_f:cons_g2_4 :: Nat -> s:0':cons_h:cons_f:cons_g


Generator Equations:
gen_s:0':cons_h:cons_f:cons_g2_4(0) <=> 0'
gen_s:0':cons_h:cons_f:cons_g2_4(+(x, 1)) <=> s(gen_s:0':cons_h:cons_f:cons_g2_4(x))


The following defined symbols remain to be analysed:
g, h, f, encArg

They will be analysed ascendingly in the following order:
h = f
h < encArg
g < f
f < encArg
g < encArg

----------------------------------------

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_s:0':cons_h:cons_f:cons_g2_4(a), gen_s:0':cons_h:cons_f:cons_g2_4(+(1, n4_4))) -> *3_4, rt in Omega(n4_4)

Induction Base:
g(gen_s:0':cons_h:cons_f:cons_g2_4(a), gen_s:0':cons_h:cons_f:cons_g2_4(+(1, 0)))

Induction Step:
g(gen_s:0':cons_h:cons_f:cons_g2_4(a), gen_s:0':cons_h:cons_f:cons_g2_4(+(1, +(n4_4, 1)))) ->_R^Omega(1)
g(gen_s:0':cons_h:cons_f:cons_g2_4(a), gen_s:0':cons_h:cons_f:cons_g2_4(+(1, n4_4))) ->_IH
*3_4

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(34)
Complex Obligation (BEST)

----------------------------------------

(35)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0', g(X, Y))
g(0', Y) -> 0'
g(X, s(Y)) -> g(X, Y)
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encode_s(x_1) -> s(encArg(x_1))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_0 -> 0'

Types:
h :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
f :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
s :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
g :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
0' :: s:0':cons_h:cons_f:cons_g
encArg :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
cons_h :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
cons_f :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
cons_g :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
encode_h :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
encode_f :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
encode_s :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
encode_g :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
encode_0 :: s:0':cons_h:cons_f:cons_g
hole_s:0':cons_h:cons_f:cons_g1_4 :: s:0':cons_h:cons_f:cons_g
gen_s:0':cons_h:cons_f:cons_g2_4 :: Nat -> s:0':cons_h:cons_f:cons_g


Generator Equations:
gen_s:0':cons_h:cons_f:cons_g2_4(0) <=> 0'
gen_s:0':cons_h:cons_f:cons_g2_4(+(x, 1)) <=> s(gen_s:0':cons_h:cons_f:cons_g2_4(x))


The following defined symbols remain to be analysed:
g, h, f, encArg

They will be analysed ascendingly in the following order:
h = f
h < encArg
g < f
f < encArg
g < encArg

----------------------------------------

(36) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(37)
BOUNDS(n^1, INF)

----------------------------------------

(38)
Obligation:
Innermost TRS:
Rules:
h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0', g(X, Y))
g(0', Y) -> 0'
g(X, s(Y)) -> g(X, Y)
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encode_s(x_1) -> s(encArg(x_1))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_0 -> 0'

Types:
h :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
f :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
s :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
g :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
0' :: s:0':cons_h:cons_f:cons_g
encArg :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
cons_h :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
cons_f :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
cons_g :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
encode_h :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
encode_f :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
encode_s :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
encode_g :: s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g -> s:0':cons_h:cons_f:cons_g
encode_0 :: s:0':cons_h:cons_f:cons_g
hole_s:0':cons_h:cons_f:cons_g1_4 :: s:0':cons_h:cons_f:cons_g
gen_s:0':cons_h:cons_f:cons_g2_4 :: Nat -> s:0':cons_h:cons_f:cons_g


Lemmas:
g(gen_s:0':cons_h:cons_f:cons_g2_4(a), gen_s:0':cons_h:cons_f:cons_g2_4(+(1, n4_4))) -> *3_4, rt in Omega(n4_4)


Generator Equations:
gen_s:0':cons_h:cons_f:cons_g2_4(0) <=> 0'
gen_s:0':cons_h:cons_f:cons_g2_4(+(x, 1)) <=> s(gen_s:0':cons_h:cons_f:cons_g2_4(x))


The following defined symbols remain to be analysed:
f, h, encArg

They will be analysed ascendingly in the following order:
h = f
h < encArg
f < encArg

----------------------------------------

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_s:0':cons_h:cons_f:cons_g2_4(n688_4)) -> gen_s:0':cons_h:cons_f:cons_g2_4(n688_4), rt in Omega(0)

Induction Base:
encArg(gen_s:0':cons_h:cons_f:cons_g2_4(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_s:0':cons_h:cons_f:cons_g2_4(+(n688_4, 1))) ->_R^Omega(0)
s(encArg(gen_s:0':cons_h:cons_f:cons_g2_4(n688_4))) ->_IH
s(gen_s:0':cons_h:cons_f:cons_g2_4(c689_4))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(40)
BOUNDS(1, INF)