/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 145 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
    (7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (8) CdtProblem
    (9) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (12) CdtProblem
    (13) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CdtProblem
    (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 115 ms]
    (16) CdtProblem
    (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 73 ms]
    (18) CdtProblem
    (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 57 ms]
    (20) CdtProblem
    (21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 44 ms]
    (22) CdtProblem
    (23) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) BOUNDS(1, 1)
(25) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(26) CpxRelTRS
    (27) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) typed CpxTrs
    (29) OrderProof [LOWER BOUND(ID), 0 ms]
    (30) typed CpxTrs
    (31) RewriteLemmaProof [LOWER BOUND(ID), 850 ms]
    (32) BEST
        (33) proven lower bound
            (34) LowerBoundPropagationProof [FINISHED, 0 ms]
            (35) BOUNDS(n^1, INF)
        (36) typed CpxTrs
            (37) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
            (38) 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:

   g(0, f(x, x)) -> x
   g(x, s(y)) -> g(f(x, y), 0)
   g(s(x), y) -> g(f(x, y), 0)
   g(f(x, y), 0) -> f(g(x, 0), g(y, 0))

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(0) -> 0
   encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))


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

(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:

   g(0, f(x, x)) -> x
   g(x, s(y)) -> g(f(x, y), 0)
   g(s(x), y) -> g(f(x, y), 0)
   g(f(x, y), 0) -> f(g(x, 0), g(y, 0))

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

   encArg(0) -> 0
   encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))

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:

   g(0, f(x, x)) -> x
   g(x, s(y)) -> g(f(x, y), 0)
   g(s(x), y) -> g(f(x, y), 0)
   g(f(x, y), 0) -> f(g(x, 0), g(y, 0))

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

   encArg(0) -> 0
   encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encode_g(z0, z1) -> g(encArg(z0), encArg(z1))
   encode_0 -> 0
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_s(z0) -> s(encArg(z0))
   g(0, f(z0, z0)) -> z0
   g(z0, s(z1)) -> g(f(z0, z1), 0)
   g(s(z0), z1) -> g(f(z0, z1), 0)
   g(f(z0, z1), 0) -> f(g(z0, 0), g(z1, 0))
Tuples:
   ENCARG(0) -> c
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_G(z0, z1) -> c4(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_0 -> c5
   ENCODE_F(z0, z1) -> c6(ENCARG(z0), ENCARG(z1))
   ENCODE_S(z0) -> c7(ENCARG(z0))
   G(0, f(z0, z0)) -> c8
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
S tuples:
   G(0, f(z0, z0)) -> c8
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
K tuples:none
Defined Rule Symbols:   g_2, encArg_1, encode_g_2, encode_0, encode_f_2, encode_s_1

Defined Pair Symbols:   ENCARG_1, ENCODE_G_2, ENCODE_0, ENCODE_F_2, ENCODE_S_1, G_2

Compound Symbols:   c, c1_2, c2_1, c3_3, c4_3, c5, c6_2, c7_1, c8, c9_1, c10_1, c11_2


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_F(z0, z1) -> c6(ENCARG(z0), ENCARG(z1))
   ENCODE_S(z0) -> c7(ENCARG(z0))
Removed 2 trailing nodes:
   ENCARG(0) -> c
   ENCODE_0 -> c5

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encode_g(z0, z1) -> g(encArg(z0), encArg(z1))
   encode_0 -> 0
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_s(z0) -> s(encArg(z0))
   g(0, f(z0, z0)) -> z0
   g(z0, s(z1)) -> g(f(z0, z1), 0)
   g(s(z0), z1) -> g(f(z0, z1), 0)
   g(f(z0, z1), 0) -> f(g(z0, 0), g(z1, 0))
Tuples:
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_G(z0, z1) -> c4(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, f(z0, z0)) -> c8
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
S tuples:
   G(0, f(z0, z0)) -> c8
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
K tuples:none
Defined Rule Symbols:   g_2, encArg_1, encode_g_2, encode_0, encode_f_2, encode_s_1

Defined Pair Symbols:   ENCARG_1, ENCODE_G_2, G_2

Compound Symbols:   c1_2, c2_1, c3_3, c4_3, c8, c9_1, c10_1, c11_2


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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encode_g(z0, z1) -> g(encArg(z0), encArg(z1))
   encode_0 -> 0
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_s(z0) -> s(encArg(z0))
   g(0, f(z0, z0)) -> z0
   g(z0, s(z1)) -> g(f(z0, z1), 0)
   g(s(z0), z1) -> g(f(z0, z1), 0)
   g(f(z0, z1), 0) -> f(g(z0, 0), g(z1, 0))
Tuples:
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, f(z0, z0)) -> c8
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
   ENCODE_G(z0, z1) -> c(G(encArg(z0), encArg(z1)))
   ENCODE_G(z0, z1) -> c(ENCARG(z0))
   ENCODE_G(z0, z1) -> c(ENCARG(z1))
S tuples:
   G(0, f(z0, z0)) -> c8
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
K tuples:none
Defined Rule Symbols:   g_2, encArg_1, encode_g_2, encode_0, encode_f_2, encode_s_1

Defined Pair Symbols:   ENCARG_1, G_2, ENCODE_G_2

Compound Symbols:   c1_2, c2_1, c3_3, c8, c9_1, c10_1, c11_2, c_1


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_G(z0, z1) -> c(ENCARG(z0))
   ENCODE_G(z0, z1) -> c(ENCARG(z1))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encode_g(z0, z1) -> g(encArg(z0), encArg(z1))
   encode_0 -> 0
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_s(z0) -> s(encArg(z0))
   g(0, f(z0, z0)) -> z0
   g(z0, s(z1)) -> g(f(z0, z1), 0)
   g(s(z0), z1) -> g(f(z0, z1), 0)
   g(f(z0, z1), 0) -> f(g(z0, 0), g(z1, 0))
Tuples:
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, f(z0, z0)) -> c8
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
   ENCODE_G(z0, z1) -> c(G(encArg(z0), encArg(z1)))
S tuples:
   G(0, f(z0, z0)) -> c8
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
K tuples:none
Defined Rule Symbols:   g_2, encArg_1, encode_g_2, encode_0, encode_f_2, encode_s_1

Defined Pair Symbols:   ENCARG_1, G_2, ENCODE_G_2

Compound Symbols:   c1_2, c2_1, c3_3, c8, c9_1, c10_1, c11_2, c_1


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

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

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   g(0, f(z0, z0)) -> z0
   g(z0, s(z1)) -> g(f(z0, z1), 0)
   g(s(z0), z1) -> g(f(z0, z1), 0)
   g(f(z0, z1), 0) -> f(g(z0, 0), g(z1, 0))
Tuples:
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, f(z0, z0)) -> c8
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
   ENCODE_G(z0, z1) -> c(G(encArg(z0), encArg(z1)))
S tuples:
   G(0, f(z0, z0)) -> c8
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
K tuples:none
Defined Rule Symbols:   encArg_1, g_2

Defined Pair Symbols:   ENCARG_1, G_2, ENCODE_G_2

Compound Symbols:   c1_2, c2_1, c3_3, c8, c9_1, c10_1, c11_2, c_1


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

(15) 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)) -> c9(G(f(z0, z1), 0))
We considered the (Usable) Rules:
   encArg(s(z0)) -> s(encArg(z0))
   g(s(z0), z1) -> g(f(z0, z1), 0)
   g(f(z0, z1), 0) -> f(g(z0, 0), g(z1, 0))
   g(z0, s(z1)) -> g(f(z0, z1), 0)
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encArg(0) -> 0
   g(0, f(z0, z0)) -> z0
And the Tuples:
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, f(z0, z0)) -> c8
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
   ENCODE_G(z0, z1) -> c(G(encArg(z0), encArg(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_G(x_1, x_2)) = [2] + x_1 + [2]x_2 + [2]x_2^2 + [2]x_1*x_2 + x_1^2
   POL(G(x_1, x_2)) = x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c8) = 0
   POL(c9(x_1)) = x_1
   POL(cons_g(x_1, x_2)) = [1] + x_1 + x_2
   POL(encArg(x_1)) = x_1
   POL(f(x_1, x_2)) = x_1 + x_2
   POL(g(x_1, x_2)) = x_2
   POL(s(x_1)) = [2] + x_1

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   g(0, f(z0, z0)) -> z0
   g(z0, s(z1)) -> g(f(z0, z1), 0)
   g(s(z0), z1) -> g(f(z0, z1), 0)
   g(f(z0, z1), 0) -> f(g(z0, 0), g(z1, 0))
Tuples:
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, f(z0, z0)) -> c8
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
   ENCODE_G(z0, z1) -> c(G(encArg(z0), encArg(z1)))
S tuples:
   G(0, f(z0, z0)) -> c8
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
K tuples:
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
Defined Rule Symbols:   encArg_1, g_2

Defined Pair Symbols:   ENCARG_1, G_2, ENCODE_G_2

Compound Symbols:   c1_2, c2_1, c3_3, c8, c9_1, c10_1, c11_2, c_1


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

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(0, f(z0, z0)) -> c8
We considered the (Usable) Rules:
   encArg(s(z0)) -> s(encArg(z0))
   g(s(z0), z1) -> g(f(z0, z1), 0)
   g(f(z0, z1), 0) -> f(g(z0, 0), g(z1, 0))
   g(z0, s(z1)) -> g(f(z0, z1), 0)
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encArg(0) -> 0
   g(0, f(z0, z0)) -> z0
And the Tuples:
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, f(z0, z0)) -> c8
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
   ENCODE_G(z0, z1) -> c(G(encArg(z0), encArg(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(ENCARG(x_1)) = [2]x_1^2
   POL(ENCODE_G(x_1, x_2)) = [2] + x_1 + x_2 + x_2^2 + [2]x_1*x_2 + [2]x_1^2
   POL(G(x_1, x_2)) = x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c8) = 0
   POL(c9(x_1)) = x_1
   POL(cons_g(x_1, x_2)) = [2] + x_1 + x_2
   POL(encArg(x_1)) = x_1
   POL(f(x_1, x_2)) = [2] + x_1 + x_2
   POL(g(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = [2] + x_1

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   g(0, f(z0, z0)) -> z0
   g(z0, s(z1)) -> g(f(z0, z1), 0)
   g(s(z0), z1) -> g(f(z0, z1), 0)
   g(f(z0, z1), 0) -> f(g(z0, 0), g(z1, 0))
Tuples:
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, f(z0, z0)) -> c8
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
   ENCODE_G(z0, z1) -> c(G(encArg(z0), encArg(z1)))
S tuples:
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
K tuples:
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(0, f(z0, z0)) -> c8
Defined Rule Symbols:   encArg_1, g_2

Defined Pair Symbols:   ENCARG_1, G_2, ENCODE_G_2

Compound Symbols:   c1_2, c2_1, c3_3, c8, c9_1, c10_1, c11_2, c_1


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

(19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
We considered the (Usable) Rules:
   encArg(s(z0)) -> s(encArg(z0))
   g(s(z0), z1) -> g(f(z0, z1), 0)
   g(f(z0, z1), 0) -> f(g(z0, 0), g(z1, 0))
   g(z0, s(z1)) -> g(f(z0, z1), 0)
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encArg(0) -> 0
   g(0, f(z0, z0)) -> z0
And the Tuples:
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, f(z0, z0)) -> c8
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
   ENCODE_G(z0, z1) -> c(G(encArg(z0), encArg(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(ENCARG(x_1)) = [2]x_1 + x_1^2
   POL(ENCODE_G(x_1, x_2)) = [2] + x_1 + [2]x_2 + x_2^2 + [2]x_1*x_2 + [2]x_1^2
   POL(G(x_1, x_2)) = x_1 + [2]x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c8) = 0
   POL(c9(x_1)) = x_1
   POL(cons_g(x_1, x_2)) = [2] + x_1 + x_2
   POL(encArg(x_1)) = x_1
   POL(f(x_1, x_2)) = x_1 + x_2
   POL(g(x_1, x_2)) = x_2
   POL(s(x_1)) = [2] + x_1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   g(0, f(z0, z0)) -> z0
   g(z0, s(z1)) -> g(f(z0, z1), 0)
   g(s(z0), z1) -> g(f(z0, z1), 0)
   g(f(z0, z1), 0) -> f(g(z0, 0), g(z1, 0))
Tuples:
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, f(z0, z0)) -> c8
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
   ENCODE_G(z0, z1) -> c(G(encArg(z0), encArg(z1)))
S tuples:
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
K tuples:
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(0, f(z0, z0)) -> c8
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
Defined Rule Symbols:   encArg_1, g_2

Defined Pair Symbols:   ENCARG_1, G_2, ENCODE_G_2

Compound Symbols:   c1_2, c2_1, c3_3, c8, c9_1, c10_1, c11_2, c_1


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

(21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
We considered the (Usable) Rules:
   encArg(s(z0)) -> s(encArg(z0))
   g(s(z0), z1) -> g(f(z0, z1), 0)
   g(f(z0, z1), 0) -> f(g(z0, 0), g(z1, 0))
   g(z0, s(z1)) -> g(f(z0, z1), 0)
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encArg(0) -> 0
   g(0, f(z0, z0)) -> z0
And the Tuples:
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, f(z0, z0)) -> c8
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
   ENCODE_G(z0, z1) -> c(G(encArg(z0), encArg(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(ENCARG(x_1)) = [2]x_1^2
   POL(ENCODE_G(x_1, x_2)) = [2] + [2]x_1 + [2]x_2 + [2]x_2^2 + x_1*x_2 + [2]x_1^2
   POL(G(x_1, x_2)) = x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c8) = 0
   POL(c9(x_1)) = x_1
   POL(cons_g(x_1, x_2)) = [2] + x_1 + x_2
   POL(encArg(x_1)) = [2]x_1
   POL(f(x_1, x_2)) = [1] + x_1 + x_2
   POL(g(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = [2] + x_1

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   g(0, f(z0, z0)) -> z0
   g(z0, s(z1)) -> g(f(z0, z1), 0)
   g(s(z0), z1) -> g(f(z0, z1), 0)
   g(f(z0, z1), 0) -> f(g(z0, 0), g(z1, 0))
Tuples:
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   G(0, f(z0, z0)) -> c8
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
   ENCODE_G(z0, z1) -> c(G(encArg(z0), encArg(z1)))
S tuples:none
K tuples:
   G(z0, s(z1)) -> c9(G(f(z0, z1), 0))
   G(0, f(z0, z0)) -> c8
   G(s(z0), z1) -> c10(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c11(G(z0, 0), G(z1, 0))
Defined Rule Symbols:   encArg_1, g_2

Defined Pair Symbols:   ENCARG_1, G_2, ENCODE_G_2

Compound Symbols:   c1_2, c2_1, c3_3, c8, c9_1, c10_1, c11_2, c_1


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

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

(24)
BOUNDS(1, 1)

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

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

(26)
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:

   g(0', f(x, x)) -> x
   g(x, s(y)) -> g(f(x, y), 0')
   g(s(x), y) -> g(f(x, y), 0')
   g(f(x, y), 0') -> f(g(x, 0'), g(y, 0'))

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

   encArg(0') -> 0'
   encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0'
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))

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

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

(28)
Obligation:
Innermost TRS:
Rules:
g(0', f(x, x)) -> x
g(x, s(y)) -> g(f(x, y), 0')
g(s(x), y) -> g(f(x, y), 0')
g(f(x, y), 0') -> f(g(x, 0'), g(y, 0'))
encArg(0') -> 0'
encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))

Types:
g :: 0':f:s:cons_g -> 0':f:s:cons_g -> 0':f:s:cons_g
0' :: 0':f:s:cons_g
f :: 0':f:s:cons_g -> 0':f:s:cons_g -> 0':f:s:cons_g
s :: 0':f:s:cons_g -> 0':f:s:cons_g
encArg :: 0':f:s:cons_g -> 0':f:s:cons_g
cons_g :: 0':f:s:cons_g -> 0':f:s:cons_g -> 0':f:s:cons_g
encode_g :: 0':f:s:cons_g -> 0':f:s:cons_g -> 0':f:s:cons_g
encode_0 :: 0':f:s:cons_g
encode_f :: 0':f:s:cons_g -> 0':f:s:cons_g -> 0':f:s:cons_g
encode_s :: 0':f:s:cons_g -> 0':f:s:cons_g
hole_0':f:s:cons_g1_3 :: 0':f:s:cons_g
gen_0':f:s:cons_g2_3 :: Nat -> 0':f:s:cons_g

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

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

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

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

(30)
Obligation:
Innermost TRS:
Rules:
g(0', f(x, x)) -> x
g(x, s(y)) -> g(f(x, y), 0')
g(s(x), y) -> g(f(x, y), 0')
g(f(x, y), 0') -> f(g(x, 0'), g(y, 0'))
encArg(0') -> 0'
encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))

Types:
g :: 0':f:s:cons_g -> 0':f:s:cons_g -> 0':f:s:cons_g
0' :: 0':f:s:cons_g
f :: 0':f:s:cons_g -> 0':f:s:cons_g -> 0':f:s:cons_g
s :: 0':f:s:cons_g -> 0':f:s:cons_g
encArg :: 0':f:s:cons_g -> 0':f:s:cons_g
cons_g :: 0':f:s:cons_g -> 0':f:s:cons_g -> 0':f:s:cons_g
encode_g :: 0':f:s:cons_g -> 0':f:s:cons_g -> 0':f:s:cons_g
encode_0 :: 0':f:s:cons_g
encode_f :: 0':f:s:cons_g -> 0':f:s:cons_g -> 0':f:s:cons_g
encode_s :: 0':f:s:cons_g -> 0':f:s:cons_g
hole_0':f:s:cons_g1_3 :: 0':f:s:cons_g
gen_0':f:s:cons_g2_3 :: Nat -> 0':f:s:cons_g


Generator Equations:
gen_0':f:s:cons_g2_3(0) <=> 0'
gen_0':f:s:cons_g2_3(+(x, 1)) <=> f(0', gen_0':f:s:cons_g2_3(x))


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

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

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_0':f:s:cons_g2_3(+(1, n4_3)), gen_0':f:s:cons_g2_3(0)) -> *3_3, rt in Omega(n4_3)

Induction Base:
g(gen_0':f:s:cons_g2_3(+(1, 0)), gen_0':f:s:cons_g2_3(0))

Induction Step:
g(gen_0':f:s:cons_g2_3(+(1, +(n4_3, 1))), gen_0':f:s:cons_g2_3(0)) ->_R^Omega(1)
f(g(0', 0'), g(gen_0':f:s:cons_g2_3(+(1, n4_3)), 0')) ->_IH
f(g(0', 0'), *3_3)

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

(32)
Complex Obligation (BEST)

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

(33)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
g(0', f(x, x)) -> x
g(x, s(y)) -> g(f(x, y), 0')
g(s(x), y) -> g(f(x, y), 0')
g(f(x, y), 0') -> f(g(x, 0'), g(y, 0'))
encArg(0') -> 0'
encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))

Types:
g :: 0':f:s:cons_g -> 0':f:s:cons_g -> 0':f:s:cons_g
0' :: 0':f:s:cons_g
f :: 0':f:s:cons_g -> 0':f:s:cons_g -> 0':f:s:cons_g
s :: 0':f:s:cons_g -> 0':f:s:cons_g
encArg :: 0':f:s:cons_g -> 0':f:s:cons_g
cons_g :: 0':f:s:cons_g -> 0':f:s:cons_g -> 0':f:s:cons_g
encode_g :: 0':f:s:cons_g -> 0':f:s:cons_g -> 0':f:s:cons_g
encode_0 :: 0':f:s:cons_g
encode_f :: 0':f:s:cons_g -> 0':f:s:cons_g -> 0':f:s:cons_g
encode_s :: 0':f:s:cons_g -> 0':f:s:cons_g
hole_0':f:s:cons_g1_3 :: 0':f:s:cons_g
gen_0':f:s:cons_g2_3 :: Nat -> 0':f:s:cons_g


Generator Equations:
gen_0':f:s:cons_g2_3(0) <=> 0'
gen_0':f:s:cons_g2_3(+(x, 1)) <=> f(0', gen_0':f:s:cons_g2_3(x))


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

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

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

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

(35)
BOUNDS(n^1, INF)

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

(36)
Obligation:
Innermost TRS:
Rules:
g(0', f(x, x)) -> x
g(x, s(y)) -> g(f(x, y), 0')
g(s(x), y) -> g(f(x, y), 0')
g(f(x, y), 0') -> f(g(x, 0'), g(y, 0'))
encArg(0') -> 0'
encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))

Types:
g :: 0':f:s:cons_g -> 0':f:s:cons_g -> 0':f:s:cons_g
0' :: 0':f:s:cons_g
f :: 0':f:s:cons_g -> 0':f:s:cons_g -> 0':f:s:cons_g
s :: 0':f:s:cons_g -> 0':f:s:cons_g
encArg :: 0':f:s:cons_g -> 0':f:s:cons_g
cons_g :: 0':f:s:cons_g -> 0':f:s:cons_g -> 0':f:s:cons_g
encode_g :: 0':f:s:cons_g -> 0':f:s:cons_g -> 0':f:s:cons_g
encode_0 :: 0':f:s:cons_g
encode_f :: 0':f:s:cons_g -> 0':f:s:cons_g -> 0':f:s:cons_g
encode_s :: 0':f:s:cons_g -> 0':f:s:cons_g
hole_0':f:s:cons_g1_3 :: 0':f:s:cons_g
gen_0':f:s:cons_g2_3 :: Nat -> 0':f:s:cons_g


Lemmas:
g(gen_0':f:s:cons_g2_3(+(1, n4_3)), gen_0':f:s:cons_g2_3(0)) -> *3_3, rt in Omega(n4_3)


Generator Equations:
gen_0':f:s:cons_g2_3(0) <=> 0'
gen_0':f:s:cons_g2_3(+(x, 1)) <=> f(0', gen_0':f:s:cons_g2_3(x))


The following defined symbols remain to be analysed:
encArg
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(37) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':f:s:cons_g2_3(n38106_3)) -> gen_0':f:s:cons_g2_3(n38106_3), rt in Omega(0)

Induction Base:
encArg(gen_0':f:s:cons_g2_3(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':f:s:cons_g2_3(+(n38106_3, 1))) ->_R^Omega(0)
f(encArg(0'), encArg(gen_0':f:s:cons_g2_3(n38106_3))) ->_R^Omega(0)
f(0', encArg(gen_0':f:s:cons_g2_3(n38106_3))) ->_IH
f(0', gen_0':f:s:cons_g2_3(c38107_3))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(38)
BOUNDS(1, INF)