/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 42 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^1)), 88 ms]
    (16) CdtProblem
    (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 138 ms]
    (18) CdtProblem
    (19) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CdtProblem
    (21) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CdtProblem
    (23) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CdtProblem
    (25) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CdtProblem
    (27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CdtProblem
    (29) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) CdtProblem
    (31) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 144 ms]
    (32) CdtProblem
    (33) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) BOUNDS(1, 1)
(35) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(36) CpxRelTRS
    (37) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) typed CpxTrs
    (39) OrderProof [LOWER BOUND(ID), 0 ms]
    (40) typed CpxTrs
    (41) RewriteLemmaProof [LOWER BOUND(ID), 409 ms]
    (42) BEST
        (43) proven lower bound
            (44) LowerBoundPropagationProof [FINISHED, 0 ms]
            (45) BOUNDS(n^1, INF)
        (46) typed CpxTrs
            (47) RewriteLemmaProof [LOWER BOUND(ID), 32 ms]
            (48) 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:

   f(f(x)) -> f(x)
   f(s(x)) -> f(x)
   g(s(0)) -> g(f(s(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(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   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:

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

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   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:

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

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   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_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_s(z0) -> s(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_0 -> 0
   f(f(z0)) -> f(z0)
   f(s(z0)) -> f(z0)
   g(s(0)) -> g(f(s(0)))
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(0) -> c1
   ENCARG(cons_f(z0)) -> c2(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c4(F(encArg(z0)), ENCARG(z0))
   ENCODE_S(z0) -> c5(ENCARG(z0))
   ENCODE_G(z0) -> c6(G(encArg(z0)), ENCARG(z0))
   ENCODE_0 -> c7
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
S tuples:
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
K tuples:none
Defined Rule Symbols:   f_1, g_1, encArg_1, encode_f_1, encode_s_1, encode_g_1, encode_0

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, ENCODE_S_1, ENCODE_G_1, ENCODE_0, F_1, G_1

Compound Symbols:   c_1, c1, c2_2, c3_2, c4_2, c5_1, c6_2, c7, c8_1, c9_1, c10_2


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   ENCODE_S(z0) -> c5(ENCARG(z0))
Removed 2 trailing nodes:
   ENCARG(0) -> c1
   ENCODE_0 -> c7

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_s(z0) -> s(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_0 -> 0
   f(f(z0)) -> f(z0)
   f(s(z0)) -> f(z0)
   g(s(0)) -> g(f(s(0)))
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c2(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c4(F(encArg(z0)), ENCARG(z0))
   ENCODE_G(z0) -> c6(G(encArg(z0)), ENCARG(z0))
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
S tuples:
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
K tuples:none
Defined Rule Symbols:   f_1, g_1, encArg_1, encode_f_1, encode_s_1, encode_g_1, encode_0

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, ENCODE_G_1, F_1, G_1

Compound Symbols:   c_1, c2_2, c3_2, c4_2, c6_2, c8_1, c9_1, c10_2


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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_s(z0) -> s(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_0 -> 0
   f(f(z0)) -> f(z0)
   f(s(z0)) -> f(z0)
   g(s(0)) -> g(f(s(0)))
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c2(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
   ENCODE_F(z0) -> c1(F(encArg(z0)))
   ENCODE_F(z0) -> c1(ENCARG(z0))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   ENCODE_G(z0) -> c1(ENCARG(z0))
S tuples:
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
K tuples:none
Defined Rule Symbols:   f_1, g_1, encArg_1, encode_f_1, encode_s_1, encode_g_1, encode_0

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_F_1, ENCODE_G_1

Compound Symbols:   c_1, c2_2, c3_2, c8_1, c9_1, c10_2, c1_1


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_F(z0) -> c1(ENCARG(z0))
   ENCODE_G(z0) -> c1(ENCARG(z0))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_s(z0) -> s(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_0 -> 0
   f(f(z0)) -> f(z0)
   f(s(z0)) -> f(z0)
   g(s(0)) -> g(f(s(0)))
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c2(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
   ENCODE_F(z0) -> c1(F(encArg(z0)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
S tuples:
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
K tuples:none
Defined Rule Symbols:   f_1, g_1, encArg_1, encode_f_1, encode_s_1, encode_g_1, encode_0

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_F_1, ENCODE_G_1

Compound Symbols:   c_1, c2_2, c3_2, c8_1, c9_1, c10_2, c1_1


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

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

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(f(z0)) -> f(z0)
   f(s(z0)) -> f(z0)
   g(s(0)) -> g(f(s(0)))
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c2(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
   ENCODE_F(z0) -> c1(F(encArg(z0)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
S tuples:
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
K tuples:none
Defined Rule Symbols:   encArg_1, f_1, g_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_F_1, ENCODE_G_1

Compound Symbols:   c_1, c2_2, c3_2, c8_1, c9_1, c10_2, c1_1


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

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
We considered the (Usable) Rules:
   encArg(s(z0)) -> s(encArg(z0))
   g(s(0)) -> g(f(s(0)))
   f(f(z0)) -> f(z0)
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(s(z0)) -> f(z0)
   encArg(0) -> 0
And the Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c2(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
   ENCODE_F(z0) -> c1(F(encArg(z0)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
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)) = 0
   POL(ENCODE_G(x_1)) = [1]
   POL(F(x_1)) = 0
   POL(G(x_1)) = x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cons_f(x_1)) = [1] + x_1
   POL(cons_g(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1]
   POL(f(x_1)) = 0
   POL(g(x_1)) = [1]
   POL(s(x_1)) = x_1

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(f(z0)) -> f(z0)
   f(s(z0)) -> f(z0)
   g(s(0)) -> g(f(s(0)))
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c2(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
   ENCODE_F(z0) -> c1(F(encArg(z0)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
S tuples:
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
K tuples:
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
Defined Rule Symbols:   encArg_1, f_1, g_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_F_1, ENCODE_G_1

Compound Symbols:   c_1, c2_2, c3_2, c8_1, c9_1, c10_2, c1_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.
   F(f(z0)) -> c8(F(z0))
We considered the (Usable) Rules:
   encArg(s(z0)) -> s(encArg(z0))
   g(s(0)) -> g(f(s(0)))
   f(f(z0)) -> f(z0)
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(s(z0)) -> f(z0)
   encArg(0) -> 0
And the Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c2(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
   ENCODE_F(z0) -> c1(F(encArg(z0)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
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)) = [2] + [2]x_1 + [2]x_1^2
   POL(ENCODE_G(x_1)) = [1] + [2]x_1^2
   POL(F(x_1)) = [2]x_1
   POL(G(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cons_f(x_1)) = [1] + x_1
   POL(cons_g(x_1)) = x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1)) = [1] + x_1
   POL(g(x_1)) = 0
   POL(s(x_1)) = x_1

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(f(z0)) -> f(z0)
   f(s(z0)) -> f(z0)
   g(s(0)) -> g(f(s(0)))
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c2(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
   ENCODE_F(z0) -> c1(F(encArg(z0)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
S tuples:
   F(s(z0)) -> c9(F(z0))
K tuples:
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
   F(f(z0)) -> c8(F(z0))
Defined Rule Symbols:   encArg_1, f_1, g_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_F_1, ENCODE_G_1

Compound Symbols:   c_1, c2_2, c3_2, c8_1, c9_1, c10_2, c1_1


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

(19) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_f(z0)) -> c2(F(encArg(z0)), ENCARG(z0)) by 
   ENCARG(cons_f(s(z0))) -> c2(F(s(encArg(z0))), ENCARG(s(z0)))
   ENCARG(cons_f(0)) -> c2(F(0), ENCARG(0))
   ENCARG(cons_f(cons_f(z0))) -> c2(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_g(z0))) -> c2(F(g(encArg(z0))), ENCARG(cons_g(z0)))

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(f(z0)) -> f(z0)
   f(s(z0)) -> f(z0)
   g(s(0)) -> g(f(s(0)))
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
   ENCODE_F(z0) -> c1(F(encArg(z0)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   ENCARG(cons_f(s(z0))) -> c2(F(s(encArg(z0))), ENCARG(s(z0)))
   ENCARG(cons_f(0)) -> c2(F(0), ENCARG(0))
   ENCARG(cons_f(cons_f(z0))) -> c2(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_g(z0))) -> c2(F(g(encArg(z0))), ENCARG(cons_g(z0)))
S tuples:
   F(s(z0)) -> c9(F(z0))
K tuples:
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
   F(f(z0)) -> c8(F(z0))
Defined Rule Symbols:   encArg_1, f_1, g_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_F_1, ENCODE_G_1

Compound Symbols:   c_1, c3_2, c8_1, c9_1, c10_2, c1_1, c2_2


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

(21) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   ENCARG(cons_f(0)) -> c2(F(0), ENCARG(0))

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(f(z0)) -> f(z0)
   f(s(z0)) -> f(z0)
   g(s(0)) -> g(f(s(0)))
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
   ENCODE_F(z0) -> c1(F(encArg(z0)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   ENCARG(cons_f(s(z0))) -> c2(F(s(encArg(z0))), ENCARG(s(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c2(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_g(z0))) -> c2(F(g(encArg(z0))), ENCARG(cons_g(z0)))
S tuples:
   F(s(z0)) -> c9(F(z0))
K tuples:
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
   F(f(z0)) -> c8(F(z0))
Defined Rule Symbols:   encArg_1, f_1, g_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_F_1, ENCODE_G_1

Compound Symbols:   c_1, c3_2, c8_1, c9_1, c10_2, c1_1, c2_2


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

(23) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0)) by 
   ENCARG(cons_g(s(z0))) -> c3(G(s(encArg(z0))), ENCARG(s(z0)))
   ENCARG(cons_g(0)) -> c3(G(0), ENCARG(0))
   ENCARG(cons_g(cons_f(z0))) -> c3(G(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_g(cons_g(z0))) -> c3(G(g(encArg(z0))), ENCARG(cons_g(z0)))

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(f(z0)) -> f(z0)
   f(s(z0)) -> f(z0)
   g(s(0)) -> g(f(s(0)))
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
   ENCODE_F(z0) -> c1(F(encArg(z0)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   ENCARG(cons_f(s(z0))) -> c2(F(s(encArg(z0))), ENCARG(s(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c2(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_g(z0))) -> c2(F(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(s(z0))) -> c3(G(s(encArg(z0))), ENCARG(s(z0)))
   ENCARG(cons_g(0)) -> c3(G(0), ENCARG(0))
   ENCARG(cons_g(cons_f(z0))) -> c3(G(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_g(cons_g(z0))) -> c3(G(g(encArg(z0))), ENCARG(cons_g(z0)))
S tuples:
   F(s(z0)) -> c9(F(z0))
K tuples:
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
   F(f(z0)) -> c8(F(z0))
Defined Rule Symbols:   encArg_1, f_1, g_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_F_1, ENCODE_G_1

Compound Symbols:   c_1, c8_1, c9_1, c10_2, c1_1, c2_2, c3_2


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

(25) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   ENCARG(cons_g(0)) -> c3(G(0), ENCARG(0))

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(f(z0)) -> f(z0)
   f(s(z0)) -> f(z0)
   g(s(0)) -> g(f(s(0)))
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
   ENCODE_F(z0) -> c1(F(encArg(z0)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   ENCARG(cons_f(s(z0))) -> c2(F(s(encArg(z0))), ENCARG(s(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c2(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_g(z0))) -> c2(F(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(s(z0))) -> c3(G(s(encArg(z0))), ENCARG(s(z0)))
   ENCARG(cons_g(cons_f(z0))) -> c3(G(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_g(cons_g(z0))) -> c3(G(g(encArg(z0))), ENCARG(cons_g(z0)))
S tuples:
   F(s(z0)) -> c9(F(z0))
K tuples:
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
   F(f(z0)) -> c8(F(z0))
Defined Rule Symbols:   encArg_1, f_1, g_1

Defined Pair Symbols:   ENCARG_1, F_1, G_1, ENCODE_F_1, ENCODE_G_1

Compound Symbols:   c_1, c8_1, c9_1, c10_2, c1_1, c2_2, c3_2


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

(27) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(0)) -> c10(G(f(s(0))), F(s(0))) by 
   G(s(0)) -> c10(G(f(0)), F(s(0)))

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

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(f(z0)) -> f(z0)
   f(s(z0)) -> f(z0)
   g(s(0)) -> g(f(s(0)))
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   ENCODE_F(z0) -> c1(F(encArg(z0)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   ENCARG(cons_f(s(z0))) -> c2(F(s(encArg(z0))), ENCARG(s(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c2(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_g(z0))) -> c2(F(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(s(z0))) -> c3(G(s(encArg(z0))), ENCARG(s(z0)))
   ENCARG(cons_g(cons_f(z0))) -> c3(G(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_g(cons_g(z0))) -> c3(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   G(s(0)) -> c10(G(f(0)), F(s(0)))
S tuples:
   F(s(z0)) -> c9(F(z0))
K tuples:
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
   F(f(z0)) -> c8(F(z0))
Defined Rule Symbols:   encArg_1, f_1, g_1

Defined Pair Symbols:   ENCARG_1, F_1, ENCODE_F_1, ENCODE_G_1, G_1

Compound Symbols:   c_1, c8_1, c9_1, c1_1, c2_2, c3_2, c10_2


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

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

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(f(z0)) -> f(z0)
   f(s(z0)) -> f(z0)
   g(s(0)) -> g(f(s(0)))
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   ENCODE_F(z0) -> c1(F(encArg(z0)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   ENCARG(cons_f(s(z0))) -> c2(F(s(encArg(z0))), ENCARG(s(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c2(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_g(z0))) -> c2(F(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(s(z0))) -> c3(G(s(encArg(z0))), ENCARG(s(z0)))
   ENCARG(cons_g(cons_f(z0))) -> c3(G(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_g(cons_g(z0))) -> c3(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   G(s(0)) -> c10(F(s(0)))
S tuples:
   F(s(z0)) -> c9(F(z0))
K tuples:
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
   F(f(z0)) -> c8(F(z0))
Defined Rule Symbols:   encArg_1, f_1, g_1

Defined Pair Symbols:   ENCARG_1, F_1, ENCODE_F_1, ENCODE_G_1, G_1

Compound Symbols:   c_1, c8_1, c9_1, c1_1, c2_2, c3_2, c10_1


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

(31) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(z0)) -> c9(F(z0))
We considered the (Usable) Rules:
   encArg(s(z0)) -> s(encArg(z0))
   g(s(0)) -> g(f(s(0)))
   f(f(z0)) -> f(z0)
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(s(z0)) -> f(z0)
   encArg(0) -> 0
And the Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   ENCODE_F(z0) -> c1(F(encArg(z0)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   ENCARG(cons_f(s(z0))) -> c2(F(s(encArg(z0))), ENCARG(s(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c2(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_g(z0))) -> c2(F(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(s(z0))) -> c3(G(s(encArg(z0))), ENCARG(s(z0)))
   ENCARG(cons_g(cons_f(z0))) -> c3(G(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_g(cons_g(z0))) -> c3(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   G(s(0)) -> c10(F(s(0)))
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)) = [2] + [2]x_1 + [2]x_1^2
   POL(ENCODE_G(x_1)) = [2] + [2]x_1 + x_1^2
   POL(F(x_1)) = x_1
   POL(G(x_1)) = [2]
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cons_f(x_1)) = [2] + x_1
   POL(cons_g(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [2]x_1
   POL(f(x_1)) = x_1
   POL(g(x_1)) = [2]
   POL(s(x_1)) = [1] + x_1

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(f(z0)) -> f(z0)
   f(s(z0)) -> f(z0)
   g(s(0)) -> g(f(s(0)))
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
   ENCODE_F(z0) -> c1(F(encArg(z0)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   ENCARG(cons_f(s(z0))) -> c2(F(s(encArg(z0))), ENCARG(s(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c2(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_g(z0))) -> c2(F(g(encArg(z0))), ENCARG(cons_g(z0)))
   ENCARG(cons_g(s(z0))) -> c3(G(s(encArg(z0))), ENCARG(s(z0)))
   ENCARG(cons_g(cons_f(z0))) -> c3(G(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_g(cons_g(z0))) -> c3(G(g(encArg(z0))), ENCARG(cons_g(z0)))
   G(s(0)) -> c10(F(s(0)))
S tuples:none
K tuples:
   G(s(0)) -> c10(G(f(s(0))), F(s(0)))
   F(f(z0)) -> c8(F(z0))
   F(s(z0)) -> c9(F(z0))
Defined Rule Symbols:   encArg_1, f_1, g_1

Defined Pair Symbols:   ENCARG_1, F_1, ENCODE_F_1, ENCODE_G_1, G_1

Compound Symbols:   c_1, c8_1, c9_1, c1_1, c2_2, c3_2, c10_1


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

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

(34)
BOUNDS(1, 1)

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

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

(36)
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:

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

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

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

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

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

(38)
Obligation:
Innermost TRS:
Rules:
f(f(x)) -> f(x)
f(s(x)) -> f(x)
g(s(0')) -> g(f(s(0')))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_g(x_1)) -> g(encArg(x_1))
encode_f(x_1) -> f(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_g(x_1) -> g(encArg(x_1))
encode_0 -> 0'

Types:
f :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
s :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
g :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
0' :: s:0':cons_f:cons_g
encArg :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
cons_f :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
cons_g :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
encode_f :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
encode_s :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
encode_g :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
encode_0 :: s:0':cons_f:cons_g
hole_s:0':cons_f:cons_g1_2 :: s:0':cons_f:cons_g
gen_s:0':cons_f:cons_g2_2 :: Nat -> s:0':cons_f:cons_g

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

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

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

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

(40)
Obligation:
Innermost TRS:
Rules:
f(f(x)) -> f(x)
f(s(x)) -> f(x)
g(s(0')) -> g(f(s(0')))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_g(x_1)) -> g(encArg(x_1))
encode_f(x_1) -> f(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_g(x_1) -> g(encArg(x_1))
encode_0 -> 0'

Types:
f :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
s :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
g :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
0' :: s:0':cons_f:cons_g
encArg :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
cons_f :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
cons_g :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
encode_f :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
encode_s :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
encode_g :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
encode_0 :: s:0':cons_f:cons_g
hole_s:0':cons_f:cons_g1_2 :: s:0':cons_f:cons_g
gen_s:0':cons_f:cons_g2_2 :: Nat -> s:0':cons_f:cons_g


Generator Equations:
gen_s:0':cons_f:cons_g2_2(0) <=> 0'
gen_s:0':cons_f:cons_g2_2(+(x, 1)) <=> s(gen_s:0':cons_f:cons_g2_2(x))


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

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

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

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_s:0':cons_f:cons_g2_2(+(1, n4_2))) -> *3_2, rt in Omega(n4_2)

Induction Base:
f(gen_s:0':cons_f:cons_g2_2(+(1, 0)))

Induction Step:
f(gen_s:0':cons_f:cons_g2_2(+(1, +(n4_2, 1)))) ->_R^Omega(1)
f(gen_s:0':cons_f:cons_g2_2(+(1, n4_2))) ->_IH
*3_2

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(42)
Complex Obligation (BEST)

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(43)
Obligation:
Proved the lower bound n^1 for the following obligation:

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

Types:
f :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
s :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
g :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
0' :: s:0':cons_f:cons_g
encArg :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
cons_f :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
cons_g :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
encode_f :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
encode_s :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
encode_g :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
encode_0 :: s:0':cons_f:cons_g
hole_s:0':cons_f:cons_g1_2 :: s:0':cons_f:cons_g
gen_s:0':cons_f:cons_g2_2 :: Nat -> s:0':cons_f:cons_g


Generator Equations:
gen_s:0':cons_f:cons_g2_2(0) <=> 0'
gen_s:0':cons_f:cons_g2_2(+(x, 1)) <=> s(gen_s:0':cons_f:cons_g2_2(x))


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

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

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(44) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(45)
BOUNDS(n^1, INF)

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(46)
Obligation:
Innermost TRS:
Rules:
f(f(x)) -> f(x)
f(s(x)) -> f(x)
g(s(0')) -> g(f(s(0')))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_g(x_1)) -> g(encArg(x_1))
encode_f(x_1) -> f(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_g(x_1) -> g(encArg(x_1))
encode_0 -> 0'

Types:
f :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
s :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
g :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
0' :: s:0':cons_f:cons_g
encArg :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
cons_f :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
cons_g :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
encode_f :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
encode_s :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
encode_g :: s:0':cons_f:cons_g -> s:0':cons_f:cons_g
encode_0 :: s:0':cons_f:cons_g
hole_s:0':cons_f:cons_g1_2 :: s:0':cons_f:cons_g
gen_s:0':cons_f:cons_g2_2 :: Nat -> s:0':cons_f:cons_g


Lemmas:
f(gen_s:0':cons_f:cons_g2_2(+(1, n4_2))) -> *3_2, rt in Omega(n4_2)


Generator Equations:
gen_s:0':cons_f:cons_g2_2(0) <=> 0'
gen_s:0':cons_f:cons_g2_2(+(x, 1)) <=> s(gen_s:0':cons_f:cons_g2_2(x))


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

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

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(47) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_s:0':cons_f:cons_g2_2(n361_2)) -> gen_s:0':cons_f:cons_g2_2(n361_2), rt in Omega(0)

Induction Base:
encArg(gen_s:0':cons_f:cons_g2_2(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_s:0':cons_f:cons_g2_2(+(n361_2, 1))) ->_R^Omega(0)
s(encArg(gen_s:0':cons_f:cons_g2_2(n361_2))) ->_IH
s(gen_s:0':cons_f:cons_g2_2(c362_2))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(48)
BOUNDS(1, INF)