/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (full) 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), 167 ms]
(4) CpxRelTRS
(5) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(6) CpxRelTRS
    (7) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (10) CdtProblem
    (11) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (12) CdtProblem
    (13) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CdtProblem
    (15) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (16) CdtProblem
    (17) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CdtProblem
    (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 65 ms]
    (20) CdtProblem
    (21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 8 ms]
    (22) CdtProblem
    (23) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 19 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), 401 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), 112 ms]
            (40) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   minus(minus(x)) -> x
   minus(h(x)) -> h(minus(x))
   minus(f(x, y)) -> f(minus(y), minus(x))

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

(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(h(x_1)) -> h(encArg(x_1))
   encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1)) -> minus(encArg(x_1))
   encode_minus(x_1) -> minus(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))


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

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


The TRS R consists of the following rules:

   minus(minus(x)) -> x
   minus(h(x)) -> h(minus(x))
   minus(f(x, y)) -> f(minus(y), minus(x))

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

   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1)) -> minus(encArg(x_1))
   encode_minus(x_1) -> minus(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

Rewrite Strategy: FULL
----------------------------------------

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

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


The TRS R consists of the following rules:

   minus(minus(x)) -> x
   minus(h(x)) -> h(minus(x))
   minus(f(x, y)) -> f(minus(y), minus(x))

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

   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1)) -> minus(encArg(x_1))
   encode_minus(x_1) -> minus(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

Rewrite Strategy: FULL
----------------------------------------

(5) NonCtorToCtorProof (UPPER BOUND(ID))
transformed non-ctor to ctor-system
----------------------------------------

(6)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   minus(h(x)) -> h(minus(x))
   minus(f(x, y)) -> f(minus(y), minus(x))
   minus(c_minus(x)) -> x

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

   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1)) -> minus(encArg(x_1))
   encode_minus(x_1) -> minus(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   minus(x0) -> c_minus(x0)

Rewrite Strategy: FULL
----------------------------------------

(7) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   minus(h(x)) -> h(minus(x))
   minus(f(x, y)) -> f(minus(y), minus(x))
   minus(c_minus(x)) -> x

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

   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1)) -> minus(encArg(x_1))
   encode_minus(x_1) -> minus(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   minus(x0) -> c_minus(x0)

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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(h(z0)) -> h(encArg(z0))
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_minus(z0)) -> minus(encArg(z0))
   encode_minus(z0) -> minus(encArg(z0))
   encode_h(z0) -> h(encArg(z0))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   minus(z0) -> c_minus(z0)
   minus(h(z0)) -> h(minus(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(z0))
   minus(c_minus(z0)) -> z0
Tuples:
   ENCARG(h(z0)) -> c(ENCARG(z0))
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_minus(z0)) -> c2(MINUS(encArg(z0)), ENCARG(z0))
   ENCODE_MINUS(z0) -> c3(MINUS(encArg(z0)), ENCARG(z0))
   ENCODE_H(z0) -> c4(ENCARG(z0))
   ENCODE_F(z0, z1) -> c5(ENCARG(z0), ENCARG(z1))
   MINUS(z0) -> c6
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   MINUS(c_minus(z0)) -> c9
S tuples:
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   MINUS(c_minus(z0)) -> c9
K tuples:none
Defined Rule Symbols:   minus_1, encArg_1, encode_minus_1, encode_h_1, encode_f_2

Defined Pair Symbols:   ENCARG_1, ENCODE_MINUS_1, ENCODE_H_1, ENCODE_F_2, MINUS_1

Compound Symbols:   c_1, c1_2, c2_2, c3_2, c4_1, c5_2, c6, c7_1, c8_2, c9


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_H(z0) -> c4(ENCARG(z0))
   ENCODE_F(z0, z1) -> c5(ENCARG(z0), ENCARG(z1))
Removed 1 trailing nodes:
   MINUS(z0) -> c6

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(h(z0)) -> h(encArg(z0))
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_minus(z0)) -> minus(encArg(z0))
   encode_minus(z0) -> minus(encArg(z0))
   encode_h(z0) -> h(encArg(z0))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   minus(z0) -> c_minus(z0)
   minus(h(z0)) -> h(minus(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(z0))
   minus(c_minus(z0)) -> z0
Tuples:
   ENCARG(h(z0)) -> c(ENCARG(z0))
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_minus(z0)) -> c2(MINUS(encArg(z0)), ENCARG(z0))
   ENCODE_MINUS(z0) -> c3(MINUS(encArg(z0)), ENCARG(z0))
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   MINUS(c_minus(z0)) -> c9
S tuples:
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   MINUS(c_minus(z0)) -> c9
K tuples:none
Defined Rule Symbols:   minus_1, encArg_1, encode_minus_1, encode_h_1, encode_f_2

Defined Pair Symbols:   ENCARG_1, ENCODE_MINUS_1, MINUS_1

Compound Symbols:   c_1, c1_2, c2_2, c3_2, c7_1, c8_2, c9


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

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(h(z0)) -> h(encArg(z0))
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_minus(z0)) -> minus(encArg(z0))
   encode_minus(z0) -> minus(encArg(z0))
   encode_h(z0) -> h(encArg(z0))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   minus(z0) -> c_minus(z0)
   minus(h(z0)) -> h(minus(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(z0))
   minus(c_minus(z0)) -> z0
Tuples:
   ENCARG(h(z0)) -> c(ENCARG(z0))
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_minus(z0)) -> c2(MINUS(encArg(z0)), ENCARG(z0))
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   MINUS(c_minus(z0)) -> c9
   ENCODE_MINUS(z0) -> c4(MINUS(encArg(z0)))
   ENCODE_MINUS(z0) -> c4(ENCARG(z0))
S tuples:
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   MINUS(c_minus(z0)) -> c9
K tuples:none
Defined Rule Symbols:   minus_1, encArg_1, encode_minus_1, encode_h_1, encode_f_2

Defined Pair Symbols:   ENCARG_1, MINUS_1, ENCODE_MINUS_1

Compound Symbols:   c_1, c1_2, c2_2, c7_1, c8_2, c9, c4_1


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

(15) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   ENCODE_MINUS(z0) -> c4(ENCARG(z0))

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(h(z0)) -> h(encArg(z0))
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_minus(z0)) -> minus(encArg(z0))
   encode_minus(z0) -> minus(encArg(z0))
   encode_h(z0) -> h(encArg(z0))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   minus(z0) -> c_minus(z0)
   minus(h(z0)) -> h(minus(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(z0))
   minus(c_minus(z0)) -> z0
Tuples:
   ENCARG(h(z0)) -> c(ENCARG(z0))
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_minus(z0)) -> c2(MINUS(encArg(z0)), ENCARG(z0))
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   MINUS(c_minus(z0)) -> c9
   ENCODE_MINUS(z0) -> c4(MINUS(encArg(z0)))
S tuples:
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   MINUS(c_minus(z0)) -> c9
K tuples:none
Defined Rule Symbols:   minus_1, encArg_1, encode_minus_1, encode_h_1, encode_f_2

Defined Pair Symbols:   ENCARG_1, MINUS_1, ENCODE_MINUS_1

Compound Symbols:   c_1, c1_2, c2_2, c7_1, c8_2, c9, c4_1


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

(17) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_minus(z0) -> minus(encArg(z0))
   encode_h(z0) -> h(encArg(z0))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(h(z0)) -> h(encArg(z0))
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_minus(z0)) -> minus(encArg(z0))
   minus(z0) -> c_minus(z0)
   minus(h(z0)) -> h(minus(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(z0))
   minus(c_minus(z0)) -> z0
Tuples:
   ENCARG(h(z0)) -> c(ENCARG(z0))
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_minus(z0)) -> c2(MINUS(encArg(z0)), ENCARG(z0))
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   MINUS(c_minus(z0)) -> c9
   ENCODE_MINUS(z0) -> c4(MINUS(encArg(z0)))
S tuples:
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   MINUS(c_minus(z0)) -> c9
K tuples:none
Defined Rule Symbols:   encArg_1, minus_1

Defined Pair Symbols:   ENCARG_1, MINUS_1, ENCODE_MINUS_1

Compound Symbols:   c_1, c1_2, c2_2, c7_1, c8_2, c9, c4_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.
   MINUS(h(z0)) -> c7(MINUS(z0))
We considered the (Usable) Rules:
   encArg(cons_minus(z0)) -> minus(encArg(z0))
   encArg(h(z0)) -> h(encArg(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(z0))
   minus(z0) -> c_minus(z0)
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   minus(h(z0)) -> h(minus(z0))
   minus(c_minus(z0)) -> z0
And the Tuples:
   ENCARG(h(z0)) -> c(ENCARG(z0))
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_minus(z0)) -> c2(MINUS(encArg(z0)), ENCARG(z0))
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   MINUS(c_minus(z0)) -> c9
   ENCODE_MINUS(z0) -> c4(MINUS(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ENCARG(x_1)) = [2]x_1 + x_1^2
   POL(ENCODE_MINUS(x_1)) = [1] + [2]x_1 + x_1^2
   POL(MINUS(x_1)) = x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9) = 0
   POL(c_minus(x_1)) = x_1
   POL(cons_minus(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [2]x_1
   POL(f(x_1, x_2)) = x_1 + x_2
   POL(h(x_1)) = [2] + x_1
   POL(minus(x_1)) = x_1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(h(z0)) -> h(encArg(z0))
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_minus(z0)) -> minus(encArg(z0))
   minus(z0) -> c_minus(z0)
   minus(h(z0)) -> h(minus(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(z0))
   minus(c_minus(z0)) -> z0
Tuples:
   ENCARG(h(z0)) -> c(ENCARG(z0))
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_minus(z0)) -> c2(MINUS(encArg(z0)), ENCARG(z0))
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   MINUS(c_minus(z0)) -> c9
   ENCODE_MINUS(z0) -> c4(MINUS(encArg(z0)))
S tuples:
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   MINUS(c_minus(z0)) -> c9
K tuples:
   MINUS(h(z0)) -> c7(MINUS(z0))
Defined Rule Symbols:   encArg_1, minus_1

Defined Pair Symbols:   ENCARG_1, MINUS_1, ENCODE_MINUS_1

Compound Symbols:   c_1, c1_2, c2_2, c7_1, c8_2, c9, c4_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.
   MINUS(c_minus(z0)) -> c9
We considered the (Usable) Rules:
   encArg(cons_minus(z0)) -> minus(encArg(z0))
   encArg(h(z0)) -> h(encArg(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(z0))
   minus(z0) -> c_minus(z0)
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   minus(h(z0)) -> h(minus(z0))
   minus(c_minus(z0)) -> z0
And the Tuples:
   ENCARG(h(z0)) -> c(ENCARG(z0))
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_minus(z0)) -> c2(MINUS(encArg(z0)), ENCARG(z0))
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   MINUS(c_minus(z0)) -> c9
   ENCODE_MINUS(z0) -> c4(MINUS(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ENCARG(x_1)) = [2] + [2]x_1 + x_1^2
   POL(ENCODE_MINUS(x_1)) = [2] + x_1 + [2]x_1^2
   POL(MINUS(x_1)) = [1] + x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9) = 0
   POL(c_minus(x_1)) = x_1
   POL(cons_minus(x_1)) = [2] + x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1, x_2)) = [1] + x_1 + x_2
   POL(h(x_1)) = x_1
   POL(minus(x_1)) = x_1

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(h(z0)) -> h(encArg(z0))
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_minus(z0)) -> minus(encArg(z0))
   minus(z0) -> c_minus(z0)
   minus(h(z0)) -> h(minus(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(z0))
   minus(c_minus(z0)) -> z0
Tuples:
   ENCARG(h(z0)) -> c(ENCARG(z0))
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_minus(z0)) -> c2(MINUS(encArg(z0)), ENCARG(z0))
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   MINUS(c_minus(z0)) -> c9
   ENCODE_MINUS(z0) -> c4(MINUS(encArg(z0)))
S tuples:
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
K tuples:
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(c_minus(z0)) -> c9
Defined Rule Symbols:   encArg_1, minus_1

Defined Pair Symbols:   ENCARG_1, MINUS_1, ENCODE_MINUS_1

Compound Symbols:   c_1, c1_2, c2_2, c7_1, c8_2, c9, c4_1


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(23) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
We considered the (Usable) Rules:
   encArg(cons_minus(z0)) -> minus(encArg(z0))
   encArg(h(z0)) -> h(encArg(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(z0))
   minus(z0) -> c_minus(z0)
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   minus(h(z0)) -> h(minus(z0))
   minus(c_minus(z0)) -> z0
And the Tuples:
   ENCARG(h(z0)) -> c(ENCARG(z0))
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_minus(z0)) -> c2(MINUS(encArg(z0)), ENCARG(z0))
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   MINUS(c_minus(z0)) -> c9
   ENCODE_MINUS(z0) -> c4(MINUS(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ENCARG(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(ENCODE_MINUS(x_1)) = [1] + [2]x_1 + x_1^2
   POL(MINUS(x_1)) = [2]x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9) = 0
   POL(c_minus(x_1)) = x_1
   POL(cons_minus(x_1)) = [2] + x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1, x_2)) = [2] + x_1 + x_2
   POL(h(x_1)) = x_1
   POL(minus(x_1)) = x_1

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(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(h(z0)) -> h(encArg(z0))
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_minus(z0)) -> minus(encArg(z0))
   minus(z0) -> c_minus(z0)
   minus(h(z0)) -> h(minus(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(z0))
   minus(c_minus(z0)) -> z0
Tuples:
   ENCARG(h(z0)) -> c(ENCARG(z0))
   ENCARG(f(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_minus(z0)) -> c2(MINUS(encArg(z0)), ENCARG(z0))
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   MINUS(c_minus(z0)) -> c9
   ENCODE_MINUS(z0) -> c4(MINUS(encArg(z0)))
S tuples:none
K tuples:
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(c_minus(z0)) -> c9
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
Defined Rule Symbols:   encArg_1, minus_1

Defined Pair Symbols:   ENCARG_1, MINUS_1, ENCODE_MINUS_1

Compound Symbols:   c_1, c1_2, c2_2, c7_1, c8_2, c9, c4_1


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

(26)
BOUNDS(1, 1)

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

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


The TRS R consists of the following rules:

   minus(minus(x)) -> x
   minus(h(x)) -> h(minus(x))
   minus(f(x, y)) -> f(minus(y), minus(x))

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

   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1)) -> minus(encArg(x_1))
   encode_minus(x_1) -> minus(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

Rewrite Strategy: FULL
----------------------------------------

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

(30)
Obligation:
TRS:
Rules:
minus(minus(x)) -> x
minus(h(x)) -> h(minus(x))
minus(f(x, y)) -> f(minus(y), minus(x))
encArg(h(x_1)) -> h(encArg(x_1))
encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(cons_minus(x_1)) -> minus(encArg(x_1))
encode_minus(x_1) -> minus(encArg(x_1))
encode_h(x_1) -> h(encArg(x_1))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

Types:
minus :: h:f:cons_minus -> h:f:cons_minus
h :: h:f:cons_minus -> h:f:cons_minus
f :: h:f:cons_minus -> h:f:cons_minus -> h:f:cons_minus
encArg :: h:f:cons_minus -> h:f:cons_minus
cons_minus :: h:f:cons_minus -> h:f:cons_minus
encode_minus :: h:f:cons_minus -> h:f:cons_minus
encode_h :: h:f:cons_minus -> h:f:cons_minus
encode_f :: h:f:cons_minus -> h:f:cons_minus -> h:f:cons_minus
hole_h:f:cons_minus1_0 :: h:f:cons_minus
gen_h:f:cons_minus2_0 :: Nat -> h:f:cons_minus

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

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

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

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

(32)
Obligation:
TRS:
Rules:
minus(minus(x)) -> x
minus(h(x)) -> h(minus(x))
minus(f(x, y)) -> f(minus(y), minus(x))
encArg(h(x_1)) -> h(encArg(x_1))
encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(cons_minus(x_1)) -> minus(encArg(x_1))
encode_minus(x_1) -> minus(encArg(x_1))
encode_h(x_1) -> h(encArg(x_1))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

Types:
minus :: h:f:cons_minus -> h:f:cons_minus
h :: h:f:cons_minus -> h:f:cons_minus
f :: h:f:cons_minus -> h:f:cons_minus -> h:f:cons_minus
encArg :: h:f:cons_minus -> h:f:cons_minus
cons_minus :: h:f:cons_minus -> h:f:cons_minus
encode_minus :: h:f:cons_minus -> h:f:cons_minus
encode_h :: h:f:cons_minus -> h:f:cons_minus
encode_f :: h:f:cons_minus -> h:f:cons_minus -> h:f:cons_minus
hole_h:f:cons_minus1_0 :: h:f:cons_minus
gen_h:f:cons_minus2_0 :: Nat -> h:f:cons_minus


Generator Equations:
gen_h:f:cons_minus2_0(0) <=> hole_h:f:cons_minus1_0
gen_h:f:cons_minus2_0(+(x, 1)) <=> h(gen_h:f:cons_minus2_0(x))


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

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

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(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_h:f:cons_minus2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
minus(gen_h:f:cons_minus2_0(+(1, 0)))

Induction Step:
minus(gen_h:f:cons_minus2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
h(minus(gen_h:f:cons_minus2_0(+(1, n4_0)))) ->_IH
h(*3_0)

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

(34)
Complex Obligation (BEST)

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

TRS:
Rules:
minus(minus(x)) -> x
minus(h(x)) -> h(minus(x))
minus(f(x, y)) -> f(minus(y), minus(x))
encArg(h(x_1)) -> h(encArg(x_1))
encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(cons_minus(x_1)) -> minus(encArg(x_1))
encode_minus(x_1) -> minus(encArg(x_1))
encode_h(x_1) -> h(encArg(x_1))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

Types:
minus :: h:f:cons_minus -> h:f:cons_minus
h :: h:f:cons_minus -> h:f:cons_minus
f :: h:f:cons_minus -> h:f:cons_minus -> h:f:cons_minus
encArg :: h:f:cons_minus -> h:f:cons_minus
cons_minus :: h:f:cons_minus -> h:f:cons_minus
encode_minus :: h:f:cons_minus -> h:f:cons_minus
encode_h :: h:f:cons_minus -> h:f:cons_minus
encode_f :: h:f:cons_minus -> h:f:cons_minus -> h:f:cons_minus
hole_h:f:cons_minus1_0 :: h:f:cons_minus
gen_h:f:cons_minus2_0 :: Nat -> h:f:cons_minus


Generator Equations:
gen_h:f:cons_minus2_0(0) <=> hole_h:f:cons_minus1_0
gen_h:f:cons_minus2_0(+(x, 1)) <=> h(gen_h:f:cons_minus2_0(x))


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

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

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

(37)
BOUNDS(n^1, INF)

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

(38)
Obligation:
TRS:
Rules:
minus(minus(x)) -> x
minus(h(x)) -> h(minus(x))
minus(f(x, y)) -> f(minus(y), minus(x))
encArg(h(x_1)) -> h(encArg(x_1))
encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(cons_minus(x_1)) -> minus(encArg(x_1))
encode_minus(x_1) -> minus(encArg(x_1))
encode_h(x_1) -> h(encArg(x_1))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

Types:
minus :: h:f:cons_minus -> h:f:cons_minus
h :: h:f:cons_minus -> h:f:cons_minus
f :: h:f:cons_minus -> h:f:cons_minus -> h:f:cons_minus
encArg :: h:f:cons_minus -> h:f:cons_minus
cons_minus :: h:f:cons_minus -> h:f:cons_minus
encode_minus :: h:f:cons_minus -> h:f:cons_minus
encode_h :: h:f:cons_minus -> h:f:cons_minus
encode_f :: h:f:cons_minus -> h:f:cons_minus -> h:f:cons_minus
hole_h:f:cons_minus1_0 :: h:f:cons_minus
gen_h:f:cons_minus2_0 :: Nat -> h:f:cons_minus


Lemmas:
minus(gen_h:f:cons_minus2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_h:f:cons_minus2_0(0) <=> hole_h:f:cons_minus1_0
gen_h:f:cons_minus2_0(+(x, 1)) <=> h(gen_h:f:cons_minus2_0(x))


The following defined symbols remain to be analysed:
encArg
----------------------------------------

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_h:f:cons_minus2_0(+(1, n840_0))) -> *3_0, rt in Omega(0)

Induction Base:
encArg(gen_h:f:cons_minus2_0(+(1, 0)))

Induction Step:
encArg(gen_h:f:cons_minus2_0(+(1, +(n840_0, 1)))) ->_R^Omega(0)
h(encArg(gen_h:f:cons_minus2_0(+(1, n840_0)))) ->_IH
h(*3_0)

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

(40)
BOUNDS(1, INF)