WORST_CASE(Omega(n^1), O(n^3))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^3).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 172 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)), 84 ms]
    (16) CdtProblem
    (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 40 ms]
    (18) CdtProblem
    (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 18 ms]
    (20) CdtProblem
    (21) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) BOUNDS(1, 1)
(23) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(24) CpxRelTRS
    (25) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) typed CpxTrs
    (27) OrderProof [LOWER BOUND(ID), 0 ms]
    (28) typed CpxTrs
    (29) RewriteLemmaProof [LOWER BOUND(ID), 424 ms]
    (30) BEST
        (31) proven lower bound
            (32) LowerBoundPropagationProof [FINISHED, 0 ms]
            (33) BOUNDS(n^1, INF)
        (34) typed CpxTrs
            (35) RewriteLemmaProof [LOWER BOUND(ID), 128 ms]
            (36) BOUNDS(1, INF)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^3).


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: 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(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 (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


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: 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^3).


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: INNERMOST
----------------------------------------

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

(6)
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(minus(z0)) -> z0
   minus(h(z0)) -> h(minus(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(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(minus(z0)) -> c6
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
S tuples:
   MINUS(minus(z0)) -> c6
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
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


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_H(z0) -> c4(ENCARG(z0))
   ENCODE_F(z0, z1) -> c5(ENCARG(z0), ENCARG(z1))

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

(8)
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(minus(z0)) -> z0
   minus(h(z0)) -> h(minus(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(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(minus(z0)) -> c6
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
S tuples:
   MINUS(minus(z0)) -> c6
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
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, c6, c7_1, c8_2


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

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

(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(minus(z0)) -> z0
   minus(h(z0)) -> h(minus(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(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(minus(z0)) -> c6
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   ENCODE_MINUS(z0) -> c4(MINUS(encArg(z0)))
   ENCODE_MINUS(z0) -> c4(ENCARG(z0))
S tuples:
   MINUS(minus(z0)) -> c6
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
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, c6, c7_1, c8_2, c4_1


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

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

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

(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(minus(z0)) -> z0
   minus(h(z0)) -> h(minus(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(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(minus(z0)) -> c6
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   ENCODE_MINUS(z0) -> c4(MINUS(encArg(z0)))
S tuples:
   MINUS(minus(z0)) -> c6
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
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, c6, c7_1, c8_2, c4_1


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

(13) 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))

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

(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))
   minus(minus(z0)) -> z0
   minus(h(z0)) -> h(minus(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(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(minus(z0)) -> c6
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   ENCODE_MINUS(z0) -> c4(MINUS(encArg(z0)))
S tuples:
   MINUS(minus(z0)) -> c6
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
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, c6, c7_1, c8_2, c4_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.
   MINUS(h(z0)) -> c7(MINUS(z0))
We considered the (Usable) Rules:
   encArg(cons_minus(z0)) -> minus(encArg(z0))
   minus(minus(z0)) -> z0
   encArg(h(z0)) -> h(encArg(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(z0))
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   minus(h(z0)) -> h(minus(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(minus(z0)) -> c6
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   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_1^2
   POL(ENCODE_MINUS(x_1)) = [1] + [2]x_1 + [2]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(c6) = 0
   POL(c7(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(cons_minus(x_1)) = [2] + x_1
   POL(encArg(x_1)) = 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

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

(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))
   minus(minus(z0)) -> z0
   minus(h(z0)) -> h(minus(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(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(minus(z0)) -> c6
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   ENCODE_MINUS(z0) -> c4(MINUS(encArg(z0)))
S tuples:
   MINUS(minus(z0)) -> c6
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
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, c6, c7_1, c8_2, c4_1


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

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MINUS(minus(z0)) -> c6
We considered the (Usable) Rules:
   encArg(cons_minus(z0)) -> minus(encArg(z0))
   minus(minus(z0)) -> z0
   encArg(h(z0)) -> h(encArg(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(z0))
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   minus(h(z0)) -> h(minus(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(minus(z0)) -> c6
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   ENCODE_MINUS(z0) -> c4(MINUS(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ENCARG(x_1)) = [1] + x_1^3
   POL(ENCODE_MINUS(x_1)) = [1] + x_1^2
   POL(MINUS(x_1)) = [1] + x_1^2
   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(c6) = 0
   POL(c7(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(cons_minus(x_1)) = [1] + 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

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

(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(minus(z0)) -> z0
   minus(h(z0)) -> h(minus(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(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(minus(z0)) -> c6
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   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(minus(z0)) -> c6
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, c6, c7_1, c8_2, 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(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
We considered the (Usable) Rules:
   encArg(cons_minus(z0)) -> minus(encArg(z0))
   minus(minus(z0)) -> z0
   encArg(h(z0)) -> h(encArg(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(z0))
   encArg(f(z0, z1)) -> f(encArg(z0), encArg(z1))
   minus(h(z0)) -> h(minus(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(minus(z0)) -> c6
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   ENCODE_MINUS(z0) -> c4(MINUS(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ENCARG(x_1)) = [1] + x_1^2
   POL(ENCODE_MINUS(x_1)) = x_1
   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(c6) = 0
   POL(c7(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(cons_minus(x_1)) = [1] + 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

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

(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(minus(z0)) -> z0
   minus(h(z0)) -> h(minus(z0))
   minus(f(z0, z1)) -> f(minus(z1), minus(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(minus(z0)) -> c6
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(f(z0, z1)) -> c8(MINUS(z1), MINUS(z0))
   ENCODE_MINUS(z0) -> c4(MINUS(encArg(z0)))
S tuples:none
K tuples:
   MINUS(h(z0)) -> c7(MINUS(z0))
   MINUS(minus(z0)) -> c6
   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, c6, c7_1, c8_2, c4_1


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(21) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(22)
BOUNDS(1, 1)

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(23) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(24)
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:

   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: INNERMOST
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(25) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(26)
Obligation:
Innermost 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

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(27) 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

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(28)
Obligation:
Innermost 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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(29) 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).
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(30)
Complex Obligation (BEST)

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(31)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost 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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(32) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(33)
BOUNDS(n^1, INF)

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(34)
Obligation:
Innermost 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
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(35) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_h:f:cons_minus2_0(+(1, n870_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, +(n870_0, 1)))) ->_R^Omega(0)
h(encArg(gen_h:f:cons_minus2_0(+(1, n870_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).
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(36)
BOUNDS(1, INF)