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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 164 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
(7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CdtProblem
(9) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CdtProblem
(11) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CdtProblem
(13) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(14) CdtProblem
(15) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CdtProblem
(17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 25 ms]
(18) CdtProblem
(19) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) BOUNDS(1, 1)


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

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


The TRS R consists of the following rules:

   ap(ap(ff, x), x) -> ap(ap(x, ap(ff, x)), ap(ap(cons, x), nil))

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(ff) -> ff
   encArg(cons) -> cons
   encArg(nil) -> nil
   encArg(cons_ap(x_1, x_2)) -> ap(encArg(x_1), encArg(x_2))
   encode_ap(x_1, x_2) -> ap(encArg(x_1), encArg(x_2))
   encode_ff -> ff
   encode_cons -> cons
   encode_nil -> nil


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

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


The TRS R consists of the following rules:

   ap(ap(ff, x), x) -> ap(ap(x, ap(ff, x)), ap(ap(cons, x), nil))

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

   encArg(ff) -> ff
   encArg(cons) -> cons
   encArg(nil) -> nil
   encArg(cons_ap(x_1, x_2)) -> ap(encArg(x_1), encArg(x_2))
   encode_ap(x_1, x_2) -> ap(encArg(x_1), encArg(x_2))
   encode_ff -> ff
   encode_cons -> cons
   encode_nil -> nil

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(1, n^1).


The TRS R consists of the following rules:

   ap(ap(ff, x), x) -> ap(ap(x, ap(ff, x)), ap(ap(cons, x), nil))

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

   encArg(ff) -> ff
   encArg(cons) -> cons
   encArg(nil) -> nil
   encArg(cons_ap(x_1, x_2)) -> ap(encArg(x_1), encArg(x_2))
   encode_ap(x_1, x_2) -> ap(encArg(x_1), encArg(x_2))
   encode_ff -> ff
   encode_cons -> cons
   encode_nil -> nil

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ff) -> ff
   encArg(cons) -> cons
   encArg(nil) -> nil
   encArg(cons_ap(z0, z1)) -> ap(encArg(z0), encArg(z1))
   encode_ap(z0, z1) -> ap(encArg(z0), encArg(z1))
   encode_ff -> ff
   encode_cons -> cons
   encode_nil -> nil
   ap(ap(ff, z0), z0) -> ap(ap(z0, ap(ff, z0)), ap(ap(cons, z0), nil))
Tuples:
   ENCARG(ff) -> c
   ENCARG(cons) -> c1
   ENCARG(nil) -> c2
   ENCARG(cons_ap(z0, z1)) -> c3(AP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_AP(z0, z1) -> c4(AP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_FF -> c5
   ENCODE_CONS -> c6
   ENCODE_NIL -> c7
   AP(ap(ff, z0), z0) -> c8(AP(ap(z0, ap(ff, z0)), ap(ap(cons, z0), nil)), AP(z0, ap(ff, z0)), AP(ff, z0), AP(ap(cons, z0), nil), AP(cons, z0))
S tuples:
   AP(ap(ff, z0), z0) -> c8(AP(ap(z0, ap(ff, z0)), ap(ap(cons, z0), nil)), AP(z0, ap(ff, z0)), AP(ff, z0), AP(ap(cons, z0), nil), AP(cons, z0))
K tuples:none
Defined Rule Symbols:   ap_2, encArg_1, encode_ap_2, encode_ff, encode_cons, encode_nil

Defined Pair Symbols:   ENCARG_1, ENCODE_AP_2, ENCODE_FF, ENCODE_CONS, ENCODE_NIL, AP_2

Compound Symbols:   c, c1, c2, c3_3, c4_3, c5, c6, c7, c8_5


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

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   ENCARG(ff) -> c
   ENCODE_CONS -> c6
   ENCARG(nil) -> c2
   ENCODE_NIL -> c7
   ENCODE_FF -> c5
   ENCARG(cons) -> c1

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ff) -> ff
   encArg(cons) -> cons
   encArg(nil) -> nil
   encArg(cons_ap(z0, z1)) -> ap(encArg(z0), encArg(z1))
   encode_ap(z0, z1) -> ap(encArg(z0), encArg(z1))
   encode_ff -> ff
   encode_cons -> cons
   encode_nil -> nil
   ap(ap(ff, z0), z0) -> ap(ap(z0, ap(ff, z0)), ap(ap(cons, z0), nil))
Tuples:
   ENCARG(cons_ap(z0, z1)) -> c3(AP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_AP(z0, z1) -> c4(AP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   AP(ap(ff, z0), z0) -> c8(AP(ap(z0, ap(ff, z0)), ap(ap(cons, z0), nil)), AP(z0, ap(ff, z0)), AP(ff, z0), AP(ap(cons, z0), nil), AP(cons, z0))
S tuples:
   AP(ap(ff, z0), z0) -> c8(AP(ap(z0, ap(ff, z0)), ap(ap(cons, z0), nil)), AP(z0, ap(ff, z0)), AP(ff, z0), AP(ap(cons, z0), nil), AP(cons, z0))
K tuples:none
Defined Rule Symbols:   ap_2, encArg_1, encode_ap_2, encode_ff, encode_cons, encode_nil

Defined Pair Symbols:   ENCARG_1, ENCODE_AP_2, AP_2

Compound Symbols:   c3_3, c4_3, c8_5


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

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing tuple parts
----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ff) -> ff
   encArg(cons) -> cons
   encArg(nil) -> nil
   encArg(cons_ap(z0, z1)) -> ap(encArg(z0), encArg(z1))
   encode_ap(z0, z1) -> ap(encArg(z0), encArg(z1))
   encode_ff -> ff
   encode_cons -> cons
   encode_nil -> nil
   ap(ap(ff, z0), z0) -> ap(ap(z0, ap(ff, z0)), ap(ap(cons, z0), nil))
Tuples:
   ENCARG(cons_ap(z0, z1)) -> c3(AP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_AP(z0, z1) -> c4(AP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   AP(ap(ff, z0), z0) -> c8
S tuples:
   AP(ap(ff, z0), z0) -> c8
K tuples:none
Defined Rule Symbols:   ap_2, encArg_1, encode_ap_2, encode_ff, encode_cons, encode_nil

Defined Pair Symbols:   ENCARG_1, ENCODE_AP_2, AP_2

Compound Symbols:   c3_3, c4_3, c8


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

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ff) -> ff
   encArg(cons) -> cons
   encArg(nil) -> nil
   encArg(cons_ap(z0, z1)) -> ap(encArg(z0), encArg(z1))
   encode_ap(z0, z1) -> ap(encArg(z0), encArg(z1))
   encode_ff -> ff
   encode_cons -> cons
   encode_nil -> nil
   ap(ap(ff, z0), z0) -> ap(ap(z0, ap(ff, z0)), ap(ap(cons, z0), nil))
Tuples:
   ENCARG(cons_ap(z0, z1)) -> c3(AP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   AP(ap(ff, z0), z0) -> c8
   ENCODE_AP(z0, z1) -> c(AP(encArg(z0), encArg(z1)))
   ENCODE_AP(z0, z1) -> c(ENCARG(z0))
   ENCODE_AP(z0, z1) -> c(ENCARG(z1))
S tuples:
   AP(ap(ff, z0), z0) -> c8
K tuples:none
Defined Rule Symbols:   ap_2, encArg_1, encode_ap_2, encode_ff, encode_cons, encode_nil

Defined Pair Symbols:   ENCARG_1, AP_2, ENCODE_AP_2

Compound Symbols:   c3_3, c8, c_1


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

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

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ff) -> ff
   encArg(cons) -> cons
   encArg(nil) -> nil
   encArg(cons_ap(z0, z1)) -> ap(encArg(z0), encArg(z1))
   encode_ap(z0, z1) -> ap(encArg(z0), encArg(z1))
   encode_ff -> ff
   encode_cons -> cons
   encode_nil -> nil
   ap(ap(ff, z0), z0) -> ap(ap(z0, ap(ff, z0)), ap(ap(cons, z0), nil))
Tuples:
   ENCARG(cons_ap(z0, z1)) -> c3(AP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   AP(ap(ff, z0), z0) -> c8
   ENCODE_AP(z0, z1) -> c(AP(encArg(z0), encArg(z1)))
S tuples:
   AP(ap(ff, z0), z0) -> c8
K tuples:none
Defined Rule Symbols:   ap_2, encArg_1, encode_ap_2, encode_ff, encode_cons, encode_nil

Defined Pair Symbols:   ENCARG_1, AP_2, ENCODE_AP_2

Compound Symbols:   c3_3, c8, c_1


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

(15) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_ap(z0, z1) -> ap(encArg(z0), encArg(z1))
   encode_ff -> ff
   encode_cons -> cons
   encode_nil -> nil

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ff) -> ff
   encArg(cons) -> cons
   encArg(nil) -> nil
   encArg(cons_ap(z0, z1)) -> ap(encArg(z0), encArg(z1))
   ap(ap(ff, z0), z0) -> ap(ap(z0, ap(ff, z0)), ap(ap(cons, z0), nil))
Tuples:
   ENCARG(cons_ap(z0, z1)) -> c3(AP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   AP(ap(ff, z0), z0) -> c8
   ENCODE_AP(z0, z1) -> c(AP(encArg(z0), encArg(z1)))
S tuples:
   AP(ap(ff, z0), z0) -> c8
K tuples:none
Defined Rule Symbols:   encArg_1, ap_2

Defined Pair Symbols:   ENCARG_1, AP_2, ENCODE_AP_2

Compound Symbols:   c3_3, c8, c_1


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

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   AP(ap(ff, z0), z0) -> c8
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons_ap(z0, z1)) -> c3(AP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   AP(ap(ff, z0), z0) -> c8
   ENCODE_AP(z0, z1) -> c(AP(encArg(z0), encArg(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(AP(x_1, x_2)) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_AP(x_1, x_2)) = [1]
   POL(ap(x_1, x_2)) = x_2
   POL(c(x_1)) = x_1
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c8) = 0
   POL(cons) = [1]
   POL(cons_ap(x_1, x_2)) = [1] + x_1 + x_2
   POL(encArg(x_1)) = [1] + x_1
   POL(ff) = [1]
   POL(nil) = 0

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ff) -> ff
   encArg(cons) -> cons
   encArg(nil) -> nil
   encArg(cons_ap(z0, z1)) -> ap(encArg(z0), encArg(z1))
   ap(ap(ff, z0), z0) -> ap(ap(z0, ap(ff, z0)), ap(ap(cons, z0), nil))
Tuples:
   ENCARG(cons_ap(z0, z1)) -> c3(AP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   AP(ap(ff, z0), z0) -> c8
   ENCODE_AP(z0, z1) -> c(AP(encArg(z0), encArg(z1)))
S tuples:none
K tuples:
   AP(ap(ff, z0), z0) -> c8
Defined Rule Symbols:   encArg_1, ap_2

Defined Pair Symbols:   ENCARG_1, AP_2, ENCODE_AP_2

Compound Symbols:   c3_3, c8, c_1


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

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

(20)
BOUNDS(1, 1)