WORST_CASE(?,O(n^1))
proof of input_a0kBz9bROh.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 198 ms]
(2) CpxRelTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) CpxTrsMatchBoundsTAProof [FINISHED, 44 ms]
(6) BOUNDS(1, n^1)


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(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(C(x1, x2)) -> C(f(x1), f(x2))
   f(Z) -> Z
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
   eqZList(C(x1, x2), Z) -> False
   eqZList(Z, C(y1, y2)) -> False
   eqZList(Z, Z) -> True
   second(C(x1, x2)) -> x2
   first(C(x1, x2)) -> x1
   g(x) -> x

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

Rewrite Strategy: PARALLEL_INNERMOST
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(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(C(x1, x2)) -> C(f(x1), f(x2))
   f(Z) -> Z
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
   eqZList(C(x1, x2), Z) -> False
   eqZList(Z, C(y1, y2)) -> False
   eqZList(Z, Z) -> True
   second(C(x1, x2)) -> x2
   first(C(x1, x2)) -> x1
   g(x) -> x

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

Rewrite Strategy: PARALLEL_INNERMOST
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(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(4)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(C(x1, x2)) -> C(f(x1), f(x2))
   f(Z) -> Z
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
   eqZList(C(x1, x2), Z) -> False
   eqZList(Z, C(y1, y2)) -> False
   eqZList(Z, Z) -> True
   second(C(x1, x2)) -> x2
   first(C(x1, x2)) -> x1
   g(x) -> x
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(5) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4, 5, 6]
transitions: 
C0(0, 0) -> 0
Z0() -> 0
False0() -> 0
True0() -> 0
f0(0) -> 1
eqZList0(0, 0) -> 2
second0(0) -> 3
first0(0) -> 4
g0(0) -> 5
and0(0, 0) -> 6
f1(0) -> 7
f1(0) -> 8
C1(7, 8) -> 1
Z1() -> 1
eqZList1(0, 0) -> 9
eqZList1(0, 0) -> 10
and1(9, 10) -> 2
False1() -> 2
True1() -> 2
False1() -> 6
True1() -> 6
C1(7, 8) -> 7
C1(7, 8) -> 8
Z1() -> 7
Z1() -> 8
and1(9, 10) -> 9
and1(9, 10) -> 10
False1() -> 9
False1() -> 10
True1() -> 9
True1() -> 10
False2() -> 2
False2() -> 9
False2() -> 10
True2() -> 2
True2() -> 9
True2() -> 10
0 -> 3
0 -> 4
0 -> 5

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(6)
BOUNDS(1, n^1)