WORST_CASE(?,O(n^1))
proof of input_DcsXBOniet.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


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

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
(3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CdtProblem
(5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
(7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTRS
(9) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(10) CpxTRS
(11) CpxTrsMatchBoundsTAProof [FINISHED, 30 ms]
(12) BOUNDS(1, n^1)


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

(0)
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:

   fst(0, Z) -> nil
   fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z)))
   from(X) -> cons(X, n__from(s(X)))
   add(0, X) -> X
   add(s(X), Y) -> s(n__add(activate(X), Y))
   len(nil) -> 0
   len(cons(X, Z)) -> s(n__len(activate(Z)))
   fst(X1, X2) -> n__fst(X1, X2)
   from(X) -> n__from(X)
   add(X1, X2) -> n__add(X1, X2)
   len(X) -> n__len(X)
   activate(n__fst(X1, X2)) -> fst(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(n__add(X1, X2)) -> add(X1, X2)
   activate(n__len(X)) -> len(X)
   activate(X) -> X

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

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fst(0, z0) -> nil
   fst(s(z0), cons(z1, z2)) -> cons(z1, n__fst(activate(z0), activate(z2)))
   fst(z0, z1) -> n__fst(z0, z1)
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   add(0, z0) -> z0
   add(s(z0), z1) -> s(n__add(activate(z0), z1))
   add(z0, z1) -> n__add(z0, z1)
   len(nil) -> 0
   len(cons(z0, z1)) -> s(n__len(activate(z1)))
   len(z0) -> n__len(z0)
   activate(n__fst(z0, z1)) -> fst(z0, z1)
   activate(n__from(z0)) -> from(z0)
   activate(n__add(z0, z1)) -> add(z0, z1)
   activate(n__len(z0)) -> len(z0)
   activate(z0) -> z0
Tuples:
   FST(0, z0) -> c
   FST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z0))
   FST(s(z0), cons(z1, z2)) -> c2(ACTIVATE(z2))
   FST(z0, z1) -> c3
   FROM(z0) -> c4
   FROM(z0) -> c5
   ADD(0, z0) -> c6
   ADD(s(z0), z1) -> c7(ACTIVATE(z0))
   ADD(z0, z1) -> c8
   LEN(nil) -> c9
   LEN(cons(z0, z1)) -> c10(ACTIVATE(z1))
   LEN(z0) -> c11
   ACTIVATE(n__fst(z0, z1)) -> c12(FST(z0, z1))
   ACTIVATE(n__from(z0)) -> c13(FROM(z0))
   ACTIVATE(n__add(z0, z1)) -> c14(ADD(z0, z1))
   ACTIVATE(n__len(z0)) -> c15(LEN(z0))
   ACTIVATE(z0) -> c16
S tuples:
   FST(0, z0) -> c
   FST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z0))
   FST(s(z0), cons(z1, z2)) -> c2(ACTIVATE(z2))
   FST(z0, z1) -> c3
   FROM(z0) -> c4
   FROM(z0) -> c5
   ADD(0, z0) -> c6
   ADD(s(z0), z1) -> c7(ACTIVATE(z0))
   ADD(z0, z1) -> c8
   LEN(nil) -> c9
   LEN(cons(z0, z1)) -> c10(ACTIVATE(z1))
   LEN(z0) -> c11
   ACTIVATE(n__fst(z0, z1)) -> c12(FST(z0, z1))
   ACTIVATE(n__from(z0)) -> c13(FROM(z0))
   ACTIVATE(n__add(z0, z1)) -> c14(ADD(z0, z1))
   ACTIVATE(n__len(z0)) -> c15(LEN(z0))
   ACTIVATE(z0) -> c16
K tuples:none
Defined Rule Symbols:   fst_2, from_1, add_2, len_1, activate_1

Defined Pair Symbols:   FST_2, FROM_1, ADD_2, LEN_1, ACTIVATE_1

Compound Symbols:   c, c1_1, c2_1, c3, c4, c5, c6, c7_1, c8, c9, c10_1, c11, c12_1, c13_1, c14_1, c15_1, c16


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 10 trailing nodes:
   ACTIVATE(z0) -> c16
   ADD(z0, z1) -> c8
   LEN(z0) -> c11
   ACTIVATE(n__from(z0)) -> c13(FROM(z0))
   ADD(0, z0) -> c6
   FROM(z0) -> c4
   FROM(z0) -> c5
   FST(z0, z1) -> c3
   LEN(nil) -> c9
   FST(0, z0) -> c

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fst(0, z0) -> nil
   fst(s(z0), cons(z1, z2)) -> cons(z1, n__fst(activate(z0), activate(z2)))
   fst(z0, z1) -> n__fst(z0, z1)
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   add(0, z0) -> z0
   add(s(z0), z1) -> s(n__add(activate(z0), z1))
   add(z0, z1) -> n__add(z0, z1)
   len(nil) -> 0
   len(cons(z0, z1)) -> s(n__len(activate(z1)))
   len(z0) -> n__len(z0)
   activate(n__fst(z0, z1)) -> fst(z0, z1)
   activate(n__from(z0)) -> from(z0)
   activate(n__add(z0, z1)) -> add(z0, z1)
   activate(n__len(z0)) -> len(z0)
   activate(z0) -> z0
Tuples:
   FST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z0))
   FST(s(z0), cons(z1, z2)) -> c2(ACTIVATE(z2))
   ADD(s(z0), z1) -> c7(ACTIVATE(z0))
   LEN(cons(z0, z1)) -> c10(ACTIVATE(z1))
   ACTIVATE(n__fst(z0, z1)) -> c12(FST(z0, z1))
   ACTIVATE(n__add(z0, z1)) -> c14(ADD(z0, z1))
   ACTIVATE(n__len(z0)) -> c15(LEN(z0))
S tuples:
   FST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z0))
   FST(s(z0), cons(z1, z2)) -> c2(ACTIVATE(z2))
   ADD(s(z0), z1) -> c7(ACTIVATE(z0))
   LEN(cons(z0, z1)) -> c10(ACTIVATE(z1))
   ACTIVATE(n__fst(z0, z1)) -> c12(FST(z0, z1))
   ACTIVATE(n__add(z0, z1)) -> c14(ADD(z0, z1))
   ACTIVATE(n__len(z0)) -> c15(LEN(z0))
K tuples:none
Defined Rule Symbols:   fst_2, from_1, add_2, len_1, activate_1

Defined Pair Symbols:   FST_2, ADD_2, LEN_1, ACTIVATE_1

Compound Symbols:   c1_1, c2_1, c7_1, c10_1, c12_1, c14_1, c15_1


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   fst(0, z0) -> nil
   fst(s(z0), cons(z1, z2)) -> cons(z1, n__fst(activate(z0), activate(z2)))
   fst(z0, z1) -> n__fst(z0, z1)
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   add(0, z0) -> z0
   add(s(z0), z1) -> s(n__add(activate(z0), z1))
   add(z0, z1) -> n__add(z0, z1)
   len(nil) -> 0
   len(cons(z0, z1)) -> s(n__len(activate(z1)))
   len(z0) -> n__len(z0)
   activate(n__fst(z0, z1)) -> fst(z0, z1)
   activate(n__from(z0)) -> from(z0)
   activate(n__add(z0, z1)) -> add(z0, z1)
   activate(n__len(z0)) -> len(z0)
   activate(z0) -> z0

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z0))
   FST(s(z0), cons(z1, z2)) -> c2(ACTIVATE(z2))
   ADD(s(z0), z1) -> c7(ACTIVATE(z0))
   LEN(cons(z0, z1)) -> c10(ACTIVATE(z1))
   ACTIVATE(n__fst(z0, z1)) -> c12(FST(z0, z1))
   ACTIVATE(n__add(z0, z1)) -> c14(ADD(z0, z1))
   ACTIVATE(n__len(z0)) -> c15(LEN(z0))
S tuples:
   FST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z0))
   FST(s(z0), cons(z1, z2)) -> c2(ACTIVATE(z2))
   ADD(s(z0), z1) -> c7(ACTIVATE(z0))
   LEN(cons(z0, z1)) -> c10(ACTIVATE(z1))
   ACTIVATE(n__fst(z0, z1)) -> c12(FST(z0, z1))
   ACTIVATE(n__add(z0, z1)) -> c14(ADD(z0, z1))
   ACTIVATE(n__len(z0)) -> c15(LEN(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   FST_2, ADD_2, LEN_1, ACTIVATE_1

Compound Symbols:   c1_1, c2_1, c7_1, c10_1, c12_1, c14_1, c15_1


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

(7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

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


The TRS R consists of the following rules:

   FST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z0))
   FST(s(z0), cons(z1, z2)) -> c2(ACTIVATE(z2))
   ADD(s(z0), z1) -> c7(ACTIVATE(z0))
   LEN(cons(z0, z1)) -> c10(ACTIVATE(z1))
   ACTIVATE(n__fst(z0, z1)) -> c12(FST(z0, z1))
   ACTIVATE(n__add(z0, z1)) -> c14(ADD(z0, z1))
   ACTIVATE(n__len(z0)) -> c15(LEN(z0))

S is empty.
Rewrite Strategy: INNERMOST
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(9) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   FST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z0))
   FST(s(z0), cons(z1, z2)) -> c2(ACTIVATE(z2))
   ADD(s(z0), z1) -> c7(ACTIVATE(z0))
   LEN(cons(z0, z1)) -> c10(ACTIVATE(z1))
   ACTIVATE(n__fst(z0, z1)) -> c12(FST(z0, z1))
   ACTIVATE(n__add(z0, z1)) -> c14(ADD(z0, z1))
   ACTIVATE(n__len(z0)) -> c15(LEN(z0))

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

(11) 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 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4]
transitions: 
s0(0) -> 0
cons0(0, 0) -> 0
c10(0) -> 0
c20(0) -> 0
c70(0) -> 0
c100(0) -> 0
n__fst0(0, 0) -> 0
c120(0) -> 0
n__add0(0, 0) -> 0
c140(0) -> 0
n__len0(0) -> 0
c150(0) -> 0
FST0(0, 0) -> 1
ADD0(0, 0) -> 2
LEN0(0) -> 3
ACTIVATE0(0) -> 4
ACTIVATE1(0) -> 5
c11(5) -> 1
ACTIVATE1(0) -> 6
c21(6) -> 1
ACTIVATE1(0) -> 7
c71(7) -> 2
ACTIVATE1(0) -> 8
c101(8) -> 3
FST1(0, 0) -> 9
c121(9) -> 4
ADD1(0, 0) -> 10
c141(10) -> 4
LEN1(0) -> 11
c151(11) -> 4
c11(5) -> 9
c21(6) -> 9
c71(7) -> 10
c101(8) -> 11
c121(9) -> 5
c121(9) -> 6
c121(9) -> 7
c121(9) -> 8
c141(10) -> 5
c141(10) -> 6
c141(10) -> 7
c141(10) -> 8
c151(11) -> 5
c151(11) -> 6
c151(11) -> 7
c151(11) -> 8

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(12)
BOUNDS(1, n^1)