WORST_CASE(?,O(n^1))
proof of input_g0IfeJWQ7m.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
(7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(8) CdtProblem
(9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CdtProblem
(11) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTRS
(13) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(14) CpxTRS
(15) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
(16) BOUNDS(1, n^1)


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(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:

   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
   from(X) -> cons(X, n__from(n__s(X)))
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
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(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(n__s(z0)))
   from(z0) -> n__from(z0)
   s(z0) -> n__s(z0)
   activate(n__from(z0)) -> from(activate(z0))
   activate(n__s(z0)) -> s(activate(z0))
   activate(z0) -> z0
Tuples:
   2ND(cons1(z0, cons(z1, z2))) -> c
   2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, activate(z1))), ACTIVATE(z1))
   FROM(z0) -> c2
   FROM(z0) -> c3
   S(z0) -> c4
   ACTIVATE(n__from(z0)) -> c5(FROM(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(S(activate(z0)), ACTIVATE(z0))
   ACTIVATE(z0) -> c7
S tuples:
   2ND(cons1(z0, cons(z1, z2))) -> c
   2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, activate(z1))), ACTIVATE(z1))
   FROM(z0) -> c2
   FROM(z0) -> c3
   S(z0) -> c4
   ACTIVATE(n__from(z0)) -> c5(FROM(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(S(activate(z0)), ACTIVATE(z0))
   ACTIVATE(z0) -> c7
K tuples:none
Defined Rule Symbols:   2nd_1, from_1, s_1, activate_1

Defined Pair Symbols:   2ND_1, FROM_1, S_1, ACTIVATE_1

Compound Symbols:   c, c1_2, c2, c3, c4, c5_2, c6_2, c7


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing nodes:
   FROM(z0) -> c3
   2ND(cons1(z0, cons(z1, z2))) -> c
   ACTIVATE(z0) -> c7
   S(z0) -> c4
   FROM(z0) -> c2

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(n__s(z0)))
   from(z0) -> n__from(z0)
   s(z0) -> n__s(z0)
   activate(n__from(z0)) -> from(activate(z0))
   activate(n__s(z0)) -> s(activate(z0))
   activate(z0) -> z0
Tuples:
   2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, activate(z1))), ACTIVATE(z1))
   ACTIVATE(n__from(z0)) -> c5(FROM(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(S(activate(z0)), ACTIVATE(z0))
S tuples:
   2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, activate(z1))), ACTIVATE(z1))
   ACTIVATE(n__from(z0)) -> c5(FROM(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:   2nd_1, from_1, s_1, activate_1

Defined Pair Symbols:   2ND_1, ACTIVATE_1

Compound Symbols:   c1_2, c5_2, c6_2


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

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(n__s(z0)))
   from(z0) -> n__from(z0)
   s(z0) -> n__s(z0)
   activate(n__from(z0)) -> from(activate(z0))
   activate(n__s(z0)) -> s(activate(z0))
   activate(z0) -> z0
Tuples:
   2ND(cons(z0, z1)) -> c1(ACTIVATE(z1))
   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))
S tuples:
   2ND(cons(z0, z1)) -> c1(ACTIVATE(z1))
   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:   2nd_1, from_1, s_1, activate_1

Defined Pair Symbols:   2ND_1, ACTIVATE_1

Compound Symbols:   c1_1, c5_1, c6_1


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   2ND(cons(z0, z1)) -> c1(ACTIVATE(z1))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(n__s(z0)))
   from(z0) -> n__from(z0)
   s(z0) -> n__s(z0)
   activate(n__from(z0)) -> from(activate(z0))
   activate(n__s(z0)) -> s(activate(z0))
   activate(z0) -> z0
Tuples:
   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))
S tuples:
   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:   2nd_1, from_1, s_1, activate_1

Defined Pair Symbols:   ACTIVATE_1

Compound Symbols:   c5_1, c6_1


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

(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(n__s(z0)))
   from(z0) -> n__from(z0)
   s(z0) -> n__s(z0)
   activate(n__from(z0)) -> from(activate(z0))
   activate(n__s(z0)) -> s(activate(z0))
   activate(z0) -> z0

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))
S tuples:
   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   ACTIVATE_1

Compound Symbols:   c5_1, c6_1


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

(12)
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:

   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))

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

(14)
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:

   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))

S is empty.
Rewrite Strategy: INNERMOST
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(15) CpxTrsMatchBoundsProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. 
The certificate found is represented by the following graph.

"[100, 101, 102, 103]
{(100,101,[ACTIVATE_1|0]), (100,102,[c5_1|1]), (100,103,[c6_1|1]), (101,101,[n__from_1|0, c5_1|0, n__s_1|0, c6_1|0]), (102,101,[ACTIVATE_1|1]), (102,102,[c5_1|1]), (102,103,[c6_1|1]), (103,101,[ACTIVATE_1|1]), (103,102,[c5_1|1]), (103,103,[c6_1|1])}"
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(16)
BOUNDS(1, n^1)