WORST_CASE(?,O(n^1))
proof of input_78HIzt01SV.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CdtProblem
(9) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxTRS
(11) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(12) CpxTRS
(13) CpxTrsMatchBoundsTAProof [FINISHED, 26 ms]
(14) 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:

   merge(nil, y) -> y
   merge(x, nil) -> x
   merge(.(x, y), .(u, v)) -> if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
   ++(nil, y) -> y
   ++(.(x, y), z) -> .(x, ++(y, z))
   if(true, x, y) -> x
   if(false, x, y) -> 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:
   merge(nil, z0) -> z0
   merge(z0, nil) -> z0
   merge(.(z0, z1), .(z2, z3)) -> if(<(z0, z2), .(z0, merge(z1, .(z2, z3))), .(z2, merge(.(z0, z1), z3)))
   ++(nil, z0) -> z0
   ++(.(z0, z1), z2) -> .(z0, ++(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z0
Tuples:
   MERGE(nil, z0) -> c
   MERGE(z0, nil) -> c1
   MERGE(.(z0, z1), .(z2, z3)) -> c2(IF(<(z0, z2), .(z0, merge(z1, .(z2, z3))), .(z2, merge(.(z0, z1), z3))), MERGE(z1, .(z2, z3)))
   MERGE(.(z0, z1), .(z2, z3)) -> c3(IF(<(z0, z2), .(z0, merge(z1, .(z2, z3))), .(z2, merge(.(z0, z1), z3))), MERGE(.(z0, z1), z3))
   ++'(nil, z0) -> c4
   ++'(.(z0, z1), z2) -> c5(++'(z1, z2))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7
S tuples:
   MERGE(nil, z0) -> c
   MERGE(z0, nil) -> c1
   MERGE(.(z0, z1), .(z2, z3)) -> c2(IF(<(z0, z2), .(z0, merge(z1, .(z2, z3))), .(z2, merge(.(z0, z1), z3))), MERGE(z1, .(z2, z3)))
   MERGE(.(z0, z1), .(z2, z3)) -> c3(IF(<(z0, z2), .(z0, merge(z1, .(z2, z3))), .(z2, merge(.(z0, z1), z3))), MERGE(.(z0, z1), z3))
   ++'(nil, z0) -> c4
   ++'(.(z0, z1), z2) -> c5(++'(z1, z2))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7
K tuples:none
Defined Rule Symbols:   merge_2, ++_2, if_3

Defined Pair Symbols:   MERGE_2, ++'_2, IF_3

Compound Symbols:   c, c1, c2_2, c3_2, c4, c5_1, c6, c7


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing nodes:
   ++'(nil, z0) -> c4
   IF(true, z0, z1) -> c6
   MERGE(z0, nil) -> c1
   IF(false, z0, z1) -> c7
   MERGE(nil, z0) -> c

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   merge(nil, z0) -> z0
   merge(z0, nil) -> z0
   merge(.(z0, z1), .(z2, z3)) -> if(<(z0, z2), .(z0, merge(z1, .(z2, z3))), .(z2, merge(.(z0, z1), z3)))
   ++(nil, z0) -> z0
   ++(.(z0, z1), z2) -> .(z0, ++(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z0
Tuples:
   MERGE(.(z0, z1), .(z2, z3)) -> c2(IF(<(z0, z2), .(z0, merge(z1, .(z2, z3))), .(z2, merge(.(z0, z1), z3))), MERGE(z1, .(z2, z3)))
   MERGE(.(z0, z1), .(z2, z3)) -> c3(IF(<(z0, z2), .(z0, merge(z1, .(z2, z3))), .(z2, merge(.(z0, z1), z3))), MERGE(.(z0, z1), z3))
   ++'(.(z0, z1), z2) -> c5(++'(z1, z2))
S tuples:
   MERGE(.(z0, z1), .(z2, z3)) -> c2(IF(<(z0, z2), .(z0, merge(z1, .(z2, z3))), .(z2, merge(.(z0, z1), z3))), MERGE(z1, .(z2, z3)))
   MERGE(.(z0, z1), .(z2, z3)) -> c3(IF(<(z0, z2), .(z0, merge(z1, .(z2, z3))), .(z2, merge(.(z0, z1), z3))), MERGE(.(z0, z1), z3))
   ++'(.(z0, z1), z2) -> c5(++'(z1, z2))
K tuples:none
Defined Rule Symbols:   merge_2, ++_2, if_3

Defined Pair Symbols:   MERGE_2, ++'_2

Compound Symbols:   c2_2, c3_2, c5_1


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(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   merge(nil, z0) -> z0
   merge(z0, nil) -> z0
   merge(.(z0, z1), .(z2, z3)) -> if(<(z0, z2), .(z0, merge(z1, .(z2, z3))), .(z2, merge(.(z0, z1), z3)))
   ++(nil, z0) -> z0
   ++(.(z0, z1), z2) -> .(z0, ++(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z0
Tuples:
   ++'(.(z0, z1), z2) -> c5(++'(z1, z2))
   MERGE(.(z0, z1), .(z2, z3)) -> c2(MERGE(z1, .(z2, z3)))
   MERGE(.(z0, z1), .(z2, z3)) -> c3(MERGE(.(z0, z1), z3))
S tuples:
   ++'(.(z0, z1), z2) -> c5(++'(z1, z2))
   MERGE(.(z0, z1), .(z2, z3)) -> c2(MERGE(z1, .(z2, z3)))
   MERGE(.(z0, z1), .(z2, z3)) -> c3(MERGE(.(z0, z1), z3))
K tuples:none
Defined Rule Symbols:   merge_2, ++_2, if_3

Defined Pair Symbols:   ++'_2, MERGE_2

Compound Symbols:   c5_1, c2_1, c3_1


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(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   merge(nil, z0) -> z0
   merge(z0, nil) -> z0
   merge(.(z0, z1), .(z2, z3)) -> if(<(z0, z2), .(z0, merge(z1, .(z2, z3))), .(z2, merge(.(z0, z1), z3)))
   ++(nil, z0) -> z0
   ++(.(z0, z1), z2) -> .(z0, ++(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z0

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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   ++'(.(z0, z1), z2) -> c5(++'(z1, z2))
   MERGE(.(z0, z1), .(z2, z3)) -> c2(MERGE(z1, .(z2, z3)))
   MERGE(.(z0, z1), .(z2, z3)) -> c3(MERGE(.(z0, z1), z3))
S tuples:
   ++'(.(z0, z1), z2) -> c5(++'(z1, z2))
   MERGE(.(z0, z1), .(z2, z3)) -> c2(MERGE(z1, .(z2, z3)))
   MERGE(.(z0, z1), .(z2, z3)) -> c3(MERGE(.(z0, z1), z3))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   ++'_2, MERGE_2

Compound Symbols:   c5_1, c2_1, c3_1


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

   ++'(.(z0, z1), z2) -> c5(++'(z1, z2))
   MERGE(.(z0, z1), .(z2, z3)) -> c2(MERGE(z1, .(z2, z3)))
   MERGE(.(z0, z1), .(z2, z3)) -> c3(MERGE(.(z0, z1), z3))

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

   ++'(.(z0, z1), z2) -> c5(++'(z1, z2))
   MERGE(.(z0, z1), .(z2, z3)) -> c2(MERGE(z1, .(z2, z3)))
   MERGE(.(z0, z1), .(z2, z3)) -> c3(MERGE(.(z0, z1), z3))

S is empty.
Rewrite Strategy: INNERMOST
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(13) 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]
transitions: 
.0(0, 0) -> 0
c50(0) -> 0
c20(0) -> 0
c30(0) -> 0
++'0(0, 0) -> 1
MERGE0(0, 0) -> 2
++'1(0, 0) -> 3
c51(3) -> 1
.1(0, 0) -> 5
MERGE1(0, 5) -> 4
c21(4) -> 2
.1(0, 0) -> 7
MERGE1(7, 0) -> 6
c31(6) -> 2
c51(3) -> 3
c21(4) -> 4
c21(4) -> 6
c31(6) -> 4
c31(6) -> 6

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(14)
BOUNDS(1, n^1)