WORST_CASE(?,O(n^1))
proof of input_CNq0TWYTPa.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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 [ComplexityIfPolyImplication, 0 ms]
(4) CdtProblem
(5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
(7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 48 ms]
(8) CdtProblem
(9) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) BOUNDS(1, 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:

   plus_x#1(0, x8) -> x8
   plus_x#1(S(x12), x14) -> S(plus_x#1(x12, x14))
   map#2(plus_x(x2), Nil) -> Nil
   map#2(plus_x(x6), Cons(x4, x2)) -> Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2))
   main(x5, x12) -> map#2(plus_x(x12), x5)

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:
   plus_x#1(0, z0) -> z0
   plus_x#1(S(z0), z1) -> S(plus_x#1(z0, z1))
   map#2(plus_x(z0), Nil) -> Nil
   map#2(plus_x(z0), Cons(z1, z2)) -> Cons(plus_x#1(z0, z1), map#2(plus_x(z0), z2))
   main(z0, z1) -> map#2(plus_x(z1), z0)
Tuples:
   PLUS_X#1(0, z0) -> c
   PLUS_X#1(S(z0), z1) -> c1(PLUS_X#1(z0, z1))
   MAP#2(plus_x(z0), Nil) -> c2
   MAP#2(plus_x(z0), Cons(z1, z2)) -> c3(PLUS_X#1(z0, z1))
   MAP#2(plus_x(z0), Cons(z1, z2)) -> c4(MAP#2(plus_x(z0), z2))
   MAIN(z0, z1) -> c5(MAP#2(plus_x(z1), z0))
S tuples:
   PLUS_X#1(0, z0) -> c
   PLUS_X#1(S(z0), z1) -> c1(PLUS_X#1(z0, z1))
   MAP#2(plus_x(z0), Nil) -> c2
   MAP#2(plus_x(z0), Cons(z1, z2)) -> c3(PLUS_X#1(z0, z1))
   MAP#2(plus_x(z0), Cons(z1, z2)) -> c4(MAP#2(plus_x(z0), z2))
   MAIN(z0, z1) -> c5(MAP#2(plus_x(z1), z0))
K tuples:none
Defined Rule Symbols:   plus_x#1_2, map#2_2, main_2

Defined Pair Symbols:   PLUS_X#1_2, MAP#2_2, MAIN_2

Compound Symbols:   c, c1_1, c2, c3_1, c4_1, c5_1


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(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   MAIN(z0, z1) -> c5(MAP#2(plus_x(z1), z0))
Removed 2 trailing nodes:
   MAP#2(plus_x(z0), Nil) -> c2
   PLUS_X#1(0, z0) -> c

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus_x#1(0, z0) -> z0
   plus_x#1(S(z0), z1) -> S(plus_x#1(z0, z1))
   map#2(plus_x(z0), Nil) -> Nil
   map#2(plus_x(z0), Cons(z1, z2)) -> Cons(plus_x#1(z0, z1), map#2(plus_x(z0), z2))
   main(z0, z1) -> map#2(plus_x(z1), z0)
Tuples:
   PLUS_X#1(S(z0), z1) -> c1(PLUS_X#1(z0, z1))
   MAP#2(plus_x(z0), Cons(z1, z2)) -> c3(PLUS_X#1(z0, z1))
   MAP#2(plus_x(z0), Cons(z1, z2)) -> c4(MAP#2(plus_x(z0), z2))
S tuples:
   PLUS_X#1(S(z0), z1) -> c1(PLUS_X#1(z0, z1))
   MAP#2(plus_x(z0), Cons(z1, z2)) -> c3(PLUS_X#1(z0, z1))
   MAP#2(plus_x(z0), Cons(z1, z2)) -> c4(MAP#2(plus_x(z0), z2))
K tuples:none
Defined Rule Symbols:   plus_x#1_2, map#2_2, main_2

Defined Pair Symbols:   PLUS_X#1_2, MAP#2_2

Compound Symbols:   c1_1, c3_1, c4_1


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(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   plus_x#1(0, z0) -> z0
   plus_x#1(S(z0), z1) -> S(plus_x#1(z0, z1))
   map#2(plus_x(z0), Nil) -> Nil
   map#2(plus_x(z0), Cons(z1, z2)) -> Cons(plus_x#1(z0, z1), map#2(plus_x(z0), z2))
   main(z0, z1) -> map#2(plus_x(z1), z0)

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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   PLUS_X#1(S(z0), z1) -> c1(PLUS_X#1(z0, z1))
   MAP#2(plus_x(z0), Cons(z1, z2)) -> c3(PLUS_X#1(z0, z1))
   MAP#2(plus_x(z0), Cons(z1, z2)) -> c4(MAP#2(plus_x(z0), z2))
S tuples:
   PLUS_X#1(S(z0), z1) -> c1(PLUS_X#1(z0, z1))
   MAP#2(plus_x(z0), Cons(z1, z2)) -> c3(PLUS_X#1(z0, z1))
   MAP#2(plus_x(z0), Cons(z1, z2)) -> c4(MAP#2(plus_x(z0), z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   PLUS_X#1_2, MAP#2_2

Compound Symbols:   c1_1, c3_1, c4_1


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(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PLUS_X#1(S(z0), z1) -> c1(PLUS_X#1(z0, z1))
   MAP#2(plus_x(z0), Cons(z1, z2)) -> c3(PLUS_X#1(z0, z1))
   MAP#2(plus_x(z0), Cons(z1, z2)) -> c4(MAP#2(plus_x(z0), z2))
We considered the (Usable) Rules:none
And the Tuples:
   PLUS_X#1(S(z0), z1) -> c1(PLUS_X#1(z0, z1))
   MAP#2(plus_x(z0), Cons(z1, z2)) -> c3(PLUS_X#1(z0, z1))
   MAP#2(plus_x(z0), Cons(z1, z2)) -> c4(MAP#2(plus_x(z0), z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(Cons(x_1, x_2)) = [3] + x_1 + x_2
   POL(MAP#2(x_1, x_2)) = [3] + [2]x_1 + [3]x_2
   POL(PLUS_X#1(x_1, x_2)) = [2]x_1 + [3]x_2
   POL(S(x_1)) = [3] + x_1
   POL(c1(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(plus_x(x_1)) = [3] + x_1

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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   PLUS_X#1(S(z0), z1) -> c1(PLUS_X#1(z0, z1))
   MAP#2(plus_x(z0), Cons(z1, z2)) -> c3(PLUS_X#1(z0, z1))
   MAP#2(plus_x(z0), Cons(z1, z2)) -> c4(MAP#2(plus_x(z0), z2))
S tuples:none
K tuples:
   PLUS_X#1(S(z0), z1) -> c1(PLUS_X#1(z0, z1))
   MAP#2(plus_x(z0), Cons(z1, z2)) -> c3(PLUS_X#1(z0, z1))
   MAP#2(plus_x(z0), Cons(z1, z2)) -> c4(MAP#2(plus_x(z0), z2))
Defined Rule Symbols:none

Defined Pair Symbols:   PLUS_X#1_2, MAP#2_2

Compound Symbols:   c1_1, c3_1, c4_1


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(9) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(10)
BOUNDS(1, 1)