WORST_CASE(?,O(n^1))
proof of input_nwjOTeb2e9.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) 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, 11 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:

   foldl(x, Cons(S(0), xs)) -> foldl(S(x), xs)
   foldl(S(0), Cons(x, xs)) -> foldl(S(x), xs)
   foldr(a, Cons(x, xs)) -> op(x, foldr(a, xs))
   foldr(a, Nil) -> a
   foldl(a, Nil) -> a
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   op(x, S(0)) -> S(x)
   op(S(0), y) -> S(y)
   fold(a, xs) -> Cons(foldl(a, xs), Cons(foldr(a, xs), Nil))

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:
   foldl(z0, Cons(S(0), z1)) -> foldl(S(z0), z1)
   foldl(S(0), Cons(z0, z1)) -> foldl(S(z0), z1)
   foldl(z0, Nil) -> z0
   foldr(z0, Cons(z1, z2)) -> op(z1, foldr(z0, z2))
   foldr(z0, Nil) -> z0
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   op(z0, S(0)) -> S(z0)
   op(S(0), z0) -> S(z0)
   fold(z0, z1) -> Cons(foldl(z0, z1), Cons(foldr(z0, z1), Nil))
Tuples:
   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDL(z0, Nil) -> c2
   FOLDR(z0, Cons(z1, z2)) -> c3(OP(z1, foldr(z0, z2)), FOLDR(z0, z2))
   FOLDR(z0, Nil) -> c4
   NOTEMPTY(Cons(z0, z1)) -> c5
   NOTEMPTY(Nil) -> c6
   OP(z0, S(0)) -> c7
   OP(S(0), z0) -> c8
   FOLD(z0, z1) -> c9(FOLDL(z0, z1))
   FOLD(z0, z1) -> c10(FOLDR(z0, z1))
S tuples:
   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDL(z0, Nil) -> c2
   FOLDR(z0, Cons(z1, z2)) -> c3(OP(z1, foldr(z0, z2)), FOLDR(z0, z2))
   FOLDR(z0, Nil) -> c4
   NOTEMPTY(Cons(z0, z1)) -> c5
   NOTEMPTY(Nil) -> c6
   OP(z0, S(0)) -> c7
   OP(S(0), z0) -> c8
   FOLD(z0, z1) -> c9(FOLDL(z0, z1))
   FOLD(z0, z1) -> c10(FOLDR(z0, z1))
K tuples:none
Defined Rule Symbols:   foldl_2, foldr_2, notEmpty_1, op_2, fold_2

Defined Pair Symbols:   FOLDL_2, FOLDR_2, NOTEMPTY_1, OP_2, FOLD_2

Compound Symbols:   c_1, c1_1, c2, c3_2, c4, c5, c6, c7, c8, c9_1, c10_1


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(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   FOLD(z0, z1) -> c9(FOLDL(z0, z1))
   FOLD(z0, z1) -> c10(FOLDR(z0, z1))
Removed 6 trailing nodes:
   NOTEMPTY(Cons(z0, z1)) -> c5
   NOTEMPTY(Nil) -> c6
   OP(z0, S(0)) -> c7
   FOLDL(z0, Nil) -> c2
   FOLDR(z0, Nil) -> c4
   OP(S(0), z0) -> c8

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   foldl(z0, Cons(S(0), z1)) -> foldl(S(z0), z1)
   foldl(S(0), Cons(z0, z1)) -> foldl(S(z0), z1)
   foldl(z0, Nil) -> z0
   foldr(z0, Cons(z1, z2)) -> op(z1, foldr(z0, z2))
   foldr(z0, Nil) -> z0
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   op(z0, S(0)) -> S(z0)
   op(S(0), z0) -> S(z0)
   fold(z0, z1) -> Cons(foldl(z0, z1), Cons(foldr(z0, z1), Nil))
Tuples:
   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDR(z0, Cons(z1, z2)) -> c3(OP(z1, foldr(z0, z2)), FOLDR(z0, z2))
S tuples:
   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDR(z0, Cons(z1, z2)) -> c3(OP(z1, foldr(z0, z2)), FOLDR(z0, z2))
K tuples:none
Defined Rule Symbols:   foldl_2, foldr_2, notEmpty_1, op_2, fold_2

Defined Pair Symbols:   FOLDL_2, FOLDR_2

Compound Symbols:   c_1, c1_1, c3_2


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(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   foldl(z0, Cons(S(0), z1)) -> foldl(S(z0), z1)
   foldl(S(0), Cons(z0, z1)) -> foldl(S(z0), z1)
   foldl(z0, Nil) -> z0
   foldr(z0, Cons(z1, z2)) -> op(z1, foldr(z0, z2))
   foldr(z0, Nil) -> z0
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   op(z0, S(0)) -> S(z0)
   op(S(0), z0) -> S(z0)
   fold(z0, z1) -> Cons(foldl(z0, z1), Cons(foldr(z0, z1), Nil))
Tuples:
   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDR(z0, Cons(z1, z2)) -> c3(FOLDR(z0, z2))
S tuples:
   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDR(z0, Cons(z1, z2)) -> c3(FOLDR(z0, z2))
K tuples:none
Defined Rule Symbols:   foldl_2, foldr_2, notEmpty_1, op_2, fold_2

Defined Pair Symbols:   FOLDL_2, FOLDR_2

Compound Symbols:   c_1, c1_1, c3_1


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(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   foldl(z0, Cons(S(0), z1)) -> foldl(S(z0), z1)
   foldl(S(0), Cons(z0, z1)) -> foldl(S(z0), z1)
   foldl(z0, Nil) -> z0
   foldr(z0, Cons(z1, z2)) -> op(z1, foldr(z0, z2))
   foldr(z0, Nil) -> z0
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   op(z0, S(0)) -> S(z0)
   op(S(0), z0) -> S(z0)
   fold(z0, z1) -> Cons(foldl(z0, z1), Cons(foldr(z0, z1), Nil))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDR(z0, Cons(z1, z2)) -> c3(FOLDR(z0, z2))
S tuples:
   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDR(z0, Cons(z1, z2)) -> c3(FOLDR(z0, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   FOLDL_2, FOLDR_2

Compound Symbols:   c_1, c1_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.
----------------------------------------

(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:

   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDR(z0, Cons(z1, z2)) -> c3(FOLDR(z0, z2))

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:

   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDR(z0, Cons(z1, z2)) -> c3(FOLDR(z0, z2))

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: 
Cons0(0, 0) -> 0
S0(0) -> 0
00() -> 0
c0(0) -> 0
c10(0) -> 0
c30(0) -> 0
FOLDL0(0, 0) -> 1
FOLDR0(0, 0) -> 2
S1(0) -> 4
FOLDL1(4, 0) -> 3
c1(3) -> 1
FOLDL1(4, 0) -> 5
c11(5) -> 1
FOLDR1(0, 0) -> 6
c31(6) -> 2
S1(4) -> 4
c1(3) -> 3
c1(3) -> 5
c11(5) -> 3
c11(5) -> 5
c31(6) -> 6

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(14)
BOUNDS(1, n^1)