WORST_CASE(Omega(n^2), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, n^2).

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (4) CdtProblem
    (5) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 58 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 38 ms]
    (12) CdtProblem
    (13) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) BOUNDS(1, 1)
(15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 0 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 253 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 16 ms]
            (28) proven lower bound
            (29) LowerBoundPropagationProof [FINISHED, 0 ms]
            (30) BOUNDS(n^2, INF)


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(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, n^2).


The TRS R consists of the following rules:

   and(tt, X) -> activate(X)
   plus(N, 0) -> N
   plus(N, s(M)) -> s(plus(N, M))
   x(N, 0) -> 0
   x(N, s(M)) -> plus(x(N, M), N)
   activate(X) -> X

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

(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

The duplicating contexts are:
x([], s(M))


The defined contexts are:
plus([], x1)


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   and(tt, X) -> activate(X)
   plus(N, 0) -> N
   plus(N, s(M)) -> s(plus(N, M))
   x(N, 0) -> 0
   x(N, s(M)) -> plus(x(N, M), N)
   activate(X) -> X

S is empty.
Rewrite Strategy: INNERMOST
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(3) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   and(tt, z0) -> activate(z0)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   x(z0, 0) -> 0
   x(z0, s(z1)) -> plus(x(z0, z1), z0)
   activate(z0) -> z0
Tuples:
   AND(tt, z0) -> c(ACTIVATE(z0))
   PLUS(z0, 0) -> c1
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, 0) -> c3
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
   ACTIVATE(z0) -> c5
S tuples:
   AND(tt, z0) -> c(ACTIVATE(z0))
   PLUS(z0, 0) -> c1
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, 0) -> c3
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
   ACTIVATE(z0) -> c5
K tuples:none
Defined Rule Symbols:   and_2, plus_2, x_2, activate_1

Defined Pair Symbols:   AND_2, PLUS_2, X_2, ACTIVATE_1

Compound Symbols:   c_1, c1, c2_1, c3, c4_2, c5


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(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   ACTIVATE(z0) -> c5
   X(z0, 0) -> c3
   AND(tt, z0) -> c(ACTIVATE(z0))
   PLUS(z0, 0) -> c1

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   and(tt, z0) -> activate(z0)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   x(z0, 0) -> 0
   x(z0, s(z1)) -> plus(x(z0, z1), z0)
   activate(z0) -> z0
Tuples:
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
S tuples:
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
K tuples:none
Defined Rule Symbols:   and_2, plus_2, x_2, activate_1

Defined Pair Symbols:   PLUS_2, X_2

Compound Symbols:   c2_1, c4_2


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   and(tt, z0) -> activate(z0)
   activate(z0) -> z0

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   x(z0, 0) -> 0
   x(z0, s(z1)) -> plus(x(z0, z1), z0)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
S tuples:
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
K tuples:none
Defined Rule Symbols:   x_2, plus_2

Defined Pair Symbols:   PLUS_2, X_2

Compound Symbols:   c2_1, c4_2


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(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(PLUS(x_1, x_2)) = 0
   POL(X(x_1, x_2)) = x_2
   POL(c2(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(plus(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = [1] + x_1
   POL(x(x_1, x_2)) = [1] + x_1 + x_2

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(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   x(z0, 0) -> 0
   x(z0, s(z1)) -> plus(x(z0, z1), z0)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
S tuples:
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
K tuples:
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
Defined Rule Symbols:   x_2, plus_2

Defined Pair Symbols:   PLUS_2, X_2

Compound Symbols:   c2_1, c4_2


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

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(PLUS(x_1, x_2)) = [2] + [2]x_2
   POL(X(x_1, x_2)) = x_2^2 + [2]x_1*x_2
   POL(c2(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(plus(x_1, x_2)) = [2] + x_2^2
   POL(s(x_1)) = [2] + x_1
   POL(x(x_1, x_2)) = x_1^2

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(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   x(z0, 0) -> 0
   x(z0, s(z1)) -> plus(x(z0, z1), z0)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
S tuples:none
K tuples:
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
Defined Rule Symbols:   x_2, plus_2

Defined Pair Symbols:   PLUS_2, X_2

Compound Symbols:   c2_1, c4_2


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(13) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(14)
BOUNDS(1, 1)

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(15) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(16)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   and(tt, X) -> activate(X)
   plus(N, 0') -> N
   plus(N, s(M)) -> s(plus(N, M))
   x(N, 0') -> 0'
   x(N, s(M)) -> plus(x(N, M), N)
   activate(X) -> X

S is empty.
Rewrite Strategy: FULL
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(17) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(18)
Obligation:
TRS:
Rules:
and(tt, X) -> activate(X)
plus(N, 0') -> N
plus(N, s(M)) -> s(plus(N, M))
x(N, 0') -> 0'
x(N, s(M)) -> plus(x(N, M), N)
activate(X) -> X

Types:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
x :: 0':s -> 0':s -> 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s

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(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
plus, x

They will be analysed ascendingly in the following order:
plus < x

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(20)
Obligation:
TRS:
Rules:
and(tt, X) -> activate(X)
plus(N, 0') -> N
plus(N, s(M)) -> s(plus(N, M))
x(N, 0') -> 0'
x(N, s(M)) -> plus(x(N, M), N)
activate(X) -> X

Types:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
x :: 0':s -> 0':s -> 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
plus, x

They will be analysed ascendingly in the following order:
plus < x

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(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) -> gen_0':s4_0(+(n6_0, a)), rt in Omega(1 + n6_0)

Induction Base:
plus(gen_0':s4_0(a), gen_0':s4_0(0)) ->_R^Omega(1)
gen_0':s4_0(a)

Induction Step:
plus(gen_0':s4_0(a), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1)
s(plus(gen_0':s4_0(a), gen_0':s4_0(n6_0))) ->_IH
s(gen_0':s4_0(+(a, c7_0)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(22)
Complex Obligation (BEST)

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(23)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
and(tt, X) -> activate(X)
plus(N, 0') -> N
plus(N, s(M)) -> s(plus(N, M))
x(N, 0') -> 0'
x(N, s(M)) -> plus(x(N, M), N)
activate(X) -> X

Types:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
x :: 0':s -> 0':s -> 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
plus, x

They will be analysed ascendingly in the following order:
plus < x

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(24) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(25)
BOUNDS(n^1, INF)

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

(26)
Obligation:
TRS:
Rules:
and(tt, X) -> activate(X)
plus(N, 0') -> N
plus(N, s(M)) -> s(plus(N, M))
x(N, 0') -> 0'
x(N, s(M)) -> plus(x(N, M), N)
activate(X) -> X

Types:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
x :: 0':s -> 0':s -> 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) -> gen_0':s4_0(+(n6_0, a)), rt in Omega(1 + n6_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
x
----------------------------------------

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
x(gen_0':s4_0(a), gen_0':s4_0(n441_0)) -> gen_0':s4_0(*(n441_0, a)), rt in Omega(1 + a*n441_0 + n441_0)

Induction Base:
x(gen_0':s4_0(a), gen_0':s4_0(0)) ->_R^Omega(1)
0'

Induction Step:
x(gen_0':s4_0(a), gen_0':s4_0(+(n441_0, 1))) ->_R^Omega(1)
plus(x(gen_0':s4_0(a), gen_0':s4_0(n441_0)), gen_0':s4_0(a)) ->_IH
plus(gen_0':s4_0(*(c442_0, a)), gen_0':s4_0(a)) ->_L^Omega(1 + a)
gen_0':s4_0(+(a, *(n441_0, a)))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(28)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
and(tt, X) -> activate(X)
plus(N, 0') -> N
plus(N, s(M)) -> s(plus(N, M))
x(N, 0') -> 0'
x(N, s(M)) -> plus(x(N, M), N)
activate(X) -> X

Types:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
x :: 0':s -> 0':s -> 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) -> gen_0':s4_0(+(n6_0, a)), rt in Omega(1 + n6_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
x
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(29) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(30)
BOUNDS(n^2, INF)