WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1, n^1).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (6) CdtProblem
    (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 42 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), 446 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   h(z, e(x)) -> h(c(z), d(z, x))
   d(z, g(0, 0)) -> e(0)
   d(z, g(x, y)) -> g(e(x), d(z, y))
   d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
   g(e(x), e(y)) -> e(g(x, y))

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 1th argument of h: d
The following defined symbols can occur below the 1th argument of d: d

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
d(z, g(0, 0)) -> e(0)
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))

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

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


The TRS R consists of the following rules:

   h(z, e(x)) -> h(c(z), d(z, x))
   g(e(x), e(y)) -> e(g(x, y))

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

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

As the TRS does not nest defined symbols, we have rc = irc.
----------------------------------------

(4)
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:

   h(z, e(x)) -> h(c(z), d(z, x))
   g(e(x), e(y)) -> e(g(x, y))

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

(5) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   h(z0, e(z1)) -> h(c(z0), d(z0, z1))
   g(e(z0), e(z1)) -> e(g(z0, z1))
Tuples:
   H(z0, e(z1)) -> c1(H(c(z0), d(z0, z1)))
   G(e(z0), e(z1)) -> c2(G(z0, z1))
S tuples:
   H(z0, e(z1)) -> c1(H(c(z0), d(z0, z1)))
   G(e(z0), e(z1)) -> c2(G(z0, z1))
K tuples:none
Defined Rule Symbols:   h_2, g_2

Defined Pair Symbols:   H_2, G_2

Compound Symbols:   c1_1, c2_1


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

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   H(z0, e(z1)) -> c1(H(c(z0), d(z0, z1)))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   h(z0, e(z1)) -> h(c(z0), d(z0, z1))
   g(e(z0), e(z1)) -> e(g(z0, z1))
Tuples:
   G(e(z0), e(z1)) -> c2(G(z0, z1))
S tuples:
   G(e(z0), e(z1)) -> c2(G(z0, z1))
K tuples:none
Defined Rule Symbols:   h_2, g_2

Defined Pair Symbols:   G_2

Compound Symbols:   c2_1


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

(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   h(z0, e(z1)) -> h(c(z0), d(z0, z1))
   g(e(z0), e(z1)) -> e(g(z0, z1))

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   G(e(z0), e(z1)) -> c2(G(z0, z1))
S tuples:
   G(e(z0), e(z1)) -> c2(G(z0, z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   G_2

Compound Symbols:   c2_1


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

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

Polynomial interpretation :

   POL(G(x_1, x_2)) = x_2
   POL(c2(x_1)) = x_1
   POL(e(x_1)) = [1] + x_1

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   G(e(z0), e(z1)) -> c2(G(z0, z1))
S tuples:none
K tuples:
   G(e(z0), e(z1)) -> c2(G(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:   G_2

Compound Symbols:   c2_1


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

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

(14)
BOUNDS(1, 1)

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

(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^1, INF).


The TRS R consists of the following rules:

   h(z, e(x)) -> h(c(z), d(z, x))
   d(z, g(0', 0')) -> e(0')
   d(z, g(x, y)) -> g(e(x), d(z, y))
   d(c(z), g(g(x, y), 0')) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
   g(e(x), e(y)) -> e(g(x, y))

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

(17) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(18)
Obligation:
TRS:
Rules:
h(z, e(x)) -> h(c(z), d(z, x))
d(z, g(0', 0')) -> e(0')
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0')) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) -> e(g(x, y))

Types:
h :: c -> e:0' -> h
e :: e:0' -> e:0'
c :: c -> c
d :: c -> e:0' -> e:0'
g :: e:0' -> e:0' -> e:0'
0' :: e:0'
hole_h1_0 :: h
hole_c2_0 :: c
hole_e:0'3_0 :: e:0'
gen_c4_0 :: Nat -> c
gen_e:0'5_0 :: Nat -> e:0'

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
h, d, g

They will be analysed ascendingly in the following order:
d < h
g < d

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

(20)
Obligation:
TRS:
Rules:
h(z, e(x)) -> h(c(z), d(z, x))
d(z, g(0', 0')) -> e(0')
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0')) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) -> e(g(x, y))

Types:
h :: c -> e:0' -> h
e :: e:0' -> e:0'
c :: c -> c
d :: c -> e:0' -> e:0'
g :: e:0' -> e:0' -> e:0'
0' :: e:0'
hole_h1_0 :: h
hole_c2_0 :: c
hole_e:0'3_0 :: e:0'
gen_c4_0 :: Nat -> c
gen_e:0'5_0 :: Nat -> e:0'


Generator Equations:
gen_c4_0(0) <=> hole_c2_0
gen_c4_0(+(x, 1)) <=> c(gen_c4_0(x))
gen_e:0'5_0(0) <=> 0'
gen_e:0'5_0(+(x, 1)) <=> e(gen_e:0'5_0(x))


The following defined symbols remain to be analysed:
g, h, d

They will be analysed ascendingly in the following order:
d < h
g < d

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_e:0'5_0(+(1, n7_0)), gen_e:0'5_0(+(1, n7_0))) -> *6_0, rt in Omega(n7_0)

Induction Base:
g(gen_e:0'5_0(+(1, 0)), gen_e:0'5_0(+(1, 0)))

Induction Step:
g(gen_e:0'5_0(+(1, +(n7_0, 1))), gen_e:0'5_0(+(1, +(n7_0, 1)))) ->_R^Omega(1)
e(g(gen_e:0'5_0(+(1, n7_0)), gen_e:0'5_0(+(1, n7_0)))) ->_IH
e(*6_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(22)
Complex Obligation (BEST)

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

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

TRS:
Rules:
h(z, e(x)) -> h(c(z), d(z, x))
d(z, g(0', 0')) -> e(0')
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0')) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) -> e(g(x, y))

Types:
h :: c -> e:0' -> h
e :: e:0' -> e:0'
c :: c -> c
d :: c -> e:0' -> e:0'
g :: e:0' -> e:0' -> e:0'
0' :: e:0'
hole_h1_0 :: h
hole_c2_0 :: c
hole_e:0'3_0 :: e:0'
gen_c4_0 :: Nat -> c
gen_e:0'5_0 :: Nat -> e:0'


Generator Equations:
gen_c4_0(0) <=> hole_c2_0
gen_c4_0(+(x, 1)) <=> c(gen_c4_0(x))
gen_e:0'5_0(0) <=> 0'
gen_e:0'5_0(+(x, 1)) <=> e(gen_e:0'5_0(x))


The following defined symbols remain to be analysed:
g, h, d

They will be analysed ascendingly in the following order:
d < h
g < d

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

(24) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(25)
BOUNDS(n^1, INF)

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

(26)
Obligation:
TRS:
Rules:
h(z, e(x)) -> h(c(z), d(z, x))
d(z, g(0', 0')) -> e(0')
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0')) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) -> e(g(x, y))

Types:
h :: c -> e:0' -> h
e :: e:0' -> e:0'
c :: c -> c
d :: c -> e:0' -> e:0'
g :: e:0' -> e:0' -> e:0'
0' :: e:0'
hole_h1_0 :: h
hole_c2_0 :: c
hole_e:0'3_0 :: e:0'
gen_c4_0 :: Nat -> c
gen_e:0'5_0 :: Nat -> e:0'


Lemmas:
g(gen_e:0'5_0(+(1, n7_0)), gen_e:0'5_0(+(1, n7_0))) -> *6_0, rt in Omega(n7_0)


Generator Equations:
gen_c4_0(0) <=> hole_c2_0
gen_c4_0(+(x, 1)) <=> c(gen_c4_0(x))
gen_e:0'5_0(0) <=> 0'
gen_e:0'5_0(+(x, 1)) <=> e(gen_e:0'5_0(x))


The following defined symbols remain to be analysed:
d, h

They will be analysed ascendingly in the following order:
d < h