WORST_CASE(Omega(n^1),?)
proof of input_7EDjBKRboe.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxTRS
(1) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CdtProblem
(3) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 4 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 528 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 276 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 218 ms]
        (20) typed CpxTrs


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

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


The TRS R consists of the following rules:

   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   incr(cons(X, Y)) -> cons(s(X), incr(Y))
   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
   hd(cons(X, Y)) -> X
   tl(cons(X, Y)) -> Y

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

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> z1
Tuples:
   NATS -> c(ADX(zeros), ZEROS)
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   HD(cons(z0, z1)) -> c4
   TL(cons(z0, z1)) -> c5
S tuples:
   NATS -> c(ADX(zeros), ZEROS)
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   HD(cons(z0, z1)) -> c4
   TL(cons(z0, z1)) -> c5
K tuples:none
Defined Rule Symbols:   nats, zeros, incr_1, adx_1, hd_1, tl_1

Defined Pair Symbols:   NATS, ZEROS, INCR_1, ADX_1, HD_1, TL_1

Compound Symbols:   c_2, c1_1, c2_1, c3_2, c4, c5


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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   NATS -> c(ADX(zeros), ZEROS)
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   HD(cons(z0, z1)) -> c4
   TL(cons(z0, z1)) -> c5

The (relative) TRS S consists of the following rules:

   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> z1

Rewrite Strategy: INNERMOST
----------------------------------------

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

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   NATS -> c(ADX(zeros), ZEROS)
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   HD(cons(z0, z1)) -> c4
   TL(cons(z0, z1)) -> c5

The (relative) TRS S consists of the following rules:

   nats -> adx(zeros)
   zeros -> cons(0', zeros)
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> z1

Rewrite Strategy: INNERMOST
----------------------------------------

(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(8)
Obligation:
Innermost TRS:
Rules:
NATS -> c(ADX(zeros), ZEROS)
ZEROS -> c1(ZEROS)
INCR(cons(z0, z1)) -> c2(INCR(z1))
ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
HD(cons(z0, z1)) -> c4
TL(cons(z0, z1)) -> c5
nats -> adx(zeros)
zeros -> cons(0', zeros)
incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
hd(cons(z0, z1)) -> z0
tl(cons(z0, z1)) -> z1

Types:
NATS :: c
c :: c3 -> c1 -> c
ADX :: cons -> c3
zeros :: cons
ZEROS :: c1
c1 :: c1 -> c1
INCR :: cons -> c2
cons :: 0':s -> cons -> cons
c2 :: c2 -> c2
c3 :: c2 -> c3 -> c3
adx :: cons -> cons
HD :: cons -> c4
c4 :: c4
TL :: cons -> c5
c5 :: c5
nats :: cons
0' :: 0':s
incr :: cons -> cons
s :: 0':s -> 0':s
hd :: cons -> 0':s
tl :: cons -> cons
hole_c1_6 :: c
hole_c32_6 :: c3
hole_c13_6 :: c1
hole_cons4_6 :: cons
hole_c25_6 :: c2
hole_0':s6_6 :: 0':s
hole_c47_6 :: c4
hole_c58_6 :: c5
gen_c39_6 :: Nat -> c3
gen_c110_6 :: Nat -> c1
gen_cons11_6 :: Nat -> cons
gen_c212_6 :: Nat -> c2
gen_0':s13_6 :: Nat -> 0':s

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
ADX, zeros, ZEROS, INCR, adx, incr

They will be analysed ascendingly in the following order:
INCR < ADX
adx < ADX
incr < adx

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

(10)
Obligation:
Innermost TRS:
Rules:
NATS -> c(ADX(zeros), ZEROS)
ZEROS -> c1(ZEROS)
INCR(cons(z0, z1)) -> c2(INCR(z1))
ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
HD(cons(z0, z1)) -> c4
TL(cons(z0, z1)) -> c5
nats -> adx(zeros)
zeros -> cons(0', zeros)
incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
hd(cons(z0, z1)) -> z0
tl(cons(z0, z1)) -> z1

Types:
NATS :: c
c :: c3 -> c1 -> c
ADX :: cons -> c3
zeros :: cons
ZEROS :: c1
c1 :: c1 -> c1
INCR :: cons -> c2
cons :: 0':s -> cons -> cons
c2 :: c2 -> c2
c3 :: c2 -> c3 -> c3
adx :: cons -> cons
HD :: cons -> c4
c4 :: c4
TL :: cons -> c5
c5 :: c5
nats :: cons
0' :: 0':s
incr :: cons -> cons
s :: 0':s -> 0':s
hd :: cons -> 0':s
tl :: cons -> cons
hole_c1_6 :: c
hole_c32_6 :: c3
hole_c13_6 :: c1
hole_cons4_6 :: cons
hole_c25_6 :: c2
hole_0':s6_6 :: 0':s
hole_c47_6 :: c4
hole_c58_6 :: c5
gen_c39_6 :: Nat -> c3
gen_c110_6 :: Nat -> c1
gen_cons11_6 :: Nat -> cons
gen_c212_6 :: Nat -> c2
gen_0':s13_6 :: Nat -> 0':s


Generator Equations:
gen_c39_6(0) <=> hole_c32_6
gen_c39_6(+(x, 1)) <=> c3(hole_c25_6, gen_c39_6(x))
gen_c110_6(0) <=> hole_c13_6
gen_c110_6(+(x, 1)) <=> c1(gen_c110_6(x))
gen_cons11_6(0) <=> hole_cons4_6
gen_cons11_6(+(x, 1)) <=> cons(0', gen_cons11_6(x))
gen_c212_6(0) <=> hole_c25_6
gen_c212_6(+(x, 1)) <=> c2(gen_c212_6(x))
gen_0':s13_6(0) <=> 0'
gen_0':s13_6(+(x, 1)) <=> s(gen_0':s13_6(x))


The following defined symbols remain to be analysed:
zeros, ADX, ZEROS, INCR, adx, incr

They will be analysed ascendingly in the following order:
INCR < ADX
adx < ADX
incr < adx

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
INCR(gen_cons11_6(+(1, n33_6))) -> *14_6, rt in Omega(n33_6)

Induction Base:
INCR(gen_cons11_6(+(1, 0)))

Induction Step:
INCR(gen_cons11_6(+(1, +(n33_6, 1)))) ->_R^Omega(1)
c2(INCR(gen_cons11_6(+(1, n33_6)))) ->_IH
c2(*14_6)

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

(12)
Complex Obligation (BEST)

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

(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
NATS -> c(ADX(zeros), ZEROS)
ZEROS -> c1(ZEROS)
INCR(cons(z0, z1)) -> c2(INCR(z1))
ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
HD(cons(z0, z1)) -> c4
TL(cons(z0, z1)) -> c5
nats -> adx(zeros)
zeros -> cons(0', zeros)
incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
hd(cons(z0, z1)) -> z0
tl(cons(z0, z1)) -> z1

Types:
NATS :: c
c :: c3 -> c1 -> c
ADX :: cons -> c3
zeros :: cons
ZEROS :: c1
c1 :: c1 -> c1
INCR :: cons -> c2
cons :: 0':s -> cons -> cons
c2 :: c2 -> c2
c3 :: c2 -> c3 -> c3
adx :: cons -> cons
HD :: cons -> c4
c4 :: c4
TL :: cons -> c5
c5 :: c5
nats :: cons
0' :: 0':s
incr :: cons -> cons
s :: 0':s -> 0':s
hd :: cons -> 0':s
tl :: cons -> cons
hole_c1_6 :: c
hole_c32_6 :: c3
hole_c13_6 :: c1
hole_cons4_6 :: cons
hole_c25_6 :: c2
hole_0':s6_6 :: 0':s
hole_c47_6 :: c4
hole_c58_6 :: c5
gen_c39_6 :: Nat -> c3
gen_c110_6 :: Nat -> c1
gen_cons11_6 :: Nat -> cons
gen_c212_6 :: Nat -> c2
gen_0':s13_6 :: Nat -> 0':s


Generator Equations:
gen_c39_6(0) <=> hole_c32_6
gen_c39_6(+(x, 1)) <=> c3(hole_c25_6, gen_c39_6(x))
gen_c110_6(0) <=> hole_c13_6
gen_c110_6(+(x, 1)) <=> c1(gen_c110_6(x))
gen_cons11_6(0) <=> hole_cons4_6
gen_cons11_6(+(x, 1)) <=> cons(0', gen_cons11_6(x))
gen_c212_6(0) <=> hole_c25_6
gen_c212_6(+(x, 1)) <=> c2(gen_c212_6(x))
gen_0':s13_6(0) <=> 0'
gen_0':s13_6(+(x, 1)) <=> s(gen_0':s13_6(x))


The following defined symbols remain to be analysed:
INCR, ADX, adx, incr

They will be analysed ascendingly in the following order:
INCR < ADX
adx < ADX
incr < adx

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
NATS -> c(ADX(zeros), ZEROS)
ZEROS -> c1(ZEROS)
INCR(cons(z0, z1)) -> c2(INCR(z1))
ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
HD(cons(z0, z1)) -> c4
TL(cons(z0, z1)) -> c5
nats -> adx(zeros)
zeros -> cons(0', zeros)
incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
hd(cons(z0, z1)) -> z0
tl(cons(z0, z1)) -> z1

Types:
NATS :: c
c :: c3 -> c1 -> c
ADX :: cons -> c3
zeros :: cons
ZEROS :: c1
c1 :: c1 -> c1
INCR :: cons -> c2
cons :: 0':s -> cons -> cons
c2 :: c2 -> c2
c3 :: c2 -> c3 -> c3
adx :: cons -> cons
HD :: cons -> c4
c4 :: c4
TL :: cons -> c5
c5 :: c5
nats :: cons
0' :: 0':s
incr :: cons -> cons
s :: 0':s -> 0':s
hd :: cons -> 0':s
tl :: cons -> cons
hole_c1_6 :: c
hole_c32_6 :: c3
hole_c13_6 :: c1
hole_cons4_6 :: cons
hole_c25_6 :: c2
hole_0':s6_6 :: 0':s
hole_c47_6 :: c4
hole_c58_6 :: c5
gen_c39_6 :: Nat -> c3
gen_c110_6 :: Nat -> c1
gen_cons11_6 :: Nat -> cons
gen_c212_6 :: Nat -> c2
gen_0':s13_6 :: Nat -> 0':s


Lemmas:
INCR(gen_cons11_6(+(1, n33_6))) -> *14_6, rt in Omega(n33_6)


Generator Equations:
gen_c39_6(0) <=> hole_c32_6
gen_c39_6(+(x, 1)) <=> c3(hole_c25_6, gen_c39_6(x))
gen_c110_6(0) <=> hole_c13_6
gen_c110_6(+(x, 1)) <=> c1(gen_c110_6(x))
gen_cons11_6(0) <=> hole_cons4_6
gen_cons11_6(+(x, 1)) <=> cons(0', gen_cons11_6(x))
gen_c212_6(0) <=> hole_c25_6
gen_c212_6(+(x, 1)) <=> c2(gen_c212_6(x))
gen_0':s13_6(0) <=> 0'
gen_0':s13_6(+(x, 1)) <=> s(gen_0':s13_6(x))


The following defined symbols remain to be analysed:
incr, ADX, adx

They will be analysed ascendingly in the following order:
adx < ADX
incr < adx

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
incr(gen_cons11_6(+(1, n291_6))) -> *14_6, rt in Omega(0)

Induction Base:
incr(gen_cons11_6(+(1, 0)))

Induction Step:
incr(gen_cons11_6(+(1, +(n291_6, 1)))) ->_R^Omega(0)
cons(s(0'), incr(gen_cons11_6(+(1, n291_6)))) ->_IH
cons(s(0'), *14_6)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(18)
Obligation:
Innermost TRS:
Rules:
NATS -> c(ADX(zeros), ZEROS)
ZEROS -> c1(ZEROS)
INCR(cons(z0, z1)) -> c2(INCR(z1))
ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
HD(cons(z0, z1)) -> c4
TL(cons(z0, z1)) -> c5
nats -> adx(zeros)
zeros -> cons(0', zeros)
incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
hd(cons(z0, z1)) -> z0
tl(cons(z0, z1)) -> z1

Types:
NATS :: c
c :: c3 -> c1 -> c
ADX :: cons -> c3
zeros :: cons
ZEROS :: c1
c1 :: c1 -> c1
INCR :: cons -> c2
cons :: 0':s -> cons -> cons
c2 :: c2 -> c2
c3 :: c2 -> c3 -> c3
adx :: cons -> cons
HD :: cons -> c4
c4 :: c4
TL :: cons -> c5
c5 :: c5
nats :: cons
0' :: 0':s
incr :: cons -> cons
s :: 0':s -> 0':s
hd :: cons -> 0':s
tl :: cons -> cons
hole_c1_6 :: c
hole_c32_6 :: c3
hole_c13_6 :: c1
hole_cons4_6 :: cons
hole_c25_6 :: c2
hole_0':s6_6 :: 0':s
hole_c47_6 :: c4
hole_c58_6 :: c5
gen_c39_6 :: Nat -> c3
gen_c110_6 :: Nat -> c1
gen_cons11_6 :: Nat -> cons
gen_c212_6 :: Nat -> c2
gen_0':s13_6 :: Nat -> 0':s


Lemmas:
INCR(gen_cons11_6(+(1, n33_6))) -> *14_6, rt in Omega(n33_6)
incr(gen_cons11_6(+(1, n291_6))) -> *14_6, rt in Omega(0)


Generator Equations:
gen_c39_6(0) <=> hole_c32_6
gen_c39_6(+(x, 1)) <=> c3(hole_c25_6, gen_c39_6(x))
gen_c110_6(0) <=> hole_c13_6
gen_c110_6(+(x, 1)) <=> c1(gen_c110_6(x))
gen_cons11_6(0) <=> hole_cons4_6
gen_cons11_6(+(x, 1)) <=> cons(0', gen_cons11_6(x))
gen_c212_6(0) <=> hole_c25_6
gen_c212_6(+(x, 1)) <=> c2(gen_c212_6(x))
gen_0':s13_6(0) <=> 0'
gen_0':s13_6(+(x, 1)) <=> s(gen_0':s13_6(x))


The following defined symbols remain to be analysed:
adx, ADX

They will be analysed ascendingly in the following order:
adx < ADX

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
adx(gen_cons11_6(+(1, n3063_6))) -> *14_6, rt in Omega(0)

Induction Base:
adx(gen_cons11_6(+(1, 0)))

Induction Step:
adx(gen_cons11_6(+(1, +(n3063_6, 1)))) ->_R^Omega(0)
incr(cons(0', adx(gen_cons11_6(+(1, n3063_6))))) ->_IH
incr(cons(0', *14_6))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(20)
Obligation:
Innermost TRS:
Rules:
NATS -> c(ADX(zeros), ZEROS)
ZEROS -> c1(ZEROS)
INCR(cons(z0, z1)) -> c2(INCR(z1))
ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
HD(cons(z0, z1)) -> c4
TL(cons(z0, z1)) -> c5
nats -> adx(zeros)
zeros -> cons(0', zeros)
incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
hd(cons(z0, z1)) -> z0
tl(cons(z0, z1)) -> z1

Types:
NATS :: c
c :: c3 -> c1 -> c
ADX :: cons -> c3
zeros :: cons
ZEROS :: c1
c1 :: c1 -> c1
INCR :: cons -> c2
cons :: 0':s -> cons -> cons
c2 :: c2 -> c2
c3 :: c2 -> c3 -> c3
adx :: cons -> cons
HD :: cons -> c4
c4 :: c4
TL :: cons -> c5
c5 :: c5
nats :: cons
0' :: 0':s
incr :: cons -> cons
s :: 0':s -> 0':s
hd :: cons -> 0':s
tl :: cons -> cons
hole_c1_6 :: c
hole_c32_6 :: c3
hole_c13_6 :: c1
hole_cons4_6 :: cons
hole_c25_6 :: c2
hole_0':s6_6 :: 0':s
hole_c47_6 :: c4
hole_c58_6 :: c5
gen_c39_6 :: Nat -> c3
gen_c110_6 :: Nat -> c1
gen_cons11_6 :: Nat -> cons
gen_c212_6 :: Nat -> c2
gen_0':s13_6 :: Nat -> 0':s


Lemmas:
INCR(gen_cons11_6(+(1, n33_6))) -> *14_6, rt in Omega(n33_6)
incr(gen_cons11_6(+(1, n291_6))) -> *14_6, rt in Omega(0)
adx(gen_cons11_6(+(1, n3063_6))) -> *14_6, rt in Omega(0)


Generator Equations:
gen_c39_6(0) <=> hole_c32_6
gen_c39_6(+(x, 1)) <=> c3(hole_c25_6, gen_c39_6(x))
gen_c110_6(0) <=> hole_c13_6
gen_c110_6(+(x, 1)) <=> c1(gen_c110_6(x))
gen_cons11_6(0) <=> hole_cons4_6
gen_cons11_6(+(x, 1)) <=> cons(0', gen_cons11_6(x))
gen_c212_6(0) <=> hole_c25_6
gen_c212_6(+(x, 1)) <=> c2(gen_c212_6(x))
gen_0':s13_6(0) <=> 0'
gen_0':s13_6(+(x, 1)) <=> s(gen_0':s13_6(x))


The following defined symbols remain to be analysed:
ADX