WORST_CASE(Omega(n^1),?)
proof of input_s4nMVHdffM.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 7 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 382 ms]
(12) typed CpxTrs
(13) RewriteLemmaProof [LOWER BOUND(ID), 55 ms]
(14) BEST
    (15) proven lower bound
        (16) LowerBoundPropagationProof [FINISHED, 0 ms]
        (17) BOUNDS(n^1, INF)
    (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 2082 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 217 ms]
        (22) BOUNDS(1, INF)


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

(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:

   f(X) -> cons(X, f(g(X)))
   g(0) -> s(0)
   g(s(X)) -> s(s(g(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)

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:
   f(z0) -> cons(z0, f(g(z0)))
   g(0) -> s(0)
   g(s(z0)) -> s(s(g(z0)))
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, z2)
Tuples:
   F(z0) -> c(F(g(z0)), G(z0))
   G(0) -> c1
   G(s(z0)) -> c2(G(z0))
   SEL(0, cons(z0, z1)) -> c3
   SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
S tuples:
   F(z0) -> c(F(g(z0)), G(z0))
   G(0) -> c1
   G(s(z0)) -> c2(G(z0))
   SEL(0, cons(z0, z1)) -> c3
   SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
K tuples:none
Defined Rule Symbols:   f_1, g_1, sel_2

Defined Pair Symbols:   F_1, G_1, SEL_2

Compound Symbols:   c_2, c1, c2_1, c3, c4_1


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

(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:

   F(z0) -> c(F(g(z0)), G(z0))
   G(0) -> c1
   G(s(z0)) -> c2(G(z0))
   SEL(0, cons(z0, z1)) -> c3
   SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))

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

   f(z0) -> cons(z0, f(g(z0)))
   g(0) -> s(0)
   g(s(z0)) -> s(s(g(z0)))
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

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:

   F(z0) -> c(F(g(z0)), G(z0))
   G(0') -> c1
   G(s(z0)) -> c2(G(z0))
   SEL(0', cons(z0, z1)) -> c3
   SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))

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

   f(z0) -> cons(z0, f(g(z0)))
   g(0') -> s(0')
   g(s(z0)) -> s(s(g(z0)))
   sel(0', cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

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

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

(8)
Obligation:
Innermost TRS:
Rules:
F(z0) -> c(F(g(z0)), G(z0))
G(0') -> c1
G(s(z0)) -> c2(G(z0))
SEL(0', cons(z0, z1)) -> c3
SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
f(z0) -> cons(z0, f(g(z0)))
g(0') -> s(0')
g(s(z0)) -> s(s(g(z0)))
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

Types:
F :: 0':s -> c
c :: c -> c1:c2 -> c
g :: 0':s -> 0':s
G :: 0':s -> c1:c2
0' :: 0':s
c1 :: c1:c2
s :: 0':s -> 0':s
c2 :: c1:c2 -> c1:c2
SEL :: 0':s -> cons -> c3:c4
cons :: 0':s -> cons -> cons
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
f :: 0':s -> cons
sel :: 0':s -> cons -> 0':s
hole_c1_5 :: c
hole_0':s2_5 :: 0':s
hole_c1:c23_5 :: c1:c2
hole_c3:c44_5 :: c3:c4
hole_cons5_5 :: cons
gen_c6_5 :: Nat -> c
gen_0':s7_5 :: Nat -> 0':s
gen_c1:c28_5 :: Nat -> c1:c2
gen_c3:c49_5 :: Nat -> c3:c4
gen_cons10_5 :: Nat -> cons

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
F, g, G, SEL, f, sel

They will be analysed ascendingly in the following order:
g < F
G < F
g < f

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

(10)
Obligation:
Innermost TRS:
Rules:
F(z0) -> c(F(g(z0)), G(z0))
G(0') -> c1
G(s(z0)) -> c2(G(z0))
SEL(0', cons(z0, z1)) -> c3
SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
f(z0) -> cons(z0, f(g(z0)))
g(0') -> s(0')
g(s(z0)) -> s(s(g(z0)))
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

Types:
F :: 0':s -> c
c :: c -> c1:c2 -> c
g :: 0':s -> 0':s
G :: 0':s -> c1:c2
0' :: 0':s
c1 :: c1:c2
s :: 0':s -> 0':s
c2 :: c1:c2 -> c1:c2
SEL :: 0':s -> cons -> c3:c4
cons :: 0':s -> cons -> cons
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
f :: 0':s -> cons
sel :: 0':s -> cons -> 0':s
hole_c1_5 :: c
hole_0':s2_5 :: 0':s
hole_c1:c23_5 :: c1:c2
hole_c3:c44_5 :: c3:c4
hole_cons5_5 :: cons
gen_c6_5 :: Nat -> c
gen_0':s7_5 :: Nat -> 0':s
gen_c1:c28_5 :: Nat -> c1:c2
gen_c3:c49_5 :: Nat -> c3:c4
gen_cons10_5 :: Nat -> cons


Generator Equations:
gen_c6_5(0) <=> hole_c1_5
gen_c6_5(+(x, 1)) <=> c(gen_c6_5(x), c1)
gen_0':s7_5(0) <=> 0'
gen_0':s7_5(+(x, 1)) <=> s(gen_0':s7_5(x))
gen_c1:c28_5(0) <=> c1
gen_c1:c28_5(+(x, 1)) <=> c2(gen_c1:c28_5(x))
gen_c3:c49_5(0) <=> c3
gen_c3:c49_5(+(x, 1)) <=> c4(gen_c3:c49_5(x))
gen_cons10_5(0) <=> hole_cons5_5
gen_cons10_5(+(x, 1)) <=> cons(0', gen_cons10_5(x))


The following defined symbols remain to be analysed:
g, F, G, SEL, f, sel

They will be analysed ascendingly in the following order:
g < F
G < F
g < f

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_0':s7_5(n12_5)) -> gen_0':s7_5(+(1, *(2, n12_5))), rt in Omega(0)

Induction Base:
g(gen_0':s7_5(0)) ->_R^Omega(0)
s(0')

Induction Step:
g(gen_0':s7_5(+(n12_5, 1))) ->_R^Omega(0)
s(s(g(gen_0':s7_5(n12_5)))) ->_IH
s(s(gen_0':s7_5(+(1, *(2, c13_5)))))

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

(12)
Obligation:
Innermost TRS:
Rules:
F(z0) -> c(F(g(z0)), G(z0))
G(0') -> c1
G(s(z0)) -> c2(G(z0))
SEL(0', cons(z0, z1)) -> c3
SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
f(z0) -> cons(z0, f(g(z0)))
g(0') -> s(0')
g(s(z0)) -> s(s(g(z0)))
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

Types:
F :: 0':s -> c
c :: c -> c1:c2 -> c
g :: 0':s -> 0':s
G :: 0':s -> c1:c2
0' :: 0':s
c1 :: c1:c2
s :: 0':s -> 0':s
c2 :: c1:c2 -> c1:c2
SEL :: 0':s -> cons -> c3:c4
cons :: 0':s -> cons -> cons
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
f :: 0':s -> cons
sel :: 0':s -> cons -> 0':s
hole_c1_5 :: c
hole_0':s2_5 :: 0':s
hole_c1:c23_5 :: c1:c2
hole_c3:c44_5 :: c3:c4
hole_cons5_5 :: cons
gen_c6_5 :: Nat -> c
gen_0':s7_5 :: Nat -> 0':s
gen_c1:c28_5 :: Nat -> c1:c2
gen_c3:c49_5 :: Nat -> c3:c4
gen_cons10_5 :: Nat -> cons


Lemmas:
g(gen_0':s7_5(n12_5)) -> gen_0':s7_5(+(1, *(2, n12_5))), rt in Omega(0)


Generator Equations:
gen_c6_5(0) <=> hole_c1_5
gen_c6_5(+(x, 1)) <=> c(gen_c6_5(x), c1)
gen_0':s7_5(0) <=> 0'
gen_0':s7_5(+(x, 1)) <=> s(gen_0':s7_5(x))
gen_c1:c28_5(0) <=> c1
gen_c1:c28_5(+(x, 1)) <=> c2(gen_c1:c28_5(x))
gen_c3:c49_5(0) <=> c3
gen_c3:c49_5(+(x, 1)) <=> c4(gen_c3:c49_5(x))
gen_cons10_5(0) <=> hole_cons5_5
gen_cons10_5(+(x, 1)) <=> cons(0', gen_cons10_5(x))


The following defined symbols remain to be analysed:
G, F, SEL, f, sel

They will be analysed ascendingly in the following order:
G < F

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
G(gen_0':s7_5(n336_5)) -> gen_c1:c28_5(n336_5), rt in Omega(1 + n336_5)

Induction Base:
G(gen_0':s7_5(0)) ->_R^Omega(1)
c1

Induction Step:
G(gen_0':s7_5(+(n336_5, 1))) ->_R^Omega(1)
c2(G(gen_0':s7_5(n336_5))) ->_IH
c2(gen_c1:c28_5(c337_5))

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

(14)
Complex Obligation (BEST)

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

(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
F(z0) -> c(F(g(z0)), G(z0))
G(0') -> c1
G(s(z0)) -> c2(G(z0))
SEL(0', cons(z0, z1)) -> c3
SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
f(z0) -> cons(z0, f(g(z0)))
g(0') -> s(0')
g(s(z0)) -> s(s(g(z0)))
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

Types:
F :: 0':s -> c
c :: c -> c1:c2 -> c
g :: 0':s -> 0':s
G :: 0':s -> c1:c2
0' :: 0':s
c1 :: c1:c2
s :: 0':s -> 0':s
c2 :: c1:c2 -> c1:c2
SEL :: 0':s -> cons -> c3:c4
cons :: 0':s -> cons -> cons
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
f :: 0':s -> cons
sel :: 0':s -> cons -> 0':s
hole_c1_5 :: c
hole_0':s2_5 :: 0':s
hole_c1:c23_5 :: c1:c2
hole_c3:c44_5 :: c3:c4
hole_cons5_5 :: cons
gen_c6_5 :: Nat -> c
gen_0':s7_5 :: Nat -> 0':s
gen_c1:c28_5 :: Nat -> c1:c2
gen_c3:c49_5 :: Nat -> c3:c4
gen_cons10_5 :: Nat -> cons


Lemmas:
g(gen_0':s7_5(n12_5)) -> gen_0':s7_5(+(1, *(2, n12_5))), rt in Omega(0)


Generator Equations:
gen_c6_5(0) <=> hole_c1_5
gen_c6_5(+(x, 1)) <=> c(gen_c6_5(x), c1)
gen_0':s7_5(0) <=> 0'
gen_0':s7_5(+(x, 1)) <=> s(gen_0':s7_5(x))
gen_c1:c28_5(0) <=> c1
gen_c1:c28_5(+(x, 1)) <=> c2(gen_c1:c28_5(x))
gen_c3:c49_5(0) <=> c3
gen_c3:c49_5(+(x, 1)) <=> c4(gen_c3:c49_5(x))
gen_cons10_5(0) <=> hole_cons5_5
gen_cons10_5(+(x, 1)) <=> cons(0', gen_cons10_5(x))


The following defined symbols remain to be analysed:
G, F, SEL, f, sel

They will be analysed ascendingly in the following order:
G < F

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

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

(17)
BOUNDS(n^1, INF)

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

(18)
Obligation:
Innermost TRS:
Rules:
F(z0) -> c(F(g(z0)), G(z0))
G(0') -> c1
G(s(z0)) -> c2(G(z0))
SEL(0', cons(z0, z1)) -> c3
SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
f(z0) -> cons(z0, f(g(z0)))
g(0') -> s(0')
g(s(z0)) -> s(s(g(z0)))
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

Types:
F :: 0':s -> c
c :: c -> c1:c2 -> c
g :: 0':s -> 0':s
G :: 0':s -> c1:c2
0' :: 0':s
c1 :: c1:c2
s :: 0':s -> 0':s
c2 :: c1:c2 -> c1:c2
SEL :: 0':s -> cons -> c3:c4
cons :: 0':s -> cons -> cons
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
f :: 0':s -> cons
sel :: 0':s -> cons -> 0':s
hole_c1_5 :: c
hole_0':s2_5 :: 0':s
hole_c1:c23_5 :: c1:c2
hole_c3:c44_5 :: c3:c4
hole_cons5_5 :: cons
gen_c6_5 :: Nat -> c
gen_0':s7_5 :: Nat -> 0':s
gen_c1:c28_5 :: Nat -> c1:c2
gen_c3:c49_5 :: Nat -> c3:c4
gen_cons10_5 :: Nat -> cons


Lemmas:
g(gen_0':s7_5(n12_5)) -> gen_0':s7_5(+(1, *(2, n12_5))), rt in Omega(0)
G(gen_0':s7_5(n336_5)) -> gen_c1:c28_5(n336_5), rt in Omega(1 + n336_5)


Generator Equations:
gen_c6_5(0) <=> hole_c1_5
gen_c6_5(+(x, 1)) <=> c(gen_c6_5(x), c1)
gen_0':s7_5(0) <=> 0'
gen_0':s7_5(+(x, 1)) <=> s(gen_0':s7_5(x))
gen_c1:c28_5(0) <=> c1
gen_c1:c28_5(+(x, 1)) <=> c2(gen_c1:c28_5(x))
gen_c3:c49_5(0) <=> c3
gen_c3:c49_5(+(x, 1)) <=> c4(gen_c3:c49_5(x))
gen_cons10_5(0) <=> hole_cons5_5
gen_cons10_5(+(x, 1)) <=> cons(0', gen_cons10_5(x))


The following defined symbols remain to be analysed:
F, SEL, f, sel
----------------------------------------

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
SEL(gen_0':s7_5(n957_5), gen_cons10_5(+(1, n957_5))) -> gen_c3:c49_5(n957_5), rt in Omega(1 + n957_5)

Induction Base:
SEL(gen_0':s7_5(0), gen_cons10_5(+(1, 0))) ->_R^Omega(1)
c3

Induction Step:
SEL(gen_0':s7_5(+(n957_5, 1)), gen_cons10_5(+(1, +(n957_5, 1)))) ->_R^Omega(1)
c4(SEL(gen_0':s7_5(n957_5), gen_cons10_5(+(1, n957_5)))) ->_IH
c4(gen_c3:c49_5(c958_5))

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

(20)
Obligation:
Innermost TRS:
Rules:
F(z0) -> c(F(g(z0)), G(z0))
G(0') -> c1
G(s(z0)) -> c2(G(z0))
SEL(0', cons(z0, z1)) -> c3
SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
f(z0) -> cons(z0, f(g(z0)))
g(0') -> s(0')
g(s(z0)) -> s(s(g(z0)))
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

Types:
F :: 0':s -> c
c :: c -> c1:c2 -> c
g :: 0':s -> 0':s
G :: 0':s -> c1:c2
0' :: 0':s
c1 :: c1:c2
s :: 0':s -> 0':s
c2 :: c1:c2 -> c1:c2
SEL :: 0':s -> cons -> c3:c4
cons :: 0':s -> cons -> cons
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
f :: 0':s -> cons
sel :: 0':s -> cons -> 0':s
hole_c1_5 :: c
hole_0':s2_5 :: 0':s
hole_c1:c23_5 :: c1:c2
hole_c3:c44_5 :: c3:c4
hole_cons5_5 :: cons
gen_c6_5 :: Nat -> c
gen_0':s7_5 :: Nat -> 0':s
gen_c1:c28_5 :: Nat -> c1:c2
gen_c3:c49_5 :: Nat -> c3:c4
gen_cons10_5 :: Nat -> cons


Lemmas:
g(gen_0':s7_5(n12_5)) -> gen_0':s7_5(+(1, *(2, n12_5))), rt in Omega(0)
G(gen_0':s7_5(n336_5)) -> gen_c1:c28_5(n336_5), rt in Omega(1 + n336_5)
SEL(gen_0':s7_5(n957_5), gen_cons10_5(+(1, n957_5))) -> gen_c3:c49_5(n957_5), rt in Omega(1 + n957_5)


Generator Equations:
gen_c6_5(0) <=> hole_c1_5
gen_c6_5(+(x, 1)) <=> c(gen_c6_5(x), c1)
gen_0':s7_5(0) <=> 0'
gen_0':s7_5(+(x, 1)) <=> s(gen_0':s7_5(x))
gen_c1:c28_5(0) <=> c1
gen_c1:c28_5(+(x, 1)) <=> c2(gen_c1:c28_5(x))
gen_c3:c49_5(0) <=> c3
gen_c3:c49_5(+(x, 1)) <=> c4(gen_c3:c49_5(x))
gen_cons10_5(0) <=> hole_cons5_5
gen_cons10_5(+(x, 1)) <=> cons(0', gen_cons10_5(x))


The following defined symbols remain to be analysed:
f, sel
----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sel(gen_0':s7_5(n1718_5), gen_cons10_5(+(1, n1718_5))) -> gen_0':s7_5(0), rt in Omega(0)

Induction Base:
sel(gen_0':s7_5(0), gen_cons10_5(+(1, 0))) ->_R^Omega(0)
0'

Induction Step:
sel(gen_0':s7_5(+(n1718_5, 1)), gen_cons10_5(+(1, +(n1718_5, 1)))) ->_R^Omega(0)
sel(gen_0':s7_5(n1718_5), gen_cons10_5(+(1, n1718_5))) ->_IH
gen_0':s7_5(0)

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

(22)
BOUNDS(1, INF)