WORST_CASE(Omega(n^1),O(n^1))
proof of input_yyAJfhDrI1.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, n^1).

(0) CpxTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsMatchBoundsTAProof [FINISHED, 26 ms]
    (4) BOUNDS(1, n^1)
(5) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) typed CpxTrs
    (13) OrderProof [LOWER BOUND(ID), 10 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 465 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 125 ms]
            (22) typed CpxTrs


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

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


The TRS R consists of the following rules:

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

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

(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(2)
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:

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

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

(3) 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, 3]
transitions: 
s0(0) -> 0
cons0(0, 0) -> 0
00() -> 0
f0(0) -> 1
g0(0) -> 2
h0(0) -> 3
f1(0) -> 1
g1(0) -> 2
s1(0) -> 2
cons1(0, 0) -> 5
g1(5) -> 4
h1(4) -> 3
g1(0) -> 4
s1(0) -> 4

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

(4)
BOUNDS(1, n^1)

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(s(z0)) -> f(z0)
   g(cons(0, z0)) -> g(z0)
   g(cons(s(z0), z1)) -> s(z0)
   h(cons(z0, z1)) -> h(g(cons(z0, z1)))
Tuples:
   F(s(z0)) -> c(F(z0))
   G(cons(0, z0)) -> c1(G(z0))
   G(cons(s(z0), z1)) -> c2
   H(cons(z0, z1)) -> c3(H(g(cons(z0, z1))), G(cons(z0, z1)))
S tuples:
   F(s(z0)) -> c(F(z0))
   G(cons(0, z0)) -> c1(G(z0))
   G(cons(s(z0), z1)) -> c2
   H(cons(z0, z1)) -> c3(H(g(cons(z0, z1))), G(cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   f_1, g_1, h_1

Defined Pair Symbols:   F_1, G_1, H_1

Compound Symbols:   c_1, c1_1, c2, c3_2


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

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

(8)
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(s(z0)) -> c(F(z0))
   G(cons(0, z0)) -> c1(G(z0))
   G(cons(s(z0), z1)) -> c2
   H(cons(z0, z1)) -> c3(H(g(cons(z0, z1))), G(cons(z0, z1)))

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

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

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

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

(10)
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(s(z0)) -> c(F(z0))
   G(cons(0', z0)) -> c1(G(z0))
   G(cons(s(z0), z1)) -> c2
   H(cons(z0, z1)) -> c3(H(g(cons(z0, z1))), G(cons(z0, z1)))

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

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

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

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

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

Types:
F :: s:0':cons -> c
s :: s:0':cons -> s:0':cons
c :: c -> c
G :: s:0':cons -> c1:c2
cons :: s:0':cons -> s:0':cons -> s:0':cons
0' :: s:0':cons
c1 :: c1:c2 -> c1:c2
c2 :: c1:c2
H :: s:0':cons -> c3
c3 :: c3 -> c1:c2 -> c3
g :: s:0':cons -> s:0':cons
f :: s:0':cons -> f
h :: s:0':cons -> h
hole_c1_4 :: c
hole_s:0':cons2_4 :: s:0':cons
hole_c1:c23_4 :: c1:c2
hole_c34_4 :: c3
hole_f5_4 :: f
hole_h6_4 :: h
gen_c7_4 :: Nat -> c
gen_s:0':cons8_4 :: Nat -> s:0':cons
gen_c1:c29_4 :: Nat -> c1:c2
gen_c310_4 :: Nat -> c3

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
F, G, H, g, f, h

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

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

(14)
Obligation:
Innermost TRS:
Rules:
F(s(z0)) -> c(F(z0))
G(cons(0', z0)) -> c1(G(z0))
G(cons(s(z0), z1)) -> c2
H(cons(z0, z1)) -> c3(H(g(cons(z0, z1))), G(cons(z0, z1)))
f(s(z0)) -> f(z0)
g(cons(0', z0)) -> g(z0)
g(cons(s(z0), z1)) -> s(z0)
h(cons(z0, z1)) -> h(g(cons(z0, z1)))

Types:
F :: s:0':cons -> c
s :: s:0':cons -> s:0':cons
c :: c -> c
G :: s:0':cons -> c1:c2
cons :: s:0':cons -> s:0':cons -> s:0':cons
0' :: s:0':cons
c1 :: c1:c2 -> c1:c2
c2 :: c1:c2
H :: s:0':cons -> c3
c3 :: c3 -> c1:c2 -> c3
g :: s:0':cons -> s:0':cons
f :: s:0':cons -> f
h :: s:0':cons -> h
hole_c1_4 :: c
hole_s:0':cons2_4 :: s:0':cons
hole_c1:c23_4 :: c1:c2
hole_c34_4 :: c3
hole_f5_4 :: f
hole_h6_4 :: h
gen_c7_4 :: Nat -> c
gen_s:0':cons8_4 :: Nat -> s:0':cons
gen_c1:c29_4 :: Nat -> c1:c2
gen_c310_4 :: Nat -> c3


Generator Equations:
gen_c7_4(0) <=> hole_c1_4
gen_c7_4(+(x, 1)) <=> c(gen_c7_4(x))
gen_s:0':cons8_4(0) <=> 0'
gen_s:0':cons8_4(+(x, 1)) <=> s(gen_s:0':cons8_4(x))
gen_c1:c29_4(0) <=> c2
gen_c1:c29_4(+(x, 1)) <=> c1(gen_c1:c29_4(x))
gen_c310_4(0) <=> hole_c34_4
gen_c310_4(+(x, 1)) <=> c3(gen_c310_4(x), c2)


The following defined symbols remain to be analysed:
F, G, H, g, f, h

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

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
F(gen_s:0':cons8_4(+(1, n12_4))) -> *11_4, rt in Omega(n12_4)

Induction Base:
F(gen_s:0':cons8_4(+(1, 0)))

Induction Step:
F(gen_s:0':cons8_4(+(1, +(n12_4, 1)))) ->_R^Omega(1)
c(F(gen_s:0':cons8_4(+(1, n12_4)))) ->_IH
c(*11_4)

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

(16)
Complex Obligation (BEST)

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

(17)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
F(s(z0)) -> c(F(z0))
G(cons(0', z0)) -> c1(G(z0))
G(cons(s(z0), z1)) -> c2
H(cons(z0, z1)) -> c3(H(g(cons(z0, z1))), G(cons(z0, z1)))
f(s(z0)) -> f(z0)
g(cons(0', z0)) -> g(z0)
g(cons(s(z0), z1)) -> s(z0)
h(cons(z0, z1)) -> h(g(cons(z0, z1)))

Types:
F :: s:0':cons -> c
s :: s:0':cons -> s:0':cons
c :: c -> c
G :: s:0':cons -> c1:c2
cons :: s:0':cons -> s:0':cons -> s:0':cons
0' :: s:0':cons
c1 :: c1:c2 -> c1:c2
c2 :: c1:c2
H :: s:0':cons -> c3
c3 :: c3 -> c1:c2 -> c3
g :: s:0':cons -> s:0':cons
f :: s:0':cons -> f
h :: s:0':cons -> h
hole_c1_4 :: c
hole_s:0':cons2_4 :: s:0':cons
hole_c1:c23_4 :: c1:c2
hole_c34_4 :: c3
hole_f5_4 :: f
hole_h6_4 :: h
gen_c7_4 :: Nat -> c
gen_s:0':cons8_4 :: Nat -> s:0':cons
gen_c1:c29_4 :: Nat -> c1:c2
gen_c310_4 :: Nat -> c3


Generator Equations:
gen_c7_4(0) <=> hole_c1_4
gen_c7_4(+(x, 1)) <=> c(gen_c7_4(x))
gen_s:0':cons8_4(0) <=> 0'
gen_s:0':cons8_4(+(x, 1)) <=> s(gen_s:0':cons8_4(x))
gen_c1:c29_4(0) <=> c2
gen_c1:c29_4(+(x, 1)) <=> c1(gen_c1:c29_4(x))
gen_c310_4(0) <=> hole_c34_4
gen_c310_4(+(x, 1)) <=> c3(gen_c310_4(x), c2)


The following defined symbols remain to be analysed:
F, G, H, g, f, h

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

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

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

(19)
BOUNDS(n^1, INF)

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

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

Types:
F :: s:0':cons -> c
s :: s:0':cons -> s:0':cons
c :: c -> c
G :: s:0':cons -> c1:c2
cons :: s:0':cons -> s:0':cons -> s:0':cons
0' :: s:0':cons
c1 :: c1:c2 -> c1:c2
c2 :: c1:c2
H :: s:0':cons -> c3
c3 :: c3 -> c1:c2 -> c3
g :: s:0':cons -> s:0':cons
f :: s:0':cons -> f
h :: s:0':cons -> h
hole_c1_4 :: c
hole_s:0':cons2_4 :: s:0':cons
hole_c1:c23_4 :: c1:c2
hole_c34_4 :: c3
hole_f5_4 :: f
hole_h6_4 :: h
gen_c7_4 :: Nat -> c
gen_s:0':cons8_4 :: Nat -> s:0':cons
gen_c1:c29_4 :: Nat -> c1:c2
gen_c310_4 :: Nat -> c3


Lemmas:
F(gen_s:0':cons8_4(+(1, n12_4))) -> *11_4, rt in Omega(n12_4)


Generator Equations:
gen_c7_4(0) <=> hole_c1_4
gen_c7_4(+(x, 1)) <=> c(gen_c7_4(x))
gen_s:0':cons8_4(0) <=> 0'
gen_s:0':cons8_4(+(x, 1)) <=> s(gen_s:0':cons8_4(x))
gen_c1:c29_4(0) <=> c2
gen_c1:c29_4(+(x, 1)) <=> c1(gen_c1:c29_4(x))
gen_c310_4(0) <=> hole_c34_4
gen_c310_4(+(x, 1)) <=> c3(gen_c310_4(x), c2)


The following defined symbols remain to be analysed:
G, H, g, f, h

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

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_s:0':cons8_4(+(1, n222_4))) -> *11_4, rt in Omega(0)

Induction Base:
f(gen_s:0':cons8_4(+(1, 0)))

Induction Step:
f(gen_s:0':cons8_4(+(1, +(n222_4, 1)))) ->_R^Omega(0)
f(gen_s:0':cons8_4(+(1, n222_4))) ->_IH
*11_4

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

(22)
Obligation:
Innermost TRS:
Rules:
F(s(z0)) -> c(F(z0))
G(cons(0', z0)) -> c1(G(z0))
G(cons(s(z0), z1)) -> c2
H(cons(z0, z1)) -> c3(H(g(cons(z0, z1))), G(cons(z0, z1)))
f(s(z0)) -> f(z0)
g(cons(0', z0)) -> g(z0)
g(cons(s(z0), z1)) -> s(z0)
h(cons(z0, z1)) -> h(g(cons(z0, z1)))

Types:
F :: s:0':cons -> c
s :: s:0':cons -> s:0':cons
c :: c -> c
G :: s:0':cons -> c1:c2
cons :: s:0':cons -> s:0':cons -> s:0':cons
0' :: s:0':cons
c1 :: c1:c2 -> c1:c2
c2 :: c1:c2
H :: s:0':cons -> c3
c3 :: c3 -> c1:c2 -> c3
g :: s:0':cons -> s:0':cons
f :: s:0':cons -> f
h :: s:0':cons -> h
hole_c1_4 :: c
hole_s:0':cons2_4 :: s:0':cons
hole_c1:c23_4 :: c1:c2
hole_c34_4 :: c3
hole_f5_4 :: f
hole_h6_4 :: h
gen_c7_4 :: Nat -> c
gen_s:0':cons8_4 :: Nat -> s:0':cons
gen_c1:c29_4 :: Nat -> c1:c2
gen_c310_4 :: Nat -> c3


Lemmas:
F(gen_s:0':cons8_4(+(1, n12_4))) -> *11_4, rt in Omega(n12_4)
f(gen_s:0':cons8_4(+(1, n222_4))) -> *11_4, rt in Omega(0)


Generator Equations:
gen_c7_4(0) <=> hole_c1_4
gen_c7_4(+(x, 1)) <=> c(gen_c7_4(x))
gen_s:0':cons8_4(0) <=> 0'
gen_s:0':cons8_4(+(x, 1)) <=> s(gen_s:0':cons8_4(x))
gen_c1:c29_4(0) <=> c2
gen_c1:c29_4(+(x, 1)) <=> c1(gen_c1:c29_4(x))
gen_c310_4(0) <=> hole_c34_4
gen_c310_4(+(x, 1)) <=> c3(gen_c310_4(x), c2)


The following defined symbols remain to be analysed:
h