WORST_CASE(Omega(n^1),O(n^1))
proof of input_Tb51zbqQr9.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 21 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), 13 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 482 ms]
    (16) typed CpxTrs
    (17) RewriteLemmaProof [LOWER BOUND(ID), 129 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 117 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 147 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), 156 ms]
            (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 170 ms]
            (30) BOUNDS(1, INF)


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

(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:

   g(f(x), y) -> f(h(x, y))
   h(x, y) -> g(x, f(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:

   g(f(x), y) -> f(h(x, y))
   h(x, y) -> g(x, f(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 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2]
transitions: 
f0(0) -> 0
g0(0, 0) -> 1
h0(0, 0) -> 2
h1(0, 0) -> 3
f1(3) -> 1
f1(0) -> 4
g1(0, 4) -> 2
h1(0, 4) -> 3
f1(3) -> 2
f2(0) -> 5
g2(0, 5) -> 3
f2(4) -> 5
h1(0, 5) -> 3
f1(3) -> 3
f2(5) -> 5

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

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

Defined Pair Symbols:   G_2, H_2

Compound Symbols:   c_1, c1_1


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

(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:

   G(f(z0), z1) -> c(H(z0, z1))
   H(z0, z1) -> c1(G(z0, f(z1)))

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

   g(f(z0), z1) -> f(h(z0, z1))
   h(z0, z1) -> g(z0, f(z1))

Rewrite Strategy: INNERMOST
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(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:

   G(f(z0), z1) -> c(H(z0, z1))
   H(z0, z1) -> c1(G(z0, f(z1)))

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

   g(f(z0), z1) -> f(h(z0, z1))
   h(z0, z1) -> g(z0, f(z1))

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

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

(12)
Obligation:
Innermost TRS:
Rules:
G(f(z0), z1) -> c(H(z0, z1))
H(z0, z1) -> c1(G(z0, f(z1)))
g(f(z0), z1) -> f(h(z0, z1))
h(z0, z1) -> g(z0, f(z1))

Types:
G :: f -> f -> c
f :: f -> f
c :: c1 -> c
H :: f -> f -> c1
c1 :: c -> c1
g :: f -> f -> f
h :: f -> f -> f
hole_c1_2 :: c
hole_f2_2 :: f
hole_c13_2 :: c1
gen_f4_2 :: Nat -> f

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

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

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

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

(14)
Obligation:
Innermost TRS:
Rules:
G(f(z0), z1) -> c(H(z0, z1))
H(z0, z1) -> c1(G(z0, f(z1)))
g(f(z0), z1) -> f(h(z0, z1))
h(z0, z1) -> g(z0, f(z1))

Types:
G :: f -> f -> c
f :: f -> f
c :: c1 -> c
H :: f -> f -> c1
c1 :: c -> c1
g :: f -> f -> f
h :: f -> f -> f
hole_c1_2 :: c
hole_f2_2 :: f
hole_c13_2 :: c1
gen_f4_2 :: Nat -> f


Generator Equations:
gen_f4_2(0) <=> hole_f2_2
gen_f4_2(+(x, 1)) <=> f(gen_f4_2(x))


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

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

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
h(gen_f4_2(n6_2), gen_f4_2(b)) -> *5_2, rt in Omega(0)

Induction Base:
h(gen_f4_2(0), gen_f4_2(b))

Induction Step:
h(gen_f4_2(+(n6_2, 1)), gen_f4_2(b)) ->_R^Omega(0)
g(gen_f4_2(+(n6_2, 1)), f(gen_f4_2(b))) ->_R^Omega(0)
f(h(gen_f4_2(n6_2), f(gen_f4_2(b)))) ->_IH
f(*5_2)

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

(16)
Obligation:
Innermost TRS:
Rules:
G(f(z0), z1) -> c(H(z0, z1))
H(z0, z1) -> c1(G(z0, f(z1)))
g(f(z0), z1) -> f(h(z0, z1))
h(z0, z1) -> g(z0, f(z1))

Types:
G :: f -> f -> c
f :: f -> f
c :: c1 -> c
H :: f -> f -> c1
c1 :: c -> c1
g :: f -> f -> f
h :: f -> f -> f
hole_c1_2 :: c
hole_f2_2 :: f
hole_c13_2 :: c1
gen_f4_2 :: Nat -> f


Lemmas:
h(gen_f4_2(n6_2), gen_f4_2(b)) -> *5_2, rt in Omega(0)


Generator Equations:
gen_f4_2(0) <=> hole_f2_2
gen_f4_2(+(x, 1)) <=> f(gen_f4_2(x))


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

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

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_f4_2(+(1, n314_2)), gen_f4_2(b)) -> *5_2, rt in Omega(0)

Induction Base:
g(gen_f4_2(+(1, 0)), gen_f4_2(b))

Induction Step:
g(gen_f4_2(+(1, +(n314_2, 1))), gen_f4_2(b)) ->_R^Omega(0)
f(h(gen_f4_2(+(1, n314_2)), gen_f4_2(b))) ->_R^Omega(0)
f(g(gen_f4_2(+(1, n314_2)), f(gen_f4_2(b)))) ->_IH
f(*5_2)

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:
G(f(z0), z1) -> c(H(z0, z1))
H(z0, z1) -> c1(G(z0, f(z1)))
g(f(z0), z1) -> f(h(z0, z1))
h(z0, z1) -> g(z0, f(z1))

Types:
G :: f -> f -> c
f :: f -> f
c :: c1 -> c
H :: f -> f -> c1
c1 :: c -> c1
g :: f -> f -> f
h :: f -> f -> f
hole_c1_2 :: c
hole_f2_2 :: f
hole_c13_2 :: c1
gen_f4_2 :: Nat -> f


Lemmas:
h(gen_f4_2(n6_2), gen_f4_2(b)) -> *5_2, rt in Omega(0)
g(gen_f4_2(+(1, n314_2)), gen_f4_2(b)) -> *5_2, rt in Omega(0)


Generator Equations:
gen_f4_2(0) <=> hole_f2_2
gen_f4_2(+(x, 1)) <=> f(gen_f4_2(x))


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

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

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
h(gen_f4_2(n2131_2), gen_f4_2(b)) -> *5_2, rt in Omega(0)

Induction Base:
h(gen_f4_2(0), gen_f4_2(b))

Induction Step:
h(gen_f4_2(+(n2131_2, 1)), gen_f4_2(b)) ->_R^Omega(0)
g(gen_f4_2(+(n2131_2, 1)), f(gen_f4_2(b))) ->_R^Omega(0)
f(h(gen_f4_2(n2131_2), f(gen_f4_2(b)))) ->_IH
f(*5_2)

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:
G(f(z0), z1) -> c(H(z0, z1))
H(z0, z1) -> c1(G(z0, f(z1)))
g(f(z0), z1) -> f(h(z0, z1))
h(z0, z1) -> g(z0, f(z1))

Types:
G :: f -> f -> c
f :: f -> f
c :: c1 -> c
H :: f -> f -> c1
c1 :: c -> c1
g :: f -> f -> f
h :: f -> f -> f
hole_c1_2 :: c
hole_f2_2 :: f
hole_c13_2 :: c1
gen_f4_2 :: Nat -> f


Lemmas:
h(gen_f4_2(n2131_2), gen_f4_2(b)) -> *5_2, rt in Omega(0)
g(gen_f4_2(+(1, n314_2)), gen_f4_2(b)) -> *5_2, rt in Omega(0)


Generator Equations:
gen_f4_2(0) <=> hole_f2_2
gen_f4_2(+(x, 1)) <=> f(gen_f4_2(x))


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

They will be analysed ascendingly in the following order:
G = H

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
H(gen_f4_2(n5465_2), gen_f4_2(b)) -> *5_2, rt in Omega(n5465_2)

Induction Base:
H(gen_f4_2(0), gen_f4_2(b))

Induction Step:
H(gen_f4_2(+(n5465_2, 1)), gen_f4_2(b)) ->_R^Omega(1)
c1(G(gen_f4_2(+(n5465_2, 1)), f(gen_f4_2(b)))) ->_R^Omega(1)
c1(c(H(gen_f4_2(n5465_2), f(gen_f4_2(b))))) ->_IH
c1(c(*5_2))

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:

Innermost TRS:
Rules:
G(f(z0), z1) -> c(H(z0, z1))
H(z0, z1) -> c1(G(z0, f(z1)))
g(f(z0), z1) -> f(h(z0, z1))
h(z0, z1) -> g(z0, f(z1))

Types:
G :: f -> f -> c
f :: f -> f
c :: c1 -> c
H :: f -> f -> c1
c1 :: c -> c1
g :: f -> f -> f
h :: f -> f -> f
hole_c1_2 :: c
hole_f2_2 :: f
hole_c13_2 :: c1
gen_f4_2 :: Nat -> f


Lemmas:
h(gen_f4_2(n2131_2), gen_f4_2(b)) -> *5_2, rt in Omega(0)
g(gen_f4_2(+(1, n314_2)), gen_f4_2(b)) -> *5_2, rt in Omega(0)


Generator Equations:
gen_f4_2(0) <=> hole_f2_2
gen_f4_2(+(x, 1)) <=> f(gen_f4_2(x))


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

They will be analysed ascendingly in the following order:
G = H

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

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

(25)
BOUNDS(n^1, INF)

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

(26)
Obligation:
Innermost TRS:
Rules:
G(f(z0), z1) -> c(H(z0, z1))
H(z0, z1) -> c1(G(z0, f(z1)))
g(f(z0), z1) -> f(h(z0, z1))
h(z0, z1) -> g(z0, f(z1))

Types:
G :: f -> f -> c
f :: f -> f
c :: c1 -> c
H :: f -> f -> c1
c1 :: c -> c1
g :: f -> f -> f
h :: f -> f -> f
hole_c1_2 :: c
hole_f2_2 :: f
hole_c13_2 :: c1
gen_f4_2 :: Nat -> f


Lemmas:
h(gen_f4_2(n2131_2), gen_f4_2(b)) -> *5_2, rt in Omega(0)
g(gen_f4_2(+(1, n314_2)), gen_f4_2(b)) -> *5_2, rt in Omega(0)
H(gen_f4_2(n5465_2), gen_f4_2(b)) -> *5_2, rt in Omega(n5465_2)


Generator Equations:
gen_f4_2(0) <=> hole_f2_2
gen_f4_2(+(x, 1)) <=> f(gen_f4_2(x))


The following defined symbols remain to be analysed:
G

They will be analysed ascendingly in the following order:
G = H

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
G(gen_f4_2(+(1, n8833_2)), gen_f4_2(b)) -> *5_2, rt in Omega(n8833_2)

Induction Base:
G(gen_f4_2(+(1, 0)), gen_f4_2(b))

Induction Step:
G(gen_f4_2(+(1, +(n8833_2, 1))), gen_f4_2(b)) ->_R^Omega(1)
c(H(gen_f4_2(+(1, n8833_2)), gen_f4_2(b))) ->_R^Omega(1)
c(c1(G(gen_f4_2(+(1, n8833_2)), f(gen_f4_2(b))))) ->_IH
c(c1(*5_2))

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

(28)
Obligation:
Innermost TRS:
Rules:
G(f(z0), z1) -> c(H(z0, z1))
H(z0, z1) -> c1(G(z0, f(z1)))
g(f(z0), z1) -> f(h(z0, z1))
h(z0, z1) -> g(z0, f(z1))

Types:
G :: f -> f -> c
f :: f -> f
c :: c1 -> c
H :: f -> f -> c1
c1 :: c -> c1
g :: f -> f -> f
h :: f -> f -> f
hole_c1_2 :: c
hole_f2_2 :: f
hole_c13_2 :: c1
gen_f4_2 :: Nat -> f


Lemmas:
h(gen_f4_2(n2131_2), gen_f4_2(b)) -> *5_2, rt in Omega(0)
g(gen_f4_2(+(1, n314_2)), gen_f4_2(b)) -> *5_2, rt in Omega(0)
H(gen_f4_2(n5465_2), gen_f4_2(b)) -> *5_2, rt in Omega(n5465_2)
G(gen_f4_2(+(1, n8833_2)), gen_f4_2(b)) -> *5_2, rt in Omega(n8833_2)


Generator Equations:
gen_f4_2(0) <=> hole_f2_2
gen_f4_2(+(x, 1)) <=> f(gen_f4_2(x))


The following defined symbols remain to be analysed:
H

They will be analysed ascendingly in the following order:
G = H

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
H(gen_f4_2(n13603_2), gen_f4_2(b)) -> *5_2, rt in Omega(n13603_2)

Induction Base:
H(gen_f4_2(0), gen_f4_2(b))

Induction Step:
H(gen_f4_2(+(n13603_2, 1)), gen_f4_2(b)) ->_R^Omega(1)
c1(G(gen_f4_2(+(n13603_2, 1)), f(gen_f4_2(b)))) ->_R^Omega(1)
c1(c(H(gen_f4_2(n13603_2), f(gen_f4_2(b))))) ->_IH
c1(c(*5_2))

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

(30)
BOUNDS(1, INF)