MAYBE
proof of input_ND62t60Gka.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CdtProblem
    (5) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxRelTRS
    (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
        (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (10) typed CpxTrs
        (11) OrderProof [LOWER BOUND(ID), 0 ms]
        (12) typed CpxTrs
    (13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
    (14) TRS for Loop Detection
(15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(16) CpxTRS
(17) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(18) CdtProblem
    (19) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (20) CdtProblem
    (21) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CdtProblem
    (23) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxTRS
    (25) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (26) CpxTRS
    (27) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CpxWeightedTrs
        (29) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (30) CpxTypedWeightedTrs
        (31) CompletionProof [UPPER BOUND(ID), 0 ms]
        (32) CpxTypedWeightedCompleteTrs
            (33) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
            (34) CpxTypedWeightedCompleteTrs
            (35) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
            (36) CpxRNTS
            (37) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
            (38) CpxRNTS
            (39) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
            (40) CpxRNTS
            (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
            (42) CpxRNTS
            (43) IntTrsBoundProof [UPPER BOUND(ID), 5691 ms]
            (44) CpxRNTS
            (45) IntTrsBoundProof [UPPER BOUND(ID), 1826 ms]
            (46) CpxRNTS
        (47) CompletionProof [UPPER BOUND(ID), 0 ms]
        (48) CpxTypedWeightedCompleteTrs
            (49) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
            (50) CpxRNTS
(51) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(52) CpxWeightedTrs
    (53) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CpxTypedWeightedTrs
    (55) CompletionProof [UPPER BOUND(ID), 0 ms]
    (56) CpxTypedWeightedCompleteTrs
        (57) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (58) CpxTypedWeightedCompleteTrs
        (59) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (60) CpxRNTS
        (61) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (62) CpxRNTS
        (63) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (64) CpxRNTS
        (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (66) CpxRNTS
        (67) IntTrsBoundProof [UPPER BOUND(ID), 1129 ms]
        (68) CpxRNTS
        (69) IntTrsBoundProof [UPPER BOUND(ID), 341 ms]
        (70) CpxRNTS
    (71) CompletionProof [UPPER BOUND(ID), 0 ms]
    (72) CpxTypedWeightedCompleteTrs
        (73) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (74) CpxRNTS


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

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


The TRS R consists of the following rules:

   g(Cons(x, xs), y) -> Cons(x, xs)
   h(Nil, y) -> h(Nil, y)
   h(Cons(x, xs), y) -> f(Cons(x, xs), y)
   g(Nil, y) -> h(Nil, y)
   f(Nil, y) -> g(Nil, y)
   f(Cons(x, xs), y) -> h(Cons(x, xs), y)
   sp1(x, y) -> f(x, y)
   r(x, y) -> x

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

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

(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   g(Cons(x, xs), y) -> Cons(x, xs)
   h(Nil, y) -> h(Nil, y)
   h(Cons(x, xs), y) -> f(Cons(x, xs), y)
   g(Nil, y) -> h(Nil, y)
   f(Nil, y) -> g(Nil, y)
   f(Cons(x, xs), y) -> h(Cons(x, xs), y)
   sp1(x, y) -> f(x, y)
   r(x, y) -> x

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

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(Cons(z0, z1), z2) -> Cons(z0, z1)
   g(Nil, z0) -> h(Nil, z0)
   h(Nil, z0) -> h(Nil, z0)
   h(Cons(z0, z1), z2) -> f(Cons(z0, z1), z2)
   f(Nil, z0) -> g(Nil, z0)
   f(Cons(z0, z1), z2) -> h(Cons(z0, z1), z2)
   sp1(z0, z1) -> f(z0, z1)
   r(z0, z1) -> z0
Tuples:
   G(Cons(z0, z1), z2) -> c
   G(Nil, z0) -> c1(H(Nil, z0))
   H(Nil, z0) -> c2(H(Nil, z0))
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
   F(Nil, z0) -> c4(G(Nil, z0))
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))
   SP1(z0, z1) -> c6(F(z0, z1))
   R(z0, z1) -> c7
S tuples:
   G(Cons(z0, z1), z2) -> c
   G(Nil, z0) -> c1(H(Nil, z0))
   H(Nil, z0) -> c2(H(Nil, z0))
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
   F(Nil, z0) -> c4(G(Nil, z0))
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))
   SP1(z0, z1) -> c6(F(z0, z1))
   R(z0, z1) -> c7
K tuples:none
Defined Rule Symbols:   g_2, h_2, f_2, sp1_2, r_2

Defined Pair Symbols:   G_2, H_2, F_2, SP1_2, R_2

Compound Symbols:   c, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7


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

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

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


The TRS R consists of the following rules:

   G(Cons(z0, z1), z2) -> c
   G(Nil, z0) -> c1(H(Nil, z0))
   H(Nil, z0) -> c2(H(Nil, z0))
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
   F(Nil, z0) -> c4(G(Nil, z0))
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))
   SP1(z0, z1) -> c6(F(z0, z1))
   R(z0, z1) -> c7

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

   g(Cons(z0, z1), z2) -> Cons(z0, z1)
   g(Nil, z0) -> h(Nil, z0)
   h(Nil, z0) -> h(Nil, z0)
   h(Cons(z0, z1), z2) -> f(Cons(z0, z1), z2)
   f(Nil, z0) -> g(Nil, z0)
   f(Cons(z0, z1), z2) -> h(Cons(z0, z1), z2)
   sp1(z0, z1) -> f(z0, z1)
   r(z0, z1) -> z0

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

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

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   G(Cons(z0, z1), z2) -> c
   G(Nil, z0) -> c1(H(Nil, z0))
   H(Nil, z0) -> c2(H(Nil, z0))
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
   F(Nil, z0) -> c4(G(Nil, z0))
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))
   SP1(z0, z1) -> c6(F(z0, z1))
   R(z0, z1) -> c7

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

   g(Cons(z0, z1), z2) -> Cons(z0, z1)
   g(Nil, z0) -> h(Nil, z0)
   h(Nil, z0) -> h(Nil, z0)
   h(Cons(z0, z1), z2) -> f(Cons(z0, z1), z2)
   f(Nil, z0) -> g(Nil, z0)
   f(Cons(z0, z1), z2) -> h(Cons(z0, z1), z2)
   sp1(z0, z1) -> f(z0, z1)
   r(z0, z1) -> z0

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

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

(10)
Obligation:
Innermost TRS:
Rules:
G(Cons(z0, z1), z2) -> c
G(Nil, z0) -> c1(H(Nil, z0))
H(Nil, z0) -> c2(H(Nil, z0))
H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
F(Nil, z0) -> c4(G(Nil, z0))
F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))
SP1(z0, z1) -> c6(F(z0, z1))
R(z0, z1) -> c7
g(Cons(z0, z1), z2) -> Cons(z0, z1)
g(Nil, z0) -> h(Nil, z0)
h(Nil, z0) -> h(Nil, z0)
h(Cons(z0, z1), z2) -> f(Cons(z0, z1), z2)
f(Nil, z0) -> g(Nil, z0)
f(Cons(z0, z1), z2) -> h(Cons(z0, z1), z2)
sp1(z0, z1) -> f(z0, z1)
r(z0, z1) -> z0

Types:
G :: Cons:Nil -> c -> c:c1
Cons :: a -> b -> Cons:Nil
c :: c:c1
Nil :: Cons:Nil
c1 :: c2:c3 -> c:c1
H :: Cons:Nil -> c -> c2:c3
c2 :: c2:c3 -> c2:c3
c3 :: c4:c5 -> c2:c3
F :: Cons:Nil -> c -> c4:c5
c4 :: c:c1 -> c4:c5
c5 :: c2:c3 -> c4:c5
SP1 :: Cons:Nil -> c -> c6
c6 :: c4:c5 -> c6
R :: d -> e -> c7
c7 :: c7
g :: Cons:Nil -> f -> Cons:Nil
h :: Cons:Nil -> f -> Cons:Nil
f :: Cons:Nil -> f -> Cons:Nil
sp1 :: Cons:Nil -> f -> Cons:Nil
r :: r -> g -> r
hole_c:c11_8 :: c:c1
hole_Cons:Nil2_8 :: Cons:Nil
hole_c3_8 :: c
hole_a4_8 :: a
hole_b5_8 :: b
hole_c2:c36_8 :: c2:c3
hole_c4:c57_8 :: c4:c5
hole_c68_8 :: c6
hole_c79_8 :: c7
hole_d10_8 :: d
hole_e11_8 :: e
hole_f12_8 :: f
hole_r13_8 :: r
hole_g14_8 :: g
gen_c2:c315_8 :: Nat -> c2:c3

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

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

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

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

(12)
Obligation:
Innermost TRS:
Rules:
G(Cons(z0, z1), z2) -> c
G(Nil, z0) -> c1(H(Nil, z0))
H(Nil, z0) -> c2(H(Nil, z0))
H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
F(Nil, z0) -> c4(G(Nil, z0))
F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))
SP1(z0, z1) -> c6(F(z0, z1))
R(z0, z1) -> c7
g(Cons(z0, z1), z2) -> Cons(z0, z1)
g(Nil, z0) -> h(Nil, z0)
h(Nil, z0) -> h(Nil, z0)
h(Cons(z0, z1), z2) -> f(Cons(z0, z1), z2)
f(Nil, z0) -> g(Nil, z0)
f(Cons(z0, z1), z2) -> h(Cons(z0, z1), z2)
sp1(z0, z1) -> f(z0, z1)
r(z0, z1) -> z0

Types:
G :: Cons:Nil -> c -> c:c1
Cons :: a -> b -> Cons:Nil
c :: c:c1
Nil :: Cons:Nil
c1 :: c2:c3 -> c:c1
H :: Cons:Nil -> c -> c2:c3
c2 :: c2:c3 -> c2:c3
c3 :: c4:c5 -> c2:c3
F :: Cons:Nil -> c -> c4:c5
c4 :: c:c1 -> c4:c5
c5 :: c2:c3 -> c4:c5
SP1 :: Cons:Nil -> c -> c6
c6 :: c4:c5 -> c6
R :: d -> e -> c7
c7 :: c7
g :: Cons:Nil -> f -> Cons:Nil
h :: Cons:Nil -> f -> Cons:Nil
f :: Cons:Nil -> f -> Cons:Nil
sp1 :: Cons:Nil -> f -> Cons:Nil
r :: r -> g -> r
hole_c:c11_8 :: c:c1
hole_Cons:Nil2_8 :: Cons:Nil
hole_c3_8 :: c
hole_a4_8 :: a
hole_b5_8 :: b
hole_c2:c36_8 :: c2:c3
hole_c4:c57_8 :: c4:c5
hole_c68_8 :: c6
hole_c79_8 :: c7
hole_d10_8 :: d
hole_e11_8 :: e
hole_f12_8 :: f
hole_r13_8 :: r
hole_g14_8 :: g
gen_c2:c315_8 :: Nat -> c2:c3


Generator Equations:
gen_c2:c315_8(0) <=> c3(c4(c))
gen_c2:c315_8(+(x, 1)) <=> c2(gen_c2:c315_8(x))


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

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

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

(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(14)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   G(Cons(z0, z1), z2) -> c
   G(Nil, z0) -> c1(H(Nil, z0))
   H(Nil, z0) -> c2(H(Nil, z0))
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
   F(Nil, z0) -> c4(G(Nil, z0))
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))
   SP1(z0, z1) -> c6(F(z0, z1))
   R(z0, z1) -> c7

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

   g(Cons(z0, z1), z2) -> Cons(z0, z1)
   g(Nil, z0) -> h(Nil, z0)
   h(Nil, z0) -> h(Nil, z0)
   h(Cons(z0, z1), z2) -> f(Cons(z0, z1), z2)
   f(Nil, z0) -> g(Nil, z0)
   f(Cons(z0, z1), z2) -> h(Cons(z0, z1), z2)
   sp1(z0, z1) -> f(z0, z1)
   r(z0, z1) -> z0

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

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

(16)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   g(Cons(x, xs), y) -> Cons(x, xs)
   h(Nil, y) -> h(Nil, y)
   h(Cons(x, xs), y) -> f(Cons(x, xs), y)
   g(Nil, y) -> h(Nil, y)
   f(Nil, y) -> g(Nil, y)
   f(Cons(x, xs), y) -> h(Cons(x, xs), y)
   sp1(x, y) -> f(x, y)
   r(x, y) -> x

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

(17) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(Cons(z0, z1), z2) -> Cons(z0, z1)
   g(Nil, z0) -> h(Nil, z0)
   h(Nil, z0) -> h(Nil, z0)
   h(Cons(z0, z1), z2) -> f(Cons(z0, z1), z2)
   f(Nil, z0) -> g(Nil, z0)
   f(Cons(z0, z1), z2) -> h(Cons(z0, z1), z2)
   sp1(z0, z1) -> f(z0, z1)
   r(z0, z1) -> z0
Tuples:
   G(Cons(z0, z1), z2) -> c
   G(Nil, z0) -> c1(H(Nil, z0))
   H(Nil, z0) -> c2(H(Nil, z0))
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
   F(Nil, z0) -> c4(G(Nil, z0))
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))
   SP1(z0, z1) -> c6(F(z0, z1))
   R(z0, z1) -> c7
S tuples:
   G(Cons(z0, z1), z2) -> c
   G(Nil, z0) -> c1(H(Nil, z0))
   H(Nil, z0) -> c2(H(Nil, z0))
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
   F(Nil, z0) -> c4(G(Nil, z0))
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))
   SP1(z0, z1) -> c6(F(z0, z1))
   R(z0, z1) -> c7
K tuples:none
Defined Rule Symbols:   g_2, h_2, f_2, sp1_2, r_2

Defined Pair Symbols:   G_2, H_2, F_2, SP1_2, R_2

Compound Symbols:   c, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7


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

(19) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 3 leading nodes:
   SP1(z0, z1) -> c6(F(z0, z1))
   F(Nil, z0) -> c4(G(Nil, z0))
   G(Nil, z0) -> c1(H(Nil, z0))
Removed 2 trailing nodes:
   G(Cons(z0, z1), z2) -> c
   R(z0, z1) -> c7

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(Cons(z0, z1), z2) -> Cons(z0, z1)
   g(Nil, z0) -> h(Nil, z0)
   h(Nil, z0) -> h(Nil, z0)
   h(Cons(z0, z1), z2) -> f(Cons(z0, z1), z2)
   f(Nil, z0) -> g(Nil, z0)
   f(Cons(z0, z1), z2) -> h(Cons(z0, z1), z2)
   sp1(z0, z1) -> f(z0, z1)
   r(z0, z1) -> z0
Tuples:
   H(Nil, z0) -> c2(H(Nil, z0))
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))
S tuples:
   H(Nil, z0) -> c2(H(Nil, z0))
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))
K tuples:none
Defined Rule Symbols:   g_2, h_2, f_2, sp1_2, r_2

Defined Pair Symbols:   H_2, F_2

Compound Symbols:   c2_1, c3_1, c5_1


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

(21) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   g(Cons(z0, z1), z2) -> Cons(z0, z1)
   g(Nil, z0) -> h(Nil, z0)
   h(Nil, z0) -> h(Nil, z0)
   h(Cons(z0, z1), z2) -> f(Cons(z0, z1), z2)
   f(Nil, z0) -> g(Nil, z0)
   f(Cons(z0, z1), z2) -> h(Cons(z0, z1), z2)
   sp1(z0, z1) -> f(z0, z1)
   r(z0, z1) -> z0

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   H(Nil, z0) -> c2(H(Nil, z0))
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))
S tuples:
   H(Nil, z0) -> c2(H(Nil, z0))
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   H_2, F_2

Compound Symbols:   c2_1, c3_1, c5_1


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

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

(24)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   H(Nil, z0) -> c2(H(Nil, z0))
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))

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

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

(26)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   H(Nil, z0) -> c2(H(Nil, z0))
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))

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

(27) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(28)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   H(Nil, z0) -> c2(H(Nil, z0)) [1]
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2)) [1]
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2)) [1]

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

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

(30)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   H(Nil, z0) -> c2(H(Nil, z0)) [1]
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2)) [1]
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2)) [1]

The TRS has the following type information:
H :: Nil:Cons -> a -> c2:c3
Nil :: Nil:Cons
c2 :: c2:c3 -> c2:c3
Cons :: b -> c -> Nil:Cons
c3 :: c5 -> c2:c3
F :: Nil:Cons -> a -> c5
c5 :: c2:c3 -> c5

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

(31) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   H_2
   F_2

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1, const2, const3, const4

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

(32)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   H(Nil, z0) -> c2(H(Nil, z0)) [1]
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2)) [1]
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2)) [1]

The TRS has the following type information:
H :: Nil:Cons -> a -> c2:c3
Nil :: Nil:Cons
c2 :: c2:c3 -> c2:c3
Cons :: b -> c -> Nil:Cons
c3 :: c5 -> c2:c3
F :: Nil:Cons -> a -> c5
c5 :: c2:c3 -> c5
const :: c2:c3
const1 :: a
const2 :: b
const3 :: c
const4 :: c5

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

(33) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(34)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   H(Nil, z0) -> c2(H(Nil, z0)) [1]
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2)) [1]
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2)) [1]

The TRS has the following type information:
H :: Nil:Cons -> a -> c2:c3
Nil :: Nil:Cons
c2 :: c2:c3 -> c2:c3
Cons :: b -> c -> Nil:Cons
c3 :: c5 -> c2:c3
F :: Nil:Cons -> a -> c5
c5 :: c2:c3 -> c5
const :: c2:c3
const1 :: a
const2 :: b
const3 :: c
const4 :: c5

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

(35) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   Nil => 0
   const => 0
   const1 => 0
   const2 => 0
   const3 => 0
   const4 => 0

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

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z') -{ 1 }-> 1 + H(1 + z0 + z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   H(z, z') -{ 1 }-> 1 + H(0, z0) :|: z0 >= 0, z = 0, z' = z0
   H(z, z') -{ 1 }-> 1 + F(1 + z0 + z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0


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

(37) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z') -{ 1 }-> 1 + H(1 + z0 + z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   H(z, z') -{ 1 }-> 1 + H(0, z') :|: z' >= 0, z = 0
   H(z, z') -{ 1 }-> 1 + F(1 + z0 + z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0


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

(39) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { H, F }

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

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z') -{ 1 }-> 1 + H(1 + z0 + z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   H(z, z') -{ 1 }-> 1 + H(0, z') :|: z' >= 0, z = 0
   H(z, z') -{ 1 }-> 1 + F(1 + z0 + z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0

Function symbols to be analyzed: {H,F}

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

(41) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z') -{ 1 }-> 1 + H(1 + z0 + z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   H(z, z') -{ 1 }-> 1 + H(0, z') :|: z' >= 0, z = 0
   H(z, z') -{ 1 }-> 1 + F(1 + z0 + z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0

Function symbols to be analyzed: {H,F}

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: H
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed SIZE bound using CoFloCo for: F
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z') -{ 1 }-> 1 + H(1 + z0 + z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   H(z, z') -{ 1 }-> 1 + H(0, z') :|: z' >= 0, z = 0
   H(z, z') -{ 1 }-> 1 + F(1 + z0 + z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0

Function symbols to be analyzed: {H,F}
Previous analysis results are:
  H: runtime: ?, size: O(1) [0]
  F: runtime: ?, size: O(1) [1]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: H
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z') -{ 1 }-> 1 + H(1 + z0 + z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   H(z, z') -{ 1 }-> 1 + H(0, z') :|: z' >= 0, z = 0
   H(z, z') -{ 1 }-> 1 + F(1 + z0 + z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0

Function symbols to be analyzed: {H,F}
Previous analysis results are:
  H: runtime: INF, size: O(1) [0]
  F: runtime: ?, size: O(1) [1]

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

(47) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   F(v0, v1) -> null_F [0]

And the following fresh constants:    null_F, const, const1, const2, const3

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

(48)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   H(Nil, z0) -> c2(H(Nil, z0)) [1]
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2)) [1]
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2)) [1]
   F(v0, v1) -> null_F [0]

The TRS has the following type information:
H :: Nil:Cons -> a -> c2:c3
Nil :: Nil:Cons
c2 :: c2:c3 -> c2:c3
Cons :: b -> c -> Nil:Cons
c3 :: c5:null_F -> c2:c3
F :: Nil:Cons -> a -> c5:null_F
c5 :: c2:c3 -> c5:null_F
null_F :: c5:null_F
const :: c2:c3
const1 :: a
const2 :: b
const3 :: c

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

(49) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   Nil => 0
   null_F => 0
   const => 0
   const1 => 0
   const2 => 0
   const3 => 0

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

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   F(z, z') -{ 1 }-> 1 + H(1 + z0 + z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   H(z, z') -{ 1 }-> 1 + H(0, z0) :|: z0 >= 0, z = 0, z' = z0
   H(z, z') -{ 1 }-> 1 + F(1 + z0 + z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0

Only complete derivations are relevant for the runtime complexity.

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

(51) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(52)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   g(Cons(x, xs), y) -> Cons(x, xs) [1]
   h(Nil, y) -> h(Nil, y) [1]
   h(Cons(x, xs), y) -> f(Cons(x, xs), y) [1]
   g(Nil, y) -> h(Nil, y) [1]
   f(Nil, y) -> g(Nil, y) [1]
   f(Cons(x, xs), y) -> h(Cons(x, xs), y) [1]
   sp1(x, y) -> f(x, y) [1]
   r(x, y) -> x [1]

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

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

(54)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   g(Cons(x, xs), y) -> Cons(x, xs) [1]
   h(Nil, y) -> h(Nil, y) [1]
   h(Cons(x, xs), y) -> f(Cons(x, xs), y) [1]
   g(Nil, y) -> h(Nil, y) [1]
   f(Nil, y) -> g(Nil, y) [1]
   f(Cons(x, xs), y) -> h(Cons(x, xs), y) [1]
   sp1(x, y) -> f(x, y) [1]
   r(x, y) -> x [1]

The TRS has the following type information:
g :: Cons:Nil -> c -> Cons:Nil
Cons :: a -> b -> Cons:Nil
h :: Cons:Nil -> c -> Cons:Nil
Nil :: Cons:Nil
f :: Cons:Nil -> c -> Cons:Nil
sp1 :: Cons:Nil -> c -> Cons:Nil
r :: r -> d -> r

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

(55) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   g_2
   h_2
   f_2
   sp1_2
   r_2

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1, const2, const3, const4

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

(56)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   g(Cons(x, xs), y) -> Cons(x, xs) [1]
   h(Nil, y) -> h(Nil, y) [1]
   h(Cons(x, xs), y) -> f(Cons(x, xs), y) [1]
   g(Nil, y) -> h(Nil, y) [1]
   f(Nil, y) -> g(Nil, y) [1]
   f(Cons(x, xs), y) -> h(Cons(x, xs), y) [1]
   sp1(x, y) -> f(x, y) [1]
   r(x, y) -> x [1]

The TRS has the following type information:
g :: Cons:Nil -> c -> Cons:Nil
Cons :: a -> b -> Cons:Nil
h :: Cons:Nil -> c -> Cons:Nil
Nil :: Cons:Nil
f :: Cons:Nil -> c -> Cons:Nil
sp1 :: Cons:Nil -> c -> Cons:Nil
r :: r -> d -> r
const :: c
const1 :: a
const2 :: b
const3 :: r
const4 :: d

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

(57) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(58)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   g(Cons(x, xs), y) -> Cons(x, xs) [1]
   h(Nil, y) -> h(Nil, y) [1]
   h(Cons(x, xs), y) -> f(Cons(x, xs), y) [1]
   g(Nil, y) -> h(Nil, y) [1]
   f(Nil, y) -> g(Nil, y) [1]
   f(Cons(x, xs), y) -> h(Cons(x, xs), y) [1]
   sp1(x, y) -> f(x, y) [1]
   r(x, y) -> x [1]

The TRS has the following type information:
g :: Cons:Nil -> c -> Cons:Nil
Cons :: a -> b -> Cons:Nil
h :: Cons:Nil -> c -> Cons:Nil
Nil :: Cons:Nil
f :: Cons:Nil -> c -> Cons:Nil
sp1 :: Cons:Nil -> c -> Cons:Nil
r :: r -> d -> r
const :: c
const1 :: a
const2 :: b
const3 :: r
const4 :: d

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

(59) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   Nil => 0
   const => 0
   const1 => 0
   const2 => 0
   const3 => 0
   const4 => 0

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

(60)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 1 }-> h(1 + x + xs, y) :|: z = 1 + x + xs, xs >= 0, x >= 0, y >= 0, z' = y
   f(z, z') -{ 1 }-> g(0, y) :|: y >= 0, z = 0, z' = y
   g(z, z') -{ 1 }-> h(0, y) :|: y >= 0, z = 0, z' = y
   g(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, y >= 0, z' = y
   h(z, z') -{ 1 }-> h(0, y) :|: y >= 0, z = 0, z' = y
   h(z, z') -{ 1 }-> f(1 + x + xs, y) :|: z = 1 + x + xs, xs >= 0, x >= 0, y >= 0, z' = y
   r(z, z') -{ 1 }-> x :|: x >= 0, y >= 0, z = x, z' = y
   sp1(z, z') -{ 1 }-> f(x, y) :|: x >= 0, y >= 0, z = x, z' = y


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

(61) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(62)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 1 }-> h(1 + x + xs, z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   f(z, z') -{ 1 }-> g(0, z') :|: z' >= 0, z = 0
   g(z, z') -{ 1 }-> h(0, z') :|: z' >= 0, z = 0
   g(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   h(z, z') -{ 1 }-> h(0, z') :|: z' >= 0, z = 0
   h(z, z') -{ 1 }-> f(1 + x + xs, z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   r(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   sp1(z, z') -{ 1 }-> f(z, z') :|: z >= 0, z' >= 0


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

(63) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { f, h, g }
   { r }
   { sp1 }

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

(64)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 1 }-> h(1 + x + xs, z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   f(z, z') -{ 1 }-> g(0, z') :|: z' >= 0, z = 0
   g(z, z') -{ 1 }-> h(0, z') :|: z' >= 0, z = 0
   g(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   h(z, z') -{ 1 }-> h(0, z') :|: z' >= 0, z = 0
   h(z, z') -{ 1 }-> f(1 + x + xs, z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   r(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   sp1(z, z') -{ 1 }-> f(z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {f,h,g}, {r}, {sp1}

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

(65) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(66)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 1 }-> h(1 + x + xs, z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   f(z, z') -{ 1 }-> g(0, z') :|: z' >= 0, z = 0
   g(z, z') -{ 1 }-> h(0, z') :|: z' >= 0, z = 0
   g(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   h(z, z') -{ 1 }-> h(0, z') :|: z' >= 0, z = 0
   h(z, z') -{ 1 }-> f(1 + x + xs, z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   r(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   sp1(z, z') -{ 1 }-> f(z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {f,h,g}, {r}, {sp1}

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

(67) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed SIZE bound using CoFloCo for: h
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed SIZE bound using CoFloCo for: g
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

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

(68)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 1 }-> h(1 + x + xs, z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   f(z, z') -{ 1 }-> g(0, z') :|: z' >= 0, z = 0
   g(z, z') -{ 1 }-> h(0, z') :|: z' >= 0, z = 0
   g(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   h(z, z') -{ 1 }-> h(0, z') :|: z' >= 0, z = 0
   h(z, z') -{ 1 }-> f(1 + x + xs, z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   r(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   sp1(z, z') -{ 1 }-> f(z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {f,h,g}, {r}, {sp1}
Previous analysis results are:
  f: runtime: ?, size: O(1) [0]
  h: runtime: ?, size: O(1) [0]
  g: runtime: ?, size: O(n^1) [z]

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

(69) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

(70)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 1 }-> h(1 + x + xs, z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   f(z, z') -{ 1 }-> g(0, z') :|: z' >= 0, z = 0
   g(z, z') -{ 1 }-> h(0, z') :|: z' >= 0, z = 0
   g(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   h(z, z') -{ 1 }-> h(0, z') :|: z' >= 0, z = 0
   h(z, z') -{ 1 }-> f(1 + x + xs, z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   r(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   sp1(z, z') -{ 1 }-> f(z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {f,h,g}, {r}, {sp1}
Previous analysis results are:
  f: runtime: INF, size: O(1) [0]
  h: runtime: ?, size: O(1) [0]
  g: runtime: ?, size: O(n^1) [z]

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

(71) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
none

And the following fresh constants:    const, const1, const2, const3, const4

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

(72)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   g(Cons(x, xs), y) -> Cons(x, xs) [1]
   h(Nil, y) -> h(Nil, y) [1]
   h(Cons(x, xs), y) -> f(Cons(x, xs), y) [1]
   g(Nil, y) -> h(Nil, y) [1]
   f(Nil, y) -> g(Nil, y) [1]
   f(Cons(x, xs), y) -> h(Cons(x, xs), y) [1]
   sp1(x, y) -> f(x, y) [1]
   r(x, y) -> x [1]

The TRS has the following type information:
g :: Cons:Nil -> c -> Cons:Nil
Cons :: a -> b -> Cons:Nil
h :: Cons:Nil -> c -> Cons:Nil
Nil :: Cons:Nil
f :: Cons:Nil -> c -> Cons:Nil
sp1 :: Cons:Nil -> c -> Cons:Nil
r :: r -> d -> r
const :: c
const1 :: a
const2 :: b
const3 :: r
const4 :: d

Rewrite Strategy: INNERMOST
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(73) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   Nil => 0
   const => 0
   const1 => 0
   const2 => 0
   const3 => 0
   const4 => 0

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(74)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 1 }-> h(1 + x + xs, y) :|: z = 1 + x + xs, xs >= 0, x >= 0, y >= 0, z' = y
   f(z, z') -{ 1 }-> g(0, y) :|: y >= 0, z = 0, z' = y
   g(z, z') -{ 1 }-> h(0, y) :|: y >= 0, z = 0, z' = y
   g(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, y >= 0, z' = y
   h(z, z') -{ 1 }-> h(0, y) :|: y >= 0, z = 0, z' = y
   h(z, z') -{ 1 }-> f(1 + x + xs, y) :|: z = 1 + x + xs, xs >= 0, x >= 0, y >= 0, z' = y
   r(z, z') -{ 1 }-> x :|: x >= 0, y >= 0, z = x, z' = y
   sp1(z, z') -{ 1 }-> f(x, y) :|: x >= 0, y >= 0, z = x, z' = y

Only complete derivations are relevant for the runtime complexity.