WORST_CASE(Omega(n^1),O(n^1))
proof of input_6IGR1DOioS.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


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

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (4) CdtProblem
    (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTRS
    (11) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxTRS
    (13) CpxTrsMatchBoundsTAProof [FINISHED, 2 ms]
    (14) BOUNDS(1, n^1)
(15) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CdtProblem
    (17) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRelTRS
    (19) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxRelTRS
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) typed CpxTrs
    (23) OrderProof [LOWER BOUND(ID), 9 ms]
    (24) typed CpxTrs
    (25) RewriteLemmaProof [LOWER BOUND(ID), 271 ms]
    (26) typed CpxTrs
    (27) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
    (28) BEST
        (29) proven lower bound
            (30) LowerBoundPropagationProof [FINISHED, 0 ms]
            (31) BOUNDS(n^1, INF)
        (32) 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:

   foldl(x, Cons(S(0), xs)) -> foldl(S(x), xs)
   foldl(S(0), Cons(x, xs)) -> foldl(S(x), xs)
   foldr(a, Cons(x, xs)) -> op(x, foldr(a, xs))
   foldr(a, Nil) -> a
   foldl(a, Nil) -> a
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   op(x, S(0)) -> S(x)
   op(S(0), y) -> S(y)
   fold(a, xs) -> Cons(foldl(a, xs), Cons(foldr(a, xs), Nil))

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

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

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   foldl(z0, Cons(S(0), z1)) -> foldl(S(z0), z1)
   foldl(S(0), Cons(z0, z1)) -> foldl(S(z0), z1)
   foldl(z0, Nil) -> z0
   foldr(z0, Cons(z1, z2)) -> op(z1, foldr(z0, z2))
   foldr(z0, Nil) -> z0
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   op(z0, S(0)) -> S(z0)
   op(S(0), z0) -> S(z0)
   fold(z0, z1) -> Cons(foldl(z0, z1), Cons(foldr(z0, z1), Nil))
Tuples:
   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDL(z0, Nil) -> c2
   FOLDR(z0, Cons(z1, z2)) -> c3(OP(z1, foldr(z0, z2)), FOLDR(z0, z2))
   FOLDR(z0, Nil) -> c4
   NOTEMPTY(Cons(z0, z1)) -> c5
   NOTEMPTY(Nil) -> c6
   OP(z0, S(0)) -> c7
   OP(S(0), z0) -> c8
   FOLD(z0, z1) -> c9(FOLDL(z0, z1))
   FOLD(z0, z1) -> c10(FOLDR(z0, z1))
S tuples:
   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDL(z0, Nil) -> c2
   FOLDR(z0, Cons(z1, z2)) -> c3(OP(z1, foldr(z0, z2)), FOLDR(z0, z2))
   FOLDR(z0, Nil) -> c4
   NOTEMPTY(Cons(z0, z1)) -> c5
   NOTEMPTY(Nil) -> c6
   OP(z0, S(0)) -> c7
   OP(S(0), z0) -> c8
   FOLD(z0, z1) -> c9(FOLDL(z0, z1))
   FOLD(z0, z1) -> c10(FOLDR(z0, z1))
K tuples:none
Defined Rule Symbols:   foldl_2, foldr_2, notEmpty_1, op_2, fold_2

Defined Pair Symbols:   FOLDL_2, FOLDR_2, NOTEMPTY_1, OP_2, FOLD_2

Compound Symbols:   c_1, c1_1, c2, c3_2, c4, c5, c6, c7, c8, c9_1, c10_1


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

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   FOLD(z0, z1) -> c9(FOLDL(z0, z1))
   FOLD(z0, z1) -> c10(FOLDR(z0, z1))
Removed 6 trailing nodes:
   FOLDR(z0, Nil) -> c4
   FOLDL(z0, Nil) -> c2
   OP(S(0), z0) -> c8
   OP(z0, S(0)) -> c7
   NOTEMPTY(Nil) -> c6
   NOTEMPTY(Cons(z0, z1)) -> c5

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   foldl(z0, Cons(S(0), z1)) -> foldl(S(z0), z1)
   foldl(S(0), Cons(z0, z1)) -> foldl(S(z0), z1)
   foldl(z0, Nil) -> z0
   foldr(z0, Cons(z1, z2)) -> op(z1, foldr(z0, z2))
   foldr(z0, Nil) -> z0
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   op(z0, S(0)) -> S(z0)
   op(S(0), z0) -> S(z0)
   fold(z0, z1) -> Cons(foldl(z0, z1), Cons(foldr(z0, z1), Nil))
Tuples:
   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDR(z0, Cons(z1, z2)) -> c3(OP(z1, foldr(z0, z2)), FOLDR(z0, z2))
S tuples:
   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDR(z0, Cons(z1, z2)) -> c3(OP(z1, foldr(z0, z2)), FOLDR(z0, z2))
K tuples:none
Defined Rule Symbols:   foldl_2, foldr_2, notEmpty_1, op_2, fold_2

Defined Pair Symbols:   FOLDL_2, FOLDR_2

Compound Symbols:   c_1, c1_1, c3_2


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

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   foldl(z0, Cons(S(0), z1)) -> foldl(S(z0), z1)
   foldl(S(0), Cons(z0, z1)) -> foldl(S(z0), z1)
   foldl(z0, Nil) -> z0
   foldr(z0, Cons(z1, z2)) -> op(z1, foldr(z0, z2))
   foldr(z0, Nil) -> z0
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   op(z0, S(0)) -> S(z0)
   op(S(0), z0) -> S(z0)
   fold(z0, z1) -> Cons(foldl(z0, z1), Cons(foldr(z0, z1), Nil))
Tuples:
   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDR(z0, Cons(z1, z2)) -> c3(FOLDR(z0, z2))
S tuples:
   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDR(z0, Cons(z1, z2)) -> c3(FOLDR(z0, z2))
K tuples:none
Defined Rule Symbols:   foldl_2, foldr_2, notEmpty_1, op_2, fold_2

Defined Pair Symbols:   FOLDL_2, FOLDR_2

Compound Symbols:   c_1, c1_1, c3_1


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   foldl(z0, Cons(S(0), z1)) -> foldl(S(z0), z1)
   foldl(S(0), Cons(z0, z1)) -> foldl(S(z0), z1)
   foldl(z0, Nil) -> z0
   foldr(z0, Cons(z1, z2)) -> op(z1, foldr(z0, z2))
   foldr(z0, Nil) -> z0
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   op(z0, S(0)) -> S(z0)
   op(S(0), z0) -> S(z0)
   fold(z0, z1) -> Cons(foldl(z0, z1), Cons(foldr(z0, z1), Nil))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDR(z0, Cons(z1, z2)) -> c3(FOLDR(z0, z2))
S tuples:
   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDR(z0, Cons(z1, z2)) -> c3(FOLDR(z0, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   FOLDL_2, FOLDR_2

Compound Symbols:   c_1, c1_1, c3_1


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

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

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDR(z0, Cons(z1, z2)) -> c3(FOLDR(z0, z2))

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

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

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDR(z0, Cons(z1, z2)) -> c3(FOLDR(z0, z2))

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

(13) 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]
transitions: 
Cons0(0, 0) -> 0
S0(0) -> 0
00() -> 0
c0(0) -> 0
c10(0) -> 0
c30(0) -> 0
FOLDL0(0, 0) -> 1
FOLDR0(0, 0) -> 2
S1(0) -> 4
FOLDL1(4, 0) -> 3
c1(3) -> 1
FOLDL1(4, 0) -> 5
c11(5) -> 1
FOLDR1(0, 0) -> 6
c31(6) -> 2
S1(4) -> 4
c1(3) -> 3
c1(3) -> 5
c11(5) -> 3
c11(5) -> 5
c31(6) -> 6

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

(14)
BOUNDS(1, n^1)

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

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   foldl(z0, Cons(S(0), z1)) -> foldl(S(z0), z1)
   foldl(S(0), Cons(z0, z1)) -> foldl(S(z0), z1)
   foldl(z0, Nil) -> z0
   foldr(z0, Cons(z1, z2)) -> op(z1, foldr(z0, z2))
   foldr(z0, Nil) -> z0
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   op(z0, S(0)) -> S(z0)
   op(S(0), z0) -> S(z0)
   fold(z0, z1) -> Cons(foldl(z0, z1), Cons(foldr(z0, z1), Nil))
Tuples:
   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDL(z0, Nil) -> c2
   FOLDR(z0, Cons(z1, z2)) -> c3(OP(z1, foldr(z0, z2)), FOLDR(z0, z2))
   FOLDR(z0, Nil) -> c4
   NOTEMPTY(Cons(z0, z1)) -> c5
   NOTEMPTY(Nil) -> c6
   OP(z0, S(0)) -> c7
   OP(S(0), z0) -> c8
   FOLD(z0, z1) -> c9(FOLDL(z0, z1))
   FOLD(z0, z1) -> c10(FOLDR(z0, z1))
S tuples:
   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDL(z0, Nil) -> c2
   FOLDR(z0, Cons(z1, z2)) -> c3(OP(z1, foldr(z0, z2)), FOLDR(z0, z2))
   FOLDR(z0, Nil) -> c4
   NOTEMPTY(Cons(z0, z1)) -> c5
   NOTEMPTY(Nil) -> c6
   OP(z0, S(0)) -> c7
   OP(S(0), z0) -> c8
   FOLD(z0, z1) -> c9(FOLDL(z0, z1))
   FOLD(z0, z1) -> c10(FOLDR(z0, z1))
K tuples:none
Defined Rule Symbols:   foldl_2, foldr_2, notEmpty_1, op_2, fold_2

Defined Pair Symbols:   FOLDL_2, FOLDR_2, NOTEMPTY_1, OP_2, FOLD_2

Compound Symbols:   c_1, c1_1, c2, c3_2, c4, c5, c6, c7, c8, c9_1, c10_1


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

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

(18)
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:

   FOLDL(z0, Cons(S(0), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDL(z0, Nil) -> c2
   FOLDR(z0, Cons(z1, z2)) -> c3(OP(z1, foldr(z0, z2)), FOLDR(z0, z2))
   FOLDR(z0, Nil) -> c4
   NOTEMPTY(Cons(z0, z1)) -> c5
   NOTEMPTY(Nil) -> c6
   OP(z0, S(0)) -> c7
   OP(S(0), z0) -> c8
   FOLD(z0, z1) -> c9(FOLDL(z0, z1))
   FOLD(z0, z1) -> c10(FOLDR(z0, z1))

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

   foldl(z0, Cons(S(0), z1)) -> foldl(S(z0), z1)
   foldl(S(0), Cons(z0, z1)) -> foldl(S(z0), z1)
   foldl(z0, Nil) -> z0
   foldr(z0, Cons(z1, z2)) -> op(z1, foldr(z0, z2))
   foldr(z0, Nil) -> z0
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   op(z0, S(0)) -> S(z0)
   op(S(0), z0) -> S(z0)
   fold(z0, z1) -> Cons(foldl(z0, z1), Cons(foldr(z0, z1), Nil))

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

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

(20)
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:

   FOLDL(z0, Cons(S(0'), z1)) -> c(FOLDL(S(z0), z1))
   FOLDL(S(0'), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
   FOLDL(z0, Nil) -> c2
   FOLDR(z0, Cons(z1, z2)) -> c3(OP(z1, foldr(z0, z2)), FOLDR(z0, z2))
   FOLDR(z0, Nil) -> c4
   NOTEMPTY(Cons(z0, z1)) -> c5
   NOTEMPTY(Nil) -> c6
   OP(z0, S(0')) -> c7
   OP(S(0'), z0) -> c8
   FOLD(z0, z1) -> c9(FOLDL(z0, z1))
   FOLD(z0, z1) -> c10(FOLDR(z0, z1))

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

   foldl(z0, Cons(S(0'), z1)) -> foldl(S(z0), z1)
   foldl(S(0'), Cons(z0, z1)) -> foldl(S(z0), z1)
   foldl(z0, Nil) -> z0
   foldr(z0, Cons(z1, z2)) -> op(z1, foldr(z0, z2))
   foldr(z0, Nil) -> z0
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   op(z0, S(0')) -> S(z0)
   op(S(0'), z0) -> S(z0)
   fold(z0, z1) -> Cons(foldl(z0, z1), Cons(foldr(z0, z1), Nil))

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

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

(22)
Obligation:
Innermost TRS:
Rules:
FOLDL(z0, Cons(S(0'), z1)) -> c(FOLDL(S(z0), z1))
FOLDL(S(0'), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
FOLDL(z0, Nil) -> c2
FOLDR(z0, Cons(z1, z2)) -> c3(OP(z1, foldr(z0, z2)), FOLDR(z0, z2))
FOLDR(z0, Nil) -> c4
NOTEMPTY(Cons(z0, z1)) -> c5
NOTEMPTY(Nil) -> c6
OP(z0, S(0')) -> c7
OP(S(0'), z0) -> c8
FOLD(z0, z1) -> c9(FOLDL(z0, z1))
FOLD(z0, z1) -> c10(FOLDR(z0, z1))
foldl(z0, Cons(S(0'), z1)) -> foldl(S(z0), z1)
foldl(S(0'), Cons(z0, z1)) -> foldl(S(z0), z1)
foldl(z0, Nil) -> z0
foldr(z0, Cons(z1, z2)) -> op(z1, foldr(z0, z2))
foldr(z0, Nil) -> z0
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
op(z0, S(0')) -> S(z0)
op(S(0'), z0) -> S(z0)
fold(z0, z1) -> Cons(foldl(z0, z1), Cons(foldr(z0, z1), Nil))

Types:
FOLDL :: 0':S -> Cons:Nil -> c:c1:c2
Cons :: 0':S -> Cons:Nil -> Cons:Nil
S :: 0':S -> 0':S
0' :: 0':S
c :: c:c1:c2 -> c:c1:c2
c1 :: c:c1:c2 -> c:c1:c2
Nil :: Cons:Nil
c2 :: c:c1:c2
FOLDR :: 0':S -> Cons:Nil -> c3:c4
c3 :: c7:c8 -> c3:c4 -> c3:c4
OP :: 0':S -> 0':S -> c7:c8
foldr :: 0':S -> Cons:Nil -> 0':S
c4 :: c3:c4
NOTEMPTY :: Cons:Nil -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
c7 :: c7:c8
c8 :: c7:c8
FOLD :: 0':S -> Cons:Nil -> c9:c10
c9 :: c:c1:c2 -> c9:c10
c10 :: c3:c4 -> c9:c10
foldl :: 0':S -> Cons:Nil -> 0':S
op :: 0':S -> 0':S -> 0':S
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
fold :: 0':S -> Cons:Nil -> Cons:Nil
hole_c:c1:c21_11 :: c:c1:c2
hole_0':S2_11 :: 0':S
hole_Cons:Nil3_11 :: Cons:Nil
hole_c3:c44_11 :: c3:c4
hole_c7:c85_11 :: c7:c8
hole_c5:c66_11 :: c5:c6
hole_c9:c107_11 :: c9:c10
hole_True:False8_11 :: True:False
gen_c:c1:c29_11 :: Nat -> c:c1:c2
gen_0':S10_11 :: Nat -> 0':S
gen_Cons:Nil11_11 :: Nat -> Cons:Nil
gen_c3:c412_11 :: Nat -> c3:c4

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

(23) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
FOLDL, FOLDR, foldr, foldl

They will be analysed ascendingly in the following order:
foldr < FOLDR

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

(24)
Obligation:
Innermost TRS:
Rules:
FOLDL(z0, Cons(S(0'), z1)) -> c(FOLDL(S(z0), z1))
FOLDL(S(0'), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
FOLDL(z0, Nil) -> c2
FOLDR(z0, Cons(z1, z2)) -> c3(OP(z1, foldr(z0, z2)), FOLDR(z0, z2))
FOLDR(z0, Nil) -> c4
NOTEMPTY(Cons(z0, z1)) -> c5
NOTEMPTY(Nil) -> c6
OP(z0, S(0')) -> c7
OP(S(0'), z0) -> c8
FOLD(z0, z1) -> c9(FOLDL(z0, z1))
FOLD(z0, z1) -> c10(FOLDR(z0, z1))
foldl(z0, Cons(S(0'), z1)) -> foldl(S(z0), z1)
foldl(S(0'), Cons(z0, z1)) -> foldl(S(z0), z1)
foldl(z0, Nil) -> z0
foldr(z0, Cons(z1, z2)) -> op(z1, foldr(z0, z2))
foldr(z0, Nil) -> z0
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
op(z0, S(0')) -> S(z0)
op(S(0'), z0) -> S(z0)
fold(z0, z1) -> Cons(foldl(z0, z1), Cons(foldr(z0, z1), Nil))

Types:
FOLDL :: 0':S -> Cons:Nil -> c:c1:c2
Cons :: 0':S -> Cons:Nil -> Cons:Nil
S :: 0':S -> 0':S
0' :: 0':S
c :: c:c1:c2 -> c:c1:c2
c1 :: c:c1:c2 -> c:c1:c2
Nil :: Cons:Nil
c2 :: c:c1:c2
FOLDR :: 0':S -> Cons:Nil -> c3:c4
c3 :: c7:c8 -> c3:c4 -> c3:c4
OP :: 0':S -> 0':S -> c7:c8
foldr :: 0':S -> Cons:Nil -> 0':S
c4 :: c3:c4
NOTEMPTY :: Cons:Nil -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
c7 :: c7:c8
c8 :: c7:c8
FOLD :: 0':S -> Cons:Nil -> c9:c10
c9 :: c:c1:c2 -> c9:c10
c10 :: c3:c4 -> c9:c10
foldl :: 0':S -> Cons:Nil -> 0':S
op :: 0':S -> 0':S -> 0':S
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
fold :: 0':S -> Cons:Nil -> Cons:Nil
hole_c:c1:c21_11 :: c:c1:c2
hole_0':S2_11 :: 0':S
hole_Cons:Nil3_11 :: Cons:Nil
hole_c3:c44_11 :: c3:c4
hole_c7:c85_11 :: c7:c8
hole_c5:c66_11 :: c5:c6
hole_c9:c107_11 :: c9:c10
hole_True:False8_11 :: True:False
gen_c:c1:c29_11 :: Nat -> c:c1:c2
gen_0':S10_11 :: Nat -> 0':S
gen_Cons:Nil11_11 :: Nat -> Cons:Nil
gen_c3:c412_11 :: Nat -> c3:c4


Generator Equations:
gen_c:c1:c29_11(0) <=> c2
gen_c:c1:c29_11(+(x, 1)) <=> c(gen_c:c1:c29_11(x))
gen_0':S10_11(0) <=> 0'
gen_0':S10_11(+(x, 1)) <=> S(gen_0':S10_11(x))
gen_Cons:Nil11_11(0) <=> Nil
gen_Cons:Nil11_11(+(x, 1)) <=> Cons(0', gen_Cons:Nil11_11(x))
gen_c3:c412_11(0) <=> c4
gen_c3:c412_11(+(x, 1)) <=> c3(c7, gen_c3:c412_11(x))


The following defined symbols remain to be analysed:
FOLDL, FOLDR, foldr, foldl

They will be analysed ascendingly in the following order:
foldr < FOLDR

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
foldr(gen_0':S10_11(1), gen_Cons:Nil11_11(n25_11)) -> gen_0':S10_11(1), rt in Omega(0)

Induction Base:
foldr(gen_0':S10_11(1), gen_Cons:Nil11_11(0)) ->_R^Omega(0)
gen_0':S10_11(1)

Induction Step:
foldr(gen_0':S10_11(1), gen_Cons:Nil11_11(+(n25_11, 1))) ->_R^Omega(0)
op(0', foldr(gen_0':S10_11(1), gen_Cons:Nil11_11(n25_11))) ->_IH
op(0', gen_0':S10_11(1)) ->_R^Omega(0)
S(0')

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

(26)
Obligation:
Innermost TRS:
Rules:
FOLDL(z0, Cons(S(0'), z1)) -> c(FOLDL(S(z0), z1))
FOLDL(S(0'), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
FOLDL(z0, Nil) -> c2
FOLDR(z0, Cons(z1, z2)) -> c3(OP(z1, foldr(z0, z2)), FOLDR(z0, z2))
FOLDR(z0, Nil) -> c4
NOTEMPTY(Cons(z0, z1)) -> c5
NOTEMPTY(Nil) -> c6
OP(z0, S(0')) -> c7
OP(S(0'), z0) -> c8
FOLD(z0, z1) -> c9(FOLDL(z0, z1))
FOLD(z0, z1) -> c10(FOLDR(z0, z1))
foldl(z0, Cons(S(0'), z1)) -> foldl(S(z0), z1)
foldl(S(0'), Cons(z0, z1)) -> foldl(S(z0), z1)
foldl(z0, Nil) -> z0
foldr(z0, Cons(z1, z2)) -> op(z1, foldr(z0, z2))
foldr(z0, Nil) -> z0
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
op(z0, S(0')) -> S(z0)
op(S(0'), z0) -> S(z0)
fold(z0, z1) -> Cons(foldl(z0, z1), Cons(foldr(z0, z1), Nil))

Types:
FOLDL :: 0':S -> Cons:Nil -> c:c1:c2
Cons :: 0':S -> Cons:Nil -> Cons:Nil
S :: 0':S -> 0':S
0' :: 0':S
c :: c:c1:c2 -> c:c1:c2
c1 :: c:c1:c2 -> c:c1:c2
Nil :: Cons:Nil
c2 :: c:c1:c2
FOLDR :: 0':S -> Cons:Nil -> c3:c4
c3 :: c7:c8 -> c3:c4 -> c3:c4
OP :: 0':S -> 0':S -> c7:c8
foldr :: 0':S -> Cons:Nil -> 0':S
c4 :: c3:c4
NOTEMPTY :: Cons:Nil -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
c7 :: c7:c8
c8 :: c7:c8
FOLD :: 0':S -> Cons:Nil -> c9:c10
c9 :: c:c1:c2 -> c9:c10
c10 :: c3:c4 -> c9:c10
foldl :: 0':S -> Cons:Nil -> 0':S
op :: 0':S -> 0':S -> 0':S
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
fold :: 0':S -> Cons:Nil -> Cons:Nil
hole_c:c1:c21_11 :: c:c1:c2
hole_0':S2_11 :: 0':S
hole_Cons:Nil3_11 :: Cons:Nil
hole_c3:c44_11 :: c3:c4
hole_c7:c85_11 :: c7:c8
hole_c5:c66_11 :: c5:c6
hole_c9:c107_11 :: c9:c10
hole_True:False8_11 :: True:False
gen_c:c1:c29_11 :: Nat -> c:c1:c2
gen_0':S10_11 :: Nat -> 0':S
gen_Cons:Nil11_11 :: Nat -> Cons:Nil
gen_c3:c412_11 :: Nat -> c3:c4


Lemmas:
foldr(gen_0':S10_11(1), gen_Cons:Nil11_11(n25_11)) -> gen_0':S10_11(1), rt in Omega(0)


Generator Equations:
gen_c:c1:c29_11(0) <=> c2
gen_c:c1:c29_11(+(x, 1)) <=> c(gen_c:c1:c29_11(x))
gen_0':S10_11(0) <=> 0'
gen_0':S10_11(+(x, 1)) <=> S(gen_0':S10_11(x))
gen_Cons:Nil11_11(0) <=> Nil
gen_Cons:Nil11_11(+(x, 1)) <=> Cons(0', gen_Cons:Nil11_11(x))
gen_c3:c412_11(0) <=> c4
gen_c3:c412_11(+(x, 1)) <=> c3(c7, gen_c3:c412_11(x))


The following defined symbols remain to be analysed:
FOLDR, foldl
----------------------------------------

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
FOLDR(gen_0':S10_11(1), gen_Cons:Nil11_11(n423_11)) -> gen_c3:c412_11(n423_11), rt in Omega(1 + n423_11)

Induction Base:
FOLDR(gen_0':S10_11(1), gen_Cons:Nil11_11(0)) ->_R^Omega(1)
c4

Induction Step:
FOLDR(gen_0':S10_11(1), gen_Cons:Nil11_11(+(n423_11, 1))) ->_R^Omega(1)
c3(OP(0', foldr(gen_0':S10_11(1), gen_Cons:Nil11_11(n423_11))), FOLDR(gen_0':S10_11(1), gen_Cons:Nil11_11(n423_11))) ->_L^Omega(0)
c3(OP(0', gen_0':S10_11(1)), FOLDR(gen_0':S10_11(1), gen_Cons:Nil11_11(n423_11))) ->_R^Omega(1)
c3(c7, FOLDR(gen_0':S10_11(1), gen_Cons:Nil11_11(n423_11))) ->_IH
c3(c7, gen_c3:c412_11(c424_11))

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

(28)
Complex Obligation (BEST)

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

(29)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
FOLDL(z0, Cons(S(0'), z1)) -> c(FOLDL(S(z0), z1))
FOLDL(S(0'), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
FOLDL(z0, Nil) -> c2
FOLDR(z0, Cons(z1, z2)) -> c3(OP(z1, foldr(z0, z2)), FOLDR(z0, z2))
FOLDR(z0, Nil) -> c4
NOTEMPTY(Cons(z0, z1)) -> c5
NOTEMPTY(Nil) -> c6
OP(z0, S(0')) -> c7
OP(S(0'), z0) -> c8
FOLD(z0, z1) -> c9(FOLDL(z0, z1))
FOLD(z0, z1) -> c10(FOLDR(z0, z1))
foldl(z0, Cons(S(0'), z1)) -> foldl(S(z0), z1)
foldl(S(0'), Cons(z0, z1)) -> foldl(S(z0), z1)
foldl(z0, Nil) -> z0
foldr(z0, Cons(z1, z2)) -> op(z1, foldr(z0, z2))
foldr(z0, Nil) -> z0
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
op(z0, S(0')) -> S(z0)
op(S(0'), z0) -> S(z0)
fold(z0, z1) -> Cons(foldl(z0, z1), Cons(foldr(z0, z1), Nil))

Types:
FOLDL :: 0':S -> Cons:Nil -> c:c1:c2
Cons :: 0':S -> Cons:Nil -> Cons:Nil
S :: 0':S -> 0':S
0' :: 0':S
c :: c:c1:c2 -> c:c1:c2
c1 :: c:c1:c2 -> c:c1:c2
Nil :: Cons:Nil
c2 :: c:c1:c2
FOLDR :: 0':S -> Cons:Nil -> c3:c4
c3 :: c7:c8 -> c3:c4 -> c3:c4
OP :: 0':S -> 0':S -> c7:c8
foldr :: 0':S -> Cons:Nil -> 0':S
c4 :: c3:c4
NOTEMPTY :: Cons:Nil -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
c7 :: c7:c8
c8 :: c7:c8
FOLD :: 0':S -> Cons:Nil -> c9:c10
c9 :: c:c1:c2 -> c9:c10
c10 :: c3:c4 -> c9:c10
foldl :: 0':S -> Cons:Nil -> 0':S
op :: 0':S -> 0':S -> 0':S
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
fold :: 0':S -> Cons:Nil -> Cons:Nil
hole_c:c1:c21_11 :: c:c1:c2
hole_0':S2_11 :: 0':S
hole_Cons:Nil3_11 :: Cons:Nil
hole_c3:c44_11 :: c3:c4
hole_c7:c85_11 :: c7:c8
hole_c5:c66_11 :: c5:c6
hole_c9:c107_11 :: c9:c10
hole_True:False8_11 :: True:False
gen_c:c1:c29_11 :: Nat -> c:c1:c2
gen_0':S10_11 :: Nat -> 0':S
gen_Cons:Nil11_11 :: Nat -> Cons:Nil
gen_c3:c412_11 :: Nat -> c3:c4


Lemmas:
foldr(gen_0':S10_11(1), gen_Cons:Nil11_11(n25_11)) -> gen_0':S10_11(1), rt in Omega(0)


Generator Equations:
gen_c:c1:c29_11(0) <=> c2
gen_c:c1:c29_11(+(x, 1)) <=> c(gen_c:c1:c29_11(x))
gen_0':S10_11(0) <=> 0'
gen_0':S10_11(+(x, 1)) <=> S(gen_0':S10_11(x))
gen_Cons:Nil11_11(0) <=> Nil
gen_Cons:Nil11_11(+(x, 1)) <=> Cons(0', gen_Cons:Nil11_11(x))
gen_c3:c412_11(0) <=> c4
gen_c3:c412_11(+(x, 1)) <=> c3(c7, gen_c3:c412_11(x))


The following defined symbols remain to be analysed:
FOLDR, foldl
----------------------------------------

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

(31)
BOUNDS(n^1, INF)

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

(32)
Obligation:
Innermost TRS:
Rules:
FOLDL(z0, Cons(S(0'), z1)) -> c(FOLDL(S(z0), z1))
FOLDL(S(0'), Cons(z0, z1)) -> c1(FOLDL(S(z0), z1))
FOLDL(z0, Nil) -> c2
FOLDR(z0, Cons(z1, z2)) -> c3(OP(z1, foldr(z0, z2)), FOLDR(z0, z2))
FOLDR(z0, Nil) -> c4
NOTEMPTY(Cons(z0, z1)) -> c5
NOTEMPTY(Nil) -> c6
OP(z0, S(0')) -> c7
OP(S(0'), z0) -> c8
FOLD(z0, z1) -> c9(FOLDL(z0, z1))
FOLD(z0, z1) -> c10(FOLDR(z0, z1))
foldl(z0, Cons(S(0'), z1)) -> foldl(S(z0), z1)
foldl(S(0'), Cons(z0, z1)) -> foldl(S(z0), z1)
foldl(z0, Nil) -> z0
foldr(z0, Cons(z1, z2)) -> op(z1, foldr(z0, z2))
foldr(z0, Nil) -> z0
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
op(z0, S(0')) -> S(z0)
op(S(0'), z0) -> S(z0)
fold(z0, z1) -> Cons(foldl(z0, z1), Cons(foldr(z0, z1), Nil))

Types:
FOLDL :: 0':S -> Cons:Nil -> c:c1:c2
Cons :: 0':S -> Cons:Nil -> Cons:Nil
S :: 0':S -> 0':S
0' :: 0':S
c :: c:c1:c2 -> c:c1:c2
c1 :: c:c1:c2 -> c:c1:c2
Nil :: Cons:Nil
c2 :: c:c1:c2
FOLDR :: 0':S -> Cons:Nil -> c3:c4
c3 :: c7:c8 -> c3:c4 -> c3:c4
OP :: 0':S -> 0':S -> c7:c8
foldr :: 0':S -> Cons:Nil -> 0':S
c4 :: c3:c4
NOTEMPTY :: Cons:Nil -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
c7 :: c7:c8
c8 :: c7:c8
FOLD :: 0':S -> Cons:Nil -> c9:c10
c9 :: c:c1:c2 -> c9:c10
c10 :: c3:c4 -> c9:c10
foldl :: 0':S -> Cons:Nil -> 0':S
op :: 0':S -> 0':S -> 0':S
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
fold :: 0':S -> Cons:Nil -> Cons:Nil
hole_c:c1:c21_11 :: c:c1:c2
hole_0':S2_11 :: 0':S
hole_Cons:Nil3_11 :: Cons:Nil
hole_c3:c44_11 :: c3:c4
hole_c7:c85_11 :: c7:c8
hole_c5:c66_11 :: c5:c6
hole_c9:c107_11 :: c9:c10
hole_True:False8_11 :: True:False
gen_c:c1:c29_11 :: Nat -> c:c1:c2
gen_0':S10_11 :: Nat -> 0':S
gen_Cons:Nil11_11 :: Nat -> Cons:Nil
gen_c3:c412_11 :: Nat -> c3:c4


Lemmas:
foldr(gen_0':S10_11(1), gen_Cons:Nil11_11(n25_11)) -> gen_0':S10_11(1), rt in Omega(0)
FOLDR(gen_0':S10_11(1), gen_Cons:Nil11_11(n423_11)) -> gen_c3:c412_11(n423_11), rt in Omega(1 + n423_11)


Generator Equations:
gen_c:c1:c29_11(0) <=> c2
gen_c:c1:c29_11(+(x, 1)) <=> c(gen_c:c1:c29_11(x))
gen_0':S10_11(0) <=> 0'
gen_0':S10_11(+(x, 1)) <=> S(gen_0':S10_11(x))
gen_Cons:Nil11_11(0) <=> Nil
gen_Cons:Nil11_11(+(x, 1)) <=> Cons(0', gen_Cons:Nil11_11(x))
gen_c3:c412_11(0) <=> c4
gen_c3:c412_11(+(x, 1)) <=> c3(c7, gen_c3:c412_11(x))


The following defined symbols remain to be analysed:
foldl