WORST_CASE(Omega(n^1),O(n^2))
proof of input_3Z43mOEiP6.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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^2).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 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) CpxRelTRS
    (11) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxWeightedTrs
    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxTypedWeightedTrs
    (15) CompletionProof [UPPER BOUND(ID), 0 ms]
    (16) CpxTypedWeightedCompleteTrs
    (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) CompleteCoflocoProof [FINISHED, 847 ms]
    (20) BOUNDS(1, n^2)
(21) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) CdtProblem
    (23) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxRelTRS
    (25) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CpxRelTRS
    (27) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) typed CpxTrs
    (29) OrderProof [LOWER BOUND(ID), 11 ms]
    (30) typed CpxTrs
    (31) RewriteLemmaProof [LOWER BOUND(ID), 313 ms]
    (32) typed CpxTrs
    (33) RewriteLemmaProof [LOWER BOUND(ID), 68 ms]
    (34) typed CpxTrs
    (35) RewriteLemmaProof [LOWER BOUND(ID), 114 ms]
    (36) BEST
        (37) proven lower bound
            (38) LowerBoundPropagationProof [FINISHED, 0 ms]
            (39) BOUNDS(n^1, INF)
        (40) typed CpxTrs
            (41) RewriteLemmaProof [LOWER BOUND(ID), 23 ms]
            (42) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   nthtail(n, l) -> cond(ge(n, length(l)), n, l)
   cond(true, n, l) -> l
   cond(false, n, l) -> tail(nthtail(s(n), l))
   tail(nil) -> nil
   tail(cons(x, l)) -> l
   length(nil) -> 0
   length(cons(x, l)) -> s(length(l))
   ge(u, 0) -> true
   ge(0, s(v)) -> false
   ge(s(u), s(v)) -> ge(u, v)

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:
   nthtail(z0, z1) -> cond(ge(z0, length(z1)), z0, z1)
   cond(true, z0, z1) -> z1
   cond(false, z0, z1) -> tail(nthtail(s(z0), z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1))
   COND(true, z0, z1) -> c1
   COND(false, z0, z1) -> c2(TAIL(nthtail(s(z0), z1)), NTHTAIL(s(z0), z1))
   TAIL(nil) -> c3
   TAIL(cons(z0, z1)) -> c4
   LENGTH(nil) -> c5
   LENGTH(cons(z0, z1)) -> c6(LENGTH(z1))
   GE(z0, 0) -> c7
   GE(0, s(z0)) -> c8
   GE(s(z0), s(z1)) -> c9(GE(z0, z1))
S tuples:
   NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1))
   COND(true, z0, z1) -> c1
   COND(false, z0, z1) -> c2(TAIL(nthtail(s(z0), z1)), NTHTAIL(s(z0), z1))
   TAIL(nil) -> c3
   TAIL(cons(z0, z1)) -> c4
   LENGTH(nil) -> c5
   LENGTH(cons(z0, z1)) -> c6(LENGTH(z1))
   GE(z0, 0) -> c7
   GE(0, s(z0)) -> c8
   GE(s(z0), s(z1)) -> c9(GE(z0, z1))
K tuples:none
Defined Rule Symbols:   nthtail_2, cond_3, tail_1, length_1, ge_2

Defined Pair Symbols:   NTHTAIL_2, COND_3, TAIL_1, LENGTH_1, GE_2

Compound Symbols:   c_3, c1, c2_2, c3, c4, c5, c6_1, c7, c8, c9_1


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   GE(0, s(z0)) -> c8
   TAIL(cons(z0, z1)) -> c4
   GE(z0, 0) -> c7
   TAIL(nil) -> c3
   LENGTH(nil) -> c5
   COND(true, z0, z1) -> c1

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   nthtail(z0, z1) -> cond(ge(z0, length(z1)), z0, z1)
   cond(true, z0, z1) -> z1
   cond(false, z0, z1) -> tail(nthtail(s(z0), z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1))
   COND(false, z0, z1) -> c2(TAIL(nthtail(s(z0), z1)), NTHTAIL(s(z0), z1))
   LENGTH(cons(z0, z1)) -> c6(LENGTH(z1))
   GE(s(z0), s(z1)) -> c9(GE(z0, z1))
S tuples:
   NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1))
   COND(false, z0, z1) -> c2(TAIL(nthtail(s(z0), z1)), NTHTAIL(s(z0), z1))
   LENGTH(cons(z0, z1)) -> c6(LENGTH(z1))
   GE(s(z0), s(z1)) -> c9(GE(z0, z1))
K tuples:none
Defined Rule Symbols:   nthtail_2, cond_3, tail_1, length_1, ge_2

Defined Pair Symbols:   NTHTAIL_2, COND_3, LENGTH_1, GE_2

Compound Symbols:   c_3, c2_2, c6_1, c9_1


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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   nthtail(z0, z1) -> cond(ge(z0, length(z1)), z0, z1)
   cond(true, z0, z1) -> z1
   cond(false, z0, z1) -> tail(nthtail(s(z0), z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1))
   LENGTH(cons(z0, z1)) -> c6(LENGTH(z1))
   GE(s(z0), s(z1)) -> c9(GE(z0, z1))
   COND(false, z0, z1) -> c2(NTHTAIL(s(z0), z1))
S tuples:
   NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1))
   LENGTH(cons(z0, z1)) -> c6(LENGTH(z1))
   GE(s(z0), s(z1)) -> c9(GE(z0, z1))
   COND(false, z0, z1) -> c2(NTHTAIL(s(z0), z1))
K tuples:none
Defined Rule Symbols:   nthtail_2, cond_3, tail_1, length_1, ge_2

Defined Pair Symbols:   NTHTAIL_2, LENGTH_1, GE_2, COND_3

Compound Symbols:   c_3, c6_1, c9_1, c2_1


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   nthtail(z0, z1) -> cond(ge(z0, length(z1)), z0, z1)
   cond(true, z0, z1) -> z1
   cond(false, z0, z1) -> tail(nthtail(s(z0), z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))
Tuples:
   NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1))
   LENGTH(cons(z0, z1)) -> c6(LENGTH(z1))
   GE(s(z0), s(z1)) -> c9(GE(z0, z1))
   COND(false, z0, z1) -> c2(NTHTAIL(s(z0), z1))
S tuples:
   NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1))
   LENGTH(cons(z0, z1)) -> c6(LENGTH(z1))
   GE(s(z0), s(z1)) -> c9(GE(z0, z1))
   COND(false, z0, z1) -> c2(NTHTAIL(s(z0), z1))
K tuples:none
Defined Rule Symbols:   ge_2, length_1

Defined Pair Symbols:   NTHTAIL_2, LENGTH_1, GE_2, COND_3

Compound Symbols:   c_3, c6_1, c9_1, c2_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 CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1))
   LENGTH(cons(z0, z1)) -> c6(LENGTH(z1))
   GE(s(z0), s(z1)) -> c9(GE(z0, z1))
   COND(false, z0, z1) -> c2(NTHTAIL(s(z0), z1))

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

   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))

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

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

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1)) [1]
   LENGTH(cons(z0, z1)) -> c6(LENGTH(z1)) [1]
   GE(s(z0), s(z1)) -> c9(GE(z0, z1)) [1]
   COND(false, z0, z1) -> c2(NTHTAIL(s(z0), z1)) [1]
   ge(z0, 0) -> true [0]
   ge(0, s(z0)) -> false [0]
   ge(s(z0), s(z1)) -> ge(z0, z1) [0]
   length(nil) -> 0 [0]
   length(cons(z0, z1)) -> s(length(z1)) [0]

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

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

(14)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1)) [1]
   LENGTH(cons(z0, z1)) -> c6(LENGTH(z1)) [1]
   GE(s(z0), s(z1)) -> c9(GE(z0, z1)) [1]
   COND(false, z0, z1) -> c2(NTHTAIL(s(z0), z1)) [1]
   ge(z0, 0) -> true [0]
   ge(0, s(z0)) -> false [0]
   ge(s(z0), s(z1)) -> ge(z0, z1) [0]
   length(nil) -> 0 [0]
   length(cons(z0, z1)) -> s(length(z1)) [0]

The TRS has the following type information:
NTHTAIL :: s:0 -> cons:nil -> c
c :: c2 -> c9 -> c6 -> c
COND :: false:true -> s:0 -> cons:nil -> c2
ge :: s:0 -> s:0 -> false:true
length :: cons:nil -> s:0
GE :: s:0 -> s:0 -> c9
LENGTH :: cons:nil -> c6
cons :: a -> cons:nil -> cons:nil
c6 :: c6 -> c6
s :: s:0 -> s:0
c9 :: c9 -> c9
false :: false:true
c2 :: c -> c2
0 :: s:0
true :: false:true
nil :: cons:nil

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

(15) 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:

   ge(v0, v1) -> null_ge [0]
   length(v0) -> null_length [0]
   LENGTH(v0) -> null_LENGTH [0]
   GE(v0, v1) -> null_GE [0]
   COND(v0, v1, v2) -> null_COND [0]

And the following fresh constants:    null_ge, null_length, null_LENGTH, null_GE, null_COND, const, const1

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

(16)
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:

   NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1)) [1]
   LENGTH(cons(z0, z1)) -> c6(LENGTH(z1)) [1]
   GE(s(z0), s(z1)) -> c9(GE(z0, z1)) [1]
   COND(false, z0, z1) -> c2(NTHTAIL(s(z0), z1)) [1]
   ge(z0, 0) -> true [0]
   ge(0, s(z0)) -> false [0]
   ge(s(z0), s(z1)) -> ge(z0, z1) [0]
   length(nil) -> 0 [0]
   length(cons(z0, z1)) -> s(length(z1)) [0]
   ge(v0, v1) -> null_ge [0]
   length(v0) -> null_length [0]
   LENGTH(v0) -> null_LENGTH [0]
   GE(v0, v1) -> null_GE [0]
   COND(v0, v1, v2) -> null_COND [0]

The TRS has the following type information:
NTHTAIL :: s:0:null_length -> cons:nil -> c
c :: c2:null_COND -> c9:null_GE -> c6:null_LENGTH -> c
COND :: false:true:null_ge -> s:0:null_length -> cons:nil -> c2:null_COND
ge :: s:0:null_length -> s:0:null_length -> false:true:null_ge
length :: cons:nil -> s:0:null_length
GE :: s:0:null_length -> s:0:null_length -> c9:null_GE
LENGTH :: cons:nil -> c6:null_LENGTH
cons :: a -> cons:nil -> cons:nil
c6 :: c6:null_LENGTH -> c6:null_LENGTH
s :: s:0:null_length -> s:0:null_length
c9 :: c9:null_GE -> c9:null_GE
false :: false:true:null_ge
c2 :: c -> c2:null_COND
0 :: s:0:null_length
true :: false:true:null_ge
nil :: cons:nil
null_ge :: false:true:null_ge
null_length :: s:0:null_length
null_LENGTH :: c6:null_LENGTH
null_GE :: c9:null_GE
null_COND :: c2:null_COND
const :: c
const1 :: a

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

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

   false => 1
   0 => 0
   true => 2
   nil => 0
   null_ge => 0
   null_length => 0
   null_LENGTH => 0
   null_GE => 0
   null_COND => 0
   const => 0
   const1 => 0

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

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   COND(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   COND(z, z', z'') -{ 1 }-> 1 + NTHTAIL(1 + z0, z1) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1
   GE(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   GE(z, z') -{ 1 }-> 1 + GE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   LENGTH(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NTHTAIL(z, z') -{ 1 }-> 1 + COND(ge(z0, length(z1)), z0, z1) + GE(z0, length(z1)) + LENGTH(z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   ge(z, z') -{ 0 }-> ge(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   ge(z, z') -{ 0 }-> 2 :|: z = z0, z0 >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z0 >= 0, z' = 1 + z0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   length(z) -{ 0 }-> 0 :|: z = 0
   length(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   length(z) -{ 0 }-> 1 + length(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Only complete derivations are relevant for the runtime complexity.

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

(19) CompleteCoflocoProof (FINISHED)
Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V1, V, V9),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V9),0,[fun3(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V9),0,[fun2(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V9),0,[fun1(V1, V, V9, Out)],[V1 >= 0,V >= 0,V9 >= 0]).
eq(start(V1, V, V9),0,[ge(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V9),0,[length(V1, Out)],[V1 >= 0]).
eq(fun(V1, V, Out),1,[length(V2, Ret00101),ge(V3, Ret00101, Ret0010),fun1(Ret0010, V3, V2, Ret001),length(V2, Ret011),fun2(V3, Ret011, Ret01),fun3(V2, Ret1)],[Out = 1 + Ret001 + Ret01 + Ret1,V1 = V3,V2 >= 0,V = V2,V3 >= 0]).
eq(fun3(V1, Out),1,[fun3(V4, Ret11)],[Out = 1 + Ret11,V4 >= 0,V5 >= 0,V1 = 1 + V4 + V5]).
eq(fun2(V1, V, Out),1,[fun2(V7, V6, Ret12)],[Out = 1 + Ret12,V6 >= 0,V1 = 1 + V7,V7 >= 0,V = 1 + V6]).
eq(fun1(V1, V, V9, Out),1,[fun(1 + V8, V10, Ret13)],[Out = 1 + Ret13,V10 >= 0,V1 = 1,V8 >= 0,V = V8,V9 = V10]).
eq(ge(V1, V, Out),0,[],[Out = 2,V1 = V11,V11 >= 0,V = 0]).
eq(ge(V1, V, Out),0,[],[Out = 1,V12 >= 0,V = 1 + V12,V1 = 0]).
eq(ge(V1, V, Out),0,[ge(V14, V13, Ret)],[Out = Ret,V13 >= 0,V1 = 1 + V14,V14 >= 0,V = 1 + V13]).
eq(length(V1, Out),0,[],[Out = 0,V1 = 0]).
eq(length(V1, Out),0,[length(V15, Ret14)],[Out = 1 + Ret14,V15 >= 0,V16 >= 0,V1 = 1 + V15 + V16]).
eq(ge(V1, V, Out),0,[],[Out = 0,V18 >= 0,V17 >= 0,V1 = V18,V = V17]).
eq(length(V1, Out),0,[],[Out = 0,V19 >= 0,V1 = V19]).
eq(fun3(V1, Out),0,[],[Out = 0,V20 >= 0,V1 = V20]).
eq(fun2(V1, V, Out),0,[],[Out = 0,V21 >= 0,V22 >= 0,V1 = V21,V = V22]).
eq(fun1(V1, V, V9, Out),0,[],[Out = 0,V23 >= 0,V9 = V25,V24 >= 0,V1 = V23,V = V24,V25 >= 0]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun3(V1,Out),[V1],[Out]).
input_output_vars(fun2(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,V9,Out),[V1,V,V9],[Out]).
input_output_vars(ge(V1,V,Out),[V1,V],[Out]).
input_output_vars(length(V1,Out),[V1],[Out]).


CoFloCo proof output:
Preprocessing Cost Relations
=====================================

#### Computed strongly connected components 
0. recursive  : [fun2/3]
1. recursive  : [fun3/2]
2. recursive  : [ge/3]
3. recursive  : [length/2]
4. recursive [non_tail] : [fun/3,fun1/4]
5. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into fun2/3
1. SCC is partially evaluated into fun3/2
2. SCC is partially evaluated into ge/3
3. SCC is partially evaluated into length/2
4. SCC is partially evaluated into fun/3
5. SCC is partially evaluated into start/3

Control-Flow Refinement of Cost Relations
=====================================

### Specialization of cost equations fun2/3 
* CE 13 is refined into CE [20] 
* CE 12 is refined into CE [21] 


### Cost equations --> "Loop" of fun2/3 
* CEs [21] --> Loop 14 
* CEs [20] --> Loop 15 

### Ranking functions of CR fun2(V1,V,Out) 
* RF of phase [14]: [V,V1]

#### Partial ranking functions of CR fun2(V1,V,Out) 
* Partial RF of phase [14]:
  - RF of loop [14:1]:
    V
    V1


### Specialization of cost equations fun3/2 
* CE 11 is refined into CE [22] 
* CE 10 is refined into CE [23] 


### Cost equations --> "Loop" of fun3/2 
* CEs [23] --> Loop 16 
* CEs [22] --> Loop 17 

### Ranking functions of CR fun3(V1,Out) 
* RF of phase [16]: [V1]

#### Partial ranking functions of CR fun3(V1,Out) 
* Partial RF of phase [16]:
  - RF of loop [16:1]:
    V1


### Specialization of cost equations ge/3 
* CE 17 is refined into CE [24] 
* CE 14 is refined into CE [25] 
* CE 15 is refined into CE [26] 
* CE 16 is refined into CE [27] 


### Cost equations --> "Loop" of ge/3 
* CEs [27] --> Loop 18 
* CEs [24] --> Loop 19 
* CEs [25] --> Loop 20 
* CEs [26] --> Loop 21 

### Ranking functions of CR ge(V1,V,Out) 
* RF of phase [18]: [V,V1]

#### Partial ranking functions of CR ge(V1,V,Out) 
* Partial RF of phase [18]:
  - RF of loop [18:1]:
    V
    V1


### Specialization of cost equations length/2 
* CE 18 is refined into CE [28] 
* CE 19 is refined into CE [29] 


### Cost equations --> "Loop" of length/2 
* CEs [29] --> Loop 22 
* CEs [28] --> Loop 23 

### Ranking functions of CR length(V1,Out) 
* RF of phase [22]: [V1]

#### Partial ranking functions of CR length(V1,Out) 
* Partial RF of phase [22]:
  - RF of loop [22:1]:
    V1


### Specialization of cost equations fun/3 
* CE 9 is refined into CE [30,31,32,33,34,35,36,37,38,39] 
* CE 8 is refined into CE [40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73] 


### Cost equations --> "Loop" of fun/3 
* CEs [45,51,61,67,73] --> Loop 24 
* CEs [41,43,44,47,49,50,53,55,57,59,60,63,65,66,69,71,72] --> Loop 25 
* CEs [40,42,46,48,52,54,56,58,62,64,68,70] --> Loop 26 
* CEs [39] --> Loop 27 
* CEs [35,37,38] --> Loop 28 
* CEs [34,36] --> Loop 29 
* CEs [31,33] --> Loop 30 
* CEs [30,32] --> Loop 31 

### Ranking functions of CR fun(V1,V,Out) 
* RF of phase [27,28,29]: [-V1+V]

#### Partial ranking functions of CR fun(V1,V,Out) 
* Partial RF of phase [27,28,29]:
  - RF of loop [27:1,28:1,29:1]:
    -V1+V


### Specialization of cost equations start/3 
* CE 1 is refined into CE [74] 
* CE 2 is refined into CE [75,76,77,78] 
* CE 3 is refined into CE [79,80,81,82,83,84,85,86,87] 
* CE 4 is refined into CE [88,89] 
* CE 5 is refined into CE [90,91] 
* CE 6 is refined into CE [92,93,94,95,96] 
* CE 7 is refined into CE [97,98] 


### Cost equations --> "Loop" of start/3 
* CEs [93] --> Loop 32 
* CEs [75,76,77,78] --> Loop 33 
* CEs [74,79,80,81,82,83,84,85,86,87,88,89,90,91,92,94,95,96,97,98] --> Loop 34 

### Ranking functions of CR start(V1,V,V9) 

#### Partial ranking functions of CR start(V1,V,V9) 


Computing Bounds
=====================================

#### Cost of chains of fun2(V1,V,Out):
* Chain [[14],15]: 1*it(14)+0
  Such that:it(14) =< Out

  with precondition: [Out>=1,V1>=Out,V>=Out] 

* Chain [15]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of fun3(V1,Out):
* Chain [[16],17]: 1*it(16)+0
  Such that:it(16) =< V1

  with precondition: [Out>=1,V1>=Out] 

* Chain [17]: 0
  with precondition: [Out=0,V1>=0] 


#### Cost of chains of ge(V1,V,Out):
* Chain [[18],21]: 0
  with precondition: [Out=1,V1>=1,V>=V1+1] 

* Chain [[18],20]: 0
  with precondition: [Out=2,V>=1,V1>=V] 

* Chain [[18],19]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

* Chain [21]: 0
  with precondition: [V1=0,Out=1,V>=1] 

* Chain [20]: 0
  with precondition: [V=0,Out=2,V1>=0] 

* Chain [19]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of length(V1,Out):
* Chain [[22],23]: 0
  with precondition: [Out>=1,V1>=Out] 

* Chain [23]: 0
  with precondition: [Out=0,V1>=0] 


#### Cost of chains of fun(V1,V,Out):
* Chain [[27,28,29],26]: 6*it(27)+2*s(11)+2*s(13)+1*s(14)+1
  Such that:aux(8) =< V
aux(9) =< -V1+V
it(27) =< aux(9)
aux(4) =< aux(8)
s(11) =< it(27)*aux(8)
s(15) =< it(27)*aux(4)
s(14) =< it(27)*aux(4)
s(13) =< s(15)

  with precondition: [V1>=1,Out>=3,V>=V1+1] 

* Chain [[27,28,29],25]: 6*it(27)+2*s(11)+2*s(13)+1*s(14)+17*s(16)+1
  Such that:aux(11) =< -V1+V
aux(12) =< V
s(16) =< aux(12)
it(27) =< aux(11)
aux(4) =< aux(12)
s(11) =< it(27)*aux(12)
s(15) =< it(27)*aux(4)
s(14) =< it(27)*aux(4)
s(13) =< s(15)

  with precondition: [V1>=1,Out>=4,V>=V1+1] 

* Chain [[27,28,29],24]: 6*it(27)+2*s(11)+2*s(13)+1*s(14)+10*s(33)+1
  Such that:aux(18) =< -V1+V
aux(19) =< V
s(33) =< aux(19)
it(27) =< aux(18)
aux(4) =< aux(19)
s(11) =< it(27)*aux(19)
s(15) =< it(27)*aux(4)
s(14) =< it(27)*aux(4)
s(13) =< s(15)

  with precondition: [V1>=1,Out>=5,V>=V1+1] 

* Chain [31,[27,28,29],26]: 6*it(27)+2*s(11)+2*s(13)+1*s(14)+3
  Such that:aux(20) =< V
it(27) =< aux(20)
aux(4) =< aux(20)
s(11) =< it(27)*aux(20)
s(15) =< it(27)*aux(4)
s(14) =< it(27)*aux(4)
s(13) =< s(15)

  with precondition: [V1=0,V>=2,Out>=5] 

* Chain [31,[27,28,29],25]: 23*it(27)+2*s(11)+2*s(13)+1*s(14)+3
  Such that:aux(21) =< V
it(27) =< aux(21)
aux(4) =< aux(21)
s(11) =< it(27)*aux(21)
s(15) =< it(27)*aux(4)
s(14) =< it(27)*aux(4)
s(13) =< s(15)

  with precondition: [V1=0,V>=2,Out>=6] 

* Chain [31,[27,28,29],24]: 16*it(27)+2*s(11)+2*s(13)+1*s(14)+3
  Such that:aux(22) =< V
it(27) =< aux(22)
aux(4) =< aux(22)
s(11) =< it(27)*aux(22)
s(15) =< it(27)*aux(4)
s(14) =< it(27)*aux(4)
s(13) =< s(15)

  with precondition: [V1=0,V>=2,Out>=7] 

* Chain [31,26]: 3
  with precondition: [V1=0,Out=3,V>=1] 

* Chain [31,25]: 16*s(16)+1*s(29)+3
  Such that:s(29) =< 1
aux(10) =< V
s(16) =< aux(10)

  with precondition: [V1=0,Out>=4,V+3>=Out] 

* Chain [31,24]: 9*s(33)+1*s(39)+3
  Such that:s(39) =< 1
aux(17) =< V
s(33) =< aux(17)

  with precondition: [V1=0,Out>=5,V+4>=Out] 

* Chain [30,[27,28,29],26]: 8*it(27)+2*s(11)+2*s(13)+1*s(14)+3
  Such that:aux(24) =< V
it(27) =< aux(24)
aux(4) =< aux(24)
s(11) =< it(27)*aux(24)
s(15) =< it(27)*aux(4)
s(14) =< it(27)*aux(4)
s(13) =< s(15)

  with precondition: [V1=0,V>=2,Out>=6] 

* Chain [30,[27,28,29],25]: 25*it(27)+2*s(11)+2*s(13)+1*s(14)+3
  Such that:aux(25) =< V
it(27) =< aux(25)
aux(4) =< aux(25)
s(11) =< it(27)*aux(25)
s(15) =< it(27)*aux(4)
s(14) =< it(27)*aux(4)
s(13) =< s(15)

  with precondition: [V1=0,V>=2,Out>=7] 

* Chain [30,[27,28,29],24]: 18*it(27)+2*s(11)+2*s(13)+1*s(14)+3
  Such that:aux(26) =< V
it(27) =< aux(26)
aux(4) =< aux(26)
s(11) =< it(27)*aux(26)
s(15) =< it(27)*aux(4)
s(14) =< it(27)*aux(4)
s(13) =< s(15)

  with precondition: [V1=0,V>=2,Out>=8] 

* Chain [30,26]: 2*s(43)+3
  Such that:aux(23) =< V
s(43) =< aux(23)

  with precondition: [V1=0,Out>=4,V+3>=Out] 

* Chain [30,25]: 18*s(16)+1*s(29)+3
  Such that:s(29) =< 1
aux(27) =< V
s(16) =< aux(27)

  with precondition: [V1=0,Out>=5,2*V+3>=Out] 

* Chain [30,24]: 11*s(33)+1*s(39)+3
  Such that:s(39) =< 1
aux(28) =< V
s(33) =< aux(28)

  with precondition: [V1=0,Out>=6,2*V+4>=Out] 

* Chain [26]: 1
  with precondition: [Out=1,V1>=0,V>=0] 

* Chain [25]: 16*s(16)+1*s(29)+1
  Such that:s(29) =< V1
aux(10) =< V
s(16) =< aux(10)

  with precondition: [V1>=0,Out>=2,V+1>=Out] 

* Chain [24]: 9*s(33)+1*s(39)+1
  Such that:s(39) =< V1
aux(17) =< V
s(33) =< aux(17)

  with precondition: [V1>=1,Out>=3,2*V+1>=Out,V+V1+1>=Out] 


#### Cost of chains of start(V1,V,V9):
* Chain [34]: 205*s(125)+12*s(127)+6*s(129)+12*s(130)+4*s(131)+3*s(141)+18*s(147)+6*s(149)+3*s(151)+6*s(152)+3
  Such that:s(144) =< -V1+V
aux(35) =< 1
aux(36) =< V1
aux(37) =< V
s(131) =< aux(35)
s(141) =< aux(36)
s(125) =< aux(37)
s(147) =< s(144)
s(126) =< aux(37)
s(149) =< s(147)*aux(37)
s(150) =< s(147)*s(126)
s(151) =< s(147)*s(126)
s(152) =< s(150)
s(127) =< s(125)*aux(37)
s(128) =< s(125)*s(126)
s(129) =< s(125)*s(126)
s(130) =< s(128)

  with precondition: [V1>=0] 

* Chain [33]: 2*s(158)+52*s(160)+18*s(164)+6*s(166)+3*s(168)+6*s(169)+2
  Such that:s(161) =< -V+V9
aux(38) =< V+1
aux(39) =< V9
s(158) =< aux(38)
s(160) =< aux(39)
s(164) =< s(161)
s(165) =< aux(39)
s(166) =< s(164)*aux(39)
s(167) =< s(164)*s(165)
s(168) =< s(164)*s(165)
s(169) =< s(167)

  with precondition: [V1=1,V>=0,V9>=0] 

* Chain [32]: 0
  with precondition: [V=0,V1>=0] 


Closed-form bounds of start(V1,V,V9): 
-------------------------------------
* Chain [34] with precondition: [V1>=0] 
    - Upper bound: 3*V1+7+nat(V)*205+nat(V)*30*nat(V)+nat(V)*15*nat(-V1+V)+nat(-V1+V)*18 
    - Complexity: n^2 
* Chain [33] with precondition: [V1=1,V>=0,V9>=0] 
    - Upper bound: 52*V9+2+15*V9*nat(-V+V9)+(2*V+2)+nat(-V+V9)*18 
    - Complexity: n^2 
* Chain [32] with precondition: [V=0,V1>=0] 
    - Upper bound: 0 
    - Complexity: constant 

### Maximum cost of start(V1,V,V9): max([nat(V9)*52+2+nat(V9)*15*nat(-V+V9)+nat(V+1)*2+nat(-V+V9)*18,3*V1+7+nat(V)*205+nat(V)*30*nat(V)+nat(V)*15*nat(-V1+V)+nat(-V1+V)*18]) 
Asymptotic class: n^2 
* Total analysis performed in 748 ms.


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

(20)
BOUNDS(1, n^2)

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

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   nthtail(z0, z1) -> cond(ge(z0, length(z1)), z0, z1)
   cond(true, z0, z1) -> z1
   cond(false, z0, z1) -> tail(nthtail(s(z0), z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1))
   COND(true, z0, z1) -> c1
   COND(false, z0, z1) -> c2(TAIL(nthtail(s(z0), z1)), NTHTAIL(s(z0), z1))
   TAIL(nil) -> c3
   TAIL(cons(z0, z1)) -> c4
   LENGTH(nil) -> c5
   LENGTH(cons(z0, z1)) -> c6(LENGTH(z1))
   GE(z0, 0) -> c7
   GE(0, s(z0)) -> c8
   GE(s(z0), s(z1)) -> c9(GE(z0, z1))
S tuples:
   NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1))
   COND(true, z0, z1) -> c1
   COND(false, z0, z1) -> c2(TAIL(nthtail(s(z0), z1)), NTHTAIL(s(z0), z1))
   TAIL(nil) -> c3
   TAIL(cons(z0, z1)) -> c4
   LENGTH(nil) -> c5
   LENGTH(cons(z0, z1)) -> c6(LENGTH(z1))
   GE(z0, 0) -> c7
   GE(0, s(z0)) -> c8
   GE(s(z0), s(z1)) -> c9(GE(z0, z1))
K tuples:none
Defined Rule Symbols:   nthtail_2, cond_3, tail_1, length_1, ge_2

Defined Pair Symbols:   NTHTAIL_2, COND_3, TAIL_1, LENGTH_1, GE_2

Compound Symbols:   c_3, c1, c2_2, c3, c4, c5, c6_1, c7, c8, c9_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 CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1))
   COND(true, z0, z1) -> c1
   COND(false, z0, z1) -> c2(TAIL(nthtail(s(z0), z1)), NTHTAIL(s(z0), z1))
   TAIL(nil) -> c3
   TAIL(cons(z0, z1)) -> c4
   LENGTH(nil) -> c5
   LENGTH(cons(z0, z1)) -> c6(LENGTH(z1))
   GE(z0, 0) -> c7
   GE(0, s(z0)) -> c8
   GE(s(z0), s(z1)) -> c9(GE(z0, z1))

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

   nthtail(z0, z1) -> cond(ge(z0, length(z1)), z0, z1)
   cond(true, z0, z1) -> z1
   cond(false, z0, z1) -> tail(nthtail(s(z0), z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)

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

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

(26)
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:

   NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1))
   COND(true, z0, z1) -> c1
   COND(false, z0, z1) -> c2(TAIL(nthtail(s(z0), z1)), NTHTAIL(s(z0), z1))
   TAIL(nil) -> c3
   TAIL(cons(z0, z1)) -> c4
   LENGTH(nil) -> c5
   LENGTH(cons(z0, z1)) -> c6(LENGTH(z1))
   GE(z0, 0') -> c7
   GE(0', s(z0)) -> c8
   GE(s(z0), s(z1)) -> c9(GE(z0, z1))

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

   nthtail(z0, z1) -> cond(ge(z0, length(z1)), z0, z1)
   cond(true, z0, z1) -> z1
   cond(false, z0, z1) -> tail(nthtail(s(z0), z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   length(nil) -> 0'
   length(cons(z0, z1)) -> s(length(z1))
   ge(z0, 0') -> true
   ge(0', s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)

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

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

(28)
Obligation:
Innermost TRS:
Rules:
NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1))
COND(true, z0, z1) -> c1
COND(false, z0, z1) -> c2(TAIL(nthtail(s(z0), z1)), NTHTAIL(s(z0), z1))
TAIL(nil) -> c3
TAIL(cons(z0, z1)) -> c4
LENGTH(nil) -> c5
LENGTH(cons(z0, z1)) -> c6(LENGTH(z1))
GE(z0, 0') -> c7
GE(0', s(z0)) -> c8
GE(s(z0), s(z1)) -> c9(GE(z0, z1))
nthtail(z0, z1) -> cond(ge(z0, length(z1)), z0, z1)
cond(true, z0, z1) -> z1
cond(false, z0, z1) -> tail(nthtail(s(z0), z1))
tail(nil) -> nil
tail(cons(z0, z1)) -> z1
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)

Types:
NTHTAIL :: s:0' -> nil:cons -> c
c :: c1:c2 -> c7:c8:c9 -> c5:c6 -> c
COND :: true:false -> s:0' -> nil:cons -> c1:c2
ge :: s:0' -> s:0' -> true:false
length :: nil:cons -> s:0'
GE :: s:0' -> s:0' -> c7:c8:c9
LENGTH :: nil:cons -> c5:c6
true :: true:false
c1 :: c1:c2
false :: true:false
c2 :: c3:c4 -> c -> c1:c2
TAIL :: nil:cons -> c3:c4
nthtail :: s:0' -> nil:cons -> nil:cons
s :: s:0' -> s:0'
nil :: nil:cons
c3 :: c3:c4
cons :: a -> nil:cons -> nil:cons
c4 :: c3:c4
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
0' :: s:0'
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
cond :: true:false -> s:0' -> nil:cons -> nil:cons
tail :: nil:cons -> nil:cons
hole_c1_10 :: c
hole_s:0'2_10 :: s:0'
hole_nil:cons3_10 :: nil:cons
hole_c1:c24_10 :: c1:c2
hole_c7:c8:c95_10 :: c7:c8:c9
hole_c5:c66_10 :: c5:c6
hole_true:false7_10 :: true:false
hole_c3:c48_10 :: c3:c4
hole_a9_10 :: a
gen_s:0'10_10 :: Nat -> s:0'
gen_nil:cons11_10 :: Nat -> nil:cons
gen_c7:c8:c912_10 :: Nat -> c7:c8:c9
gen_c5:c613_10 :: Nat -> c5:c6

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

(29) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
NTHTAIL, ge, length, GE, LENGTH, nthtail

They will be analysed ascendingly in the following order:
ge < NTHTAIL
length < NTHTAIL
GE < NTHTAIL
LENGTH < NTHTAIL
nthtail < NTHTAIL
ge < nthtail
length < nthtail

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

(30)
Obligation:
Innermost TRS:
Rules:
NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1))
COND(true, z0, z1) -> c1
COND(false, z0, z1) -> c2(TAIL(nthtail(s(z0), z1)), NTHTAIL(s(z0), z1))
TAIL(nil) -> c3
TAIL(cons(z0, z1)) -> c4
LENGTH(nil) -> c5
LENGTH(cons(z0, z1)) -> c6(LENGTH(z1))
GE(z0, 0') -> c7
GE(0', s(z0)) -> c8
GE(s(z0), s(z1)) -> c9(GE(z0, z1))
nthtail(z0, z1) -> cond(ge(z0, length(z1)), z0, z1)
cond(true, z0, z1) -> z1
cond(false, z0, z1) -> tail(nthtail(s(z0), z1))
tail(nil) -> nil
tail(cons(z0, z1)) -> z1
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)

Types:
NTHTAIL :: s:0' -> nil:cons -> c
c :: c1:c2 -> c7:c8:c9 -> c5:c6 -> c
COND :: true:false -> s:0' -> nil:cons -> c1:c2
ge :: s:0' -> s:0' -> true:false
length :: nil:cons -> s:0'
GE :: s:0' -> s:0' -> c7:c8:c9
LENGTH :: nil:cons -> c5:c6
true :: true:false
c1 :: c1:c2
false :: true:false
c2 :: c3:c4 -> c -> c1:c2
TAIL :: nil:cons -> c3:c4
nthtail :: s:0' -> nil:cons -> nil:cons
s :: s:0' -> s:0'
nil :: nil:cons
c3 :: c3:c4
cons :: a -> nil:cons -> nil:cons
c4 :: c3:c4
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
0' :: s:0'
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
cond :: true:false -> s:0' -> nil:cons -> nil:cons
tail :: nil:cons -> nil:cons
hole_c1_10 :: c
hole_s:0'2_10 :: s:0'
hole_nil:cons3_10 :: nil:cons
hole_c1:c24_10 :: c1:c2
hole_c7:c8:c95_10 :: c7:c8:c9
hole_c5:c66_10 :: c5:c6
hole_true:false7_10 :: true:false
hole_c3:c48_10 :: c3:c4
hole_a9_10 :: a
gen_s:0'10_10 :: Nat -> s:0'
gen_nil:cons11_10 :: Nat -> nil:cons
gen_c7:c8:c912_10 :: Nat -> c7:c8:c9
gen_c5:c613_10 :: Nat -> c5:c6


Generator Equations:
gen_s:0'10_10(0) <=> 0'
gen_s:0'10_10(+(x, 1)) <=> s(gen_s:0'10_10(x))
gen_nil:cons11_10(0) <=> nil
gen_nil:cons11_10(+(x, 1)) <=> cons(hole_a9_10, gen_nil:cons11_10(x))
gen_c7:c8:c912_10(0) <=> c7
gen_c7:c8:c912_10(+(x, 1)) <=> c9(gen_c7:c8:c912_10(x))
gen_c5:c613_10(0) <=> c5
gen_c5:c613_10(+(x, 1)) <=> c6(gen_c5:c613_10(x))


The following defined symbols remain to be analysed:
ge, NTHTAIL, length, GE, LENGTH, nthtail

They will be analysed ascendingly in the following order:
ge < NTHTAIL
length < NTHTAIL
GE < NTHTAIL
LENGTH < NTHTAIL
nthtail < NTHTAIL
ge < nthtail
length < nthtail

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_s:0'10_10(n15_10), gen_s:0'10_10(n15_10)) -> true, rt in Omega(0)

Induction Base:
ge(gen_s:0'10_10(0), gen_s:0'10_10(0)) ->_R^Omega(0)
true

Induction Step:
ge(gen_s:0'10_10(+(n15_10, 1)), gen_s:0'10_10(+(n15_10, 1))) ->_R^Omega(0)
ge(gen_s:0'10_10(n15_10), gen_s:0'10_10(n15_10)) ->_IH
true

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

(32)
Obligation:
Innermost TRS:
Rules:
NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1))
COND(true, z0, z1) -> c1
COND(false, z0, z1) -> c2(TAIL(nthtail(s(z0), z1)), NTHTAIL(s(z0), z1))
TAIL(nil) -> c3
TAIL(cons(z0, z1)) -> c4
LENGTH(nil) -> c5
LENGTH(cons(z0, z1)) -> c6(LENGTH(z1))
GE(z0, 0') -> c7
GE(0', s(z0)) -> c8
GE(s(z0), s(z1)) -> c9(GE(z0, z1))
nthtail(z0, z1) -> cond(ge(z0, length(z1)), z0, z1)
cond(true, z0, z1) -> z1
cond(false, z0, z1) -> tail(nthtail(s(z0), z1))
tail(nil) -> nil
tail(cons(z0, z1)) -> z1
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)

Types:
NTHTAIL :: s:0' -> nil:cons -> c
c :: c1:c2 -> c7:c8:c9 -> c5:c6 -> c
COND :: true:false -> s:0' -> nil:cons -> c1:c2
ge :: s:0' -> s:0' -> true:false
length :: nil:cons -> s:0'
GE :: s:0' -> s:0' -> c7:c8:c9
LENGTH :: nil:cons -> c5:c6
true :: true:false
c1 :: c1:c2
false :: true:false
c2 :: c3:c4 -> c -> c1:c2
TAIL :: nil:cons -> c3:c4
nthtail :: s:0' -> nil:cons -> nil:cons
s :: s:0' -> s:0'
nil :: nil:cons
c3 :: c3:c4
cons :: a -> nil:cons -> nil:cons
c4 :: c3:c4
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
0' :: s:0'
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
cond :: true:false -> s:0' -> nil:cons -> nil:cons
tail :: nil:cons -> nil:cons
hole_c1_10 :: c
hole_s:0'2_10 :: s:0'
hole_nil:cons3_10 :: nil:cons
hole_c1:c24_10 :: c1:c2
hole_c7:c8:c95_10 :: c7:c8:c9
hole_c5:c66_10 :: c5:c6
hole_true:false7_10 :: true:false
hole_c3:c48_10 :: c3:c4
hole_a9_10 :: a
gen_s:0'10_10 :: Nat -> s:0'
gen_nil:cons11_10 :: Nat -> nil:cons
gen_c7:c8:c912_10 :: Nat -> c7:c8:c9
gen_c5:c613_10 :: Nat -> c5:c6


Lemmas:
ge(gen_s:0'10_10(n15_10), gen_s:0'10_10(n15_10)) -> true, rt in Omega(0)


Generator Equations:
gen_s:0'10_10(0) <=> 0'
gen_s:0'10_10(+(x, 1)) <=> s(gen_s:0'10_10(x))
gen_nil:cons11_10(0) <=> nil
gen_nil:cons11_10(+(x, 1)) <=> cons(hole_a9_10, gen_nil:cons11_10(x))
gen_c7:c8:c912_10(0) <=> c7
gen_c7:c8:c912_10(+(x, 1)) <=> c9(gen_c7:c8:c912_10(x))
gen_c5:c613_10(0) <=> c5
gen_c5:c613_10(+(x, 1)) <=> c6(gen_c5:c613_10(x))


The following defined symbols remain to be analysed:
length, NTHTAIL, GE, LENGTH, nthtail

They will be analysed ascendingly in the following order:
length < NTHTAIL
GE < NTHTAIL
LENGTH < NTHTAIL
nthtail < NTHTAIL
length < nthtail

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

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_nil:cons11_10(n345_10)) -> gen_s:0'10_10(n345_10), rt in Omega(0)

Induction Base:
length(gen_nil:cons11_10(0)) ->_R^Omega(0)
0'

Induction Step:
length(gen_nil:cons11_10(+(n345_10, 1))) ->_R^Omega(0)
s(length(gen_nil:cons11_10(n345_10))) ->_IH
s(gen_s:0'10_10(c346_10))

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

(34)
Obligation:
Innermost TRS:
Rules:
NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1))
COND(true, z0, z1) -> c1
COND(false, z0, z1) -> c2(TAIL(nthtail(s(z0), z1)), NTHTAIL(s(z0), z1))
TAIL(nil) -> c3
TAIL(cons(z0, z1)) -> c4
LENGTH(nil) -> c5
LENGTH(cons(z0, z1)) -> c6(LENGTH(z1))
GE(z0, 0') -> c7
GE(0', s(z0)) -> c8
GE(s(z0), s(z1)) -> c9(GE(z0, z1))
nthtail(z0, z1) -> cond(ge(z0, length(z1)), z0, z1)
cond(true, z0, z1) -> z1
cond(false, z0, z1) -> tail(nthtail(s(z0), z1))
tail(nil) -> nil
tail(cons(z0, z1)) -> z1
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)

Types:
NTHTAIL :: s:0' -> nil:cons -> c
c :: c1:c2 -> c7:c8:c9 -> c5:c6 -> c
COND :: true:false -> s:0' -> nil:cons -> c1:c2
ge :: s:0' -> s:0' -> true:false
length :: nil:cons -> s:0'
GE :: s:0' -> s:0' -> c7:c8:c9
LENGTH :: nil:cons -> c5:c6
true :: true:false
c1 :: c1:c2
false :: true:false
c2 :: c3:c4 -> c -> c1:c2
TAIL :: nil:cons -> c3:c4
nthtail :: s:0' -> nil:cons -> nil:cons
s :: s:0' -> s:0'
nil :: nil:cons
c3 :: c3:c4
cons :: a -> nil:cons -> nil:cons
c4 :: c3:c4
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
0' :: s:0'
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
cond :: true:false -> s:0' -> nil:cons -> nil:cons
tail :: nil:cons -> nil:cons
hole_c1_10 :: c
hole_s:0'2_10 :: s:0'
hole_nil:cons3_10 :: nil:cons
hole_c1:c24_10 :: c1:c2
hole_c7:c8:c95_10 :: c7:c8:c9
hole_c5:c66_10 :: c5:c6
hole_true:false7_10 :: true:false
hole_c3:c48_10 :: c3:c4
hole_a9_10 :: a
gen_s:0'10_10 :: Nat -> s:0'
gen_nil:cons11_10 :: Nat -> nil:cons
gen_c7:c8:c912_10 :: Nat -> c7:c8:c9
gen_c5:c613_10 :: Nat -> c5:c6


Lemmas:
ge(gen_s:0'10_10(n15_10), gen_s:0'10_10(n15_10)) -> true, rt in Omega(0)
length(gen_nil:cons11_10(n345_10)) -> gen_s:0'10_10(n345_10), rt in Omega(0)


Generator Equations:
gen_s:0'10_10(0) <=> 0'
gen_s:0'10_10(+(x, 1)) <=> s(gen_s:0'10_10(x))
gen_nil:cons11_10(0) <=> nil
gen_nil:cons11_10(+(x, 1)) <=> cons(hole_a9_10, gen_nil:cons11_10(x))
gen_c7:c8:c912_10(0) <=> c7
gen_c7:c8:c912_10(+(x, 1)) <=> c9(gen_c7:c8:c912_10(x))
gen_c5:c613_10(0) <=> c5
gen_c5:c613_10(+(x, 1)) <=> c6(gen_c5:c613_10(x))


The following defined symbols remain to be analysed:
GE, NTHTAIL, LENGTH, nthtail

They will be analysed ascendingly in the following order:
GE < NTHTAIL
LENGTH < NTHTAIL
nthtail < NTHTAIL

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

(35) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
GE(gen_s:0'10_10(n634_10), gen_s:0'10_10(n634_10)) -> gen_c7:c8:c912_10(n634_10), rt in Omega(1 + n634_10)

Induction Base:
GE(gen_s:0'10_10(0), gen_s:0'10_10(0)) ->_R^Omega(1)
c7

Induction Step:
GE(gen_s:0'10_10(+(n634_10, 1)), gen_s:0'10_10(+(n634_10, 1))) ->_R^Omega(1)
c9(GE(gen_s:0'10_10(n634_10), gen_s:0'10_10(n634_10))) ->_IH
c9(gen_c7:c8:c912_10(c635_10))

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

(36)
Complex Obligation (BEST)

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

(37)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1))
COND(true, z0, z1) -> c1
COND(false, z0, z1) -> c2(TAIL(nthtail(s(z0), z1)), NTHTAIL(s(z0), z1))
TAIL(nil) -> c3
TAIL(cons(z0, z1)) -> c4
LENGTH(nil) -> c5
LENGTH(cons(z0, z1)) -> c6(LENGTH(z1))
GE(z0, 0') -> c7
GE(0', s(z0)) -> c8
GE(s(z0), s(z1)) -> c9(GE(z0, z1))
nthtail(z0, z1) -> cond(ge(z0, length(z1)), z0, z1)
cond(true, z0, z1) -> z1
cond(false, z0, z1) -> tail(nthtail(s(z0), z1))
tail(nil) -> nil
tail(cons(z0, z1)) -> z1
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)

Types:
NTHTAIL :: s:0' -> nil:cons -> c
c :: c1:c2 -> c7:c8:c9 -> c5:c6 -> c
COND :: true:false -> s:0' -> nil:cons -> c1:c2
ge :: s:0' -> s:0' -> true:false
length :: nil:cons -> s:0'
GE :: s:0' -> s:0' -> c7:c8:c9
LENGTH :: nil:cons -> c5:c6
true :: true:false
c1 :: c1:c2
false :: true:false
c2 :: c3:c4 -> c -> c1:c2
TAIL :: nil:cons -> c3:c4
nthtail :: s:0' -> nil:cons -> nil:cons
s :: s:0' -> s:0'
nil :: nil:cons
c3 :: c3:c4
cons :: a -> nil:cons -> nil:cons
c4 :: c3:c4
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
0' :: s:0'
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
cond :: true:false -> s:0' -> nil:cons -> nil:cons
tail :: nil:cons -> nil:cons
hole_c1_10 :: c
hole_s:0'2_10 :: s:0'
hole_nil:cons3_10 :: nil:cons
hole_c1:c24_10 :: c1:c2
hole_c7:c8:c95_10 :: c7:c8:c9
hole_c5:c66_10 :: c5:c6
hole_true:false7_10 :: true:false
hole_c3:c48_10 :: c3:c4
hole_a9_10 :: a
gen_s:0'10_10 :: Nat -> s:0'
gen_nil:cons11_10 :: Nat -> nil:cons
gen_c7:c8:c912_10 :: Nat -> c7:c8:c9
gen_c5:c613_10 :: Nat -> c5:c6


Lemmas:
ge(gen_s:0'10_10(n15_10), gen_s:0'10_10(n15_10)) -> true, rt in Omega(0)
length(gen_nil:cons11_10(n345_10)) -> gen_s:0'10_10(n345_10), rt in Omega(0)


Generator Equations:
gen_s:0'10_10(0) <=> 0'
gen_s:0'10_10(+(x, 1)) <=> s(gen_s:0'10_10(x))
gen_nil:cons11_10(0) <=> nil
gen_nil:cons11_10(+(x, 1)) <=> cons(hole_a9_10, gen_nil:cons11_10(x))
gen_c7:c8:c912_10(0) <=> c7
gen_c7:c8:c912_10(+(x, 1)) <=> c9(gen_c7:c8:c912_10(x))
gen_c5:c613_10(0) <=> c5
gen_c5:c613_10(+(x, 1)) <=> c6(gen_c5:c613_10(x))


The following defined symbols remain to be analysed:
GE, NTHTAIL, LENGTH, nthtail

They will be analysed ascendingly in the following order:
GE < NTHTAIL
LENGTH < NTHTAIL
nthtail < NTHTAIL

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

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

(39)
BOUNDS(n^1, INF)

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

(40)
Obligation:
Innermost TRS:
Rules:
NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1))
COND(true, z0, z1) -> c1
COND(false, z0, z1) -> c2(TAIL(nthtail(s(z0), z1)), NTHTAIL(s(z0), z1))
TAIL(nil) -> c3
TAIL(cons(z0, z1)) -> c4
LENGTH(nil) -> c5
LENGTH(cons(z0, z1)) -> c6(LENGTH(z1))
GE(z0, 0') -> c7
GE(0', s(z0)) -> c8
GE(s(z0), s(z1)) -> c9(GE(z0, z1))
nthtail(z0, z1) -> cond(ge(z0, length(z1)), z0, z1)
cond(true, z0, z1) -> z1
cond(false, z0, z1) -> tail(nthtail(s(z0), z1))
tail(nil) -> nil
tail(cons(z0, z1)) -> z1
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)

Types:
NTHTAIL :: s:0' -> nil:cons -> c
c :: c1:c2 -> c7:c8:c9 -> c5:c6 -> c
COND :: true:false -> s:0' -> nil:cons -> c1:c2
ge :: s:0' -> s:0' -> true:false
length :: nil:cons -> s:0'
GE :: s:0' -> s:0' -> c7:c8:c9
LENGTH :: nil:cons -> c5:c6
true :: true:false
c1 :: c1:c2
false :: true:false
c2 :: c3:c4 -> c -> c1:c2
TAIL :: nil:cons -> c3:c4
nthtail :: s:0' -> nil:cons -> nil:cons
s :: s:0' -> s:0'
nil :: nil:cons
c3 :: c3:c4
cons :: a -> nil:cons -> nil:cons
c4 :: c3:c4
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
0' :: s:0'
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
cond :: true:false -> s:0' -> nil:cons -> nil:cons
tail :: nil:cons -> nil:cons
hole_c1_10 :: c
hole_s:0'2_10 :: s:0'
hole_nil:cons3_10 :: nil:cons
hole_c1:c24_10 :: c1:c2
hole_c7:c8:c95_10 :: c7:c8:c9
hole_c5:c66_10 :: c5:c6
hole_true:false7_10 :: true:false
hole_c3:c48_10 :: c3:c4
hole_a9_10 :: a
gen_s:0'10_10 :: Nat -> s:0'
gen_nil:cons11_10 :: Nat -> nil:cons
gen_c7:c8:c912_10 :: Nat -> c7:c8:c9
gen_c5:c613_10 :: Nat -> c5:c6


Lemmas:
ge(gen_s:0'10_10(n15_10), gen_s:0'10_10(n15_10)) -> true, rt in Omega(0)
length(gen_nil:cons11_10(n345_10)) -> gen_s:0'10_10(n345_10), rt in Omega(0)
GE(gen_s:0'10_10(n634_10), gen_s:0'10_10(n634_10)) -> gen_c7:c8:c912_10(n634_10), rt in Omega(1 + n634_10)


Generator Equations:
gen_s:0'10_10(0) <=> 0'
gen_s:0'10_10(+(x, 1)) <=> s(gen_s:0'10_10(x))
gen_nil:cons11_10(0) <=> nil
gen_nil:cons11_10(+(x, 1)) <=> cons(hole_a9_10, gen_nil:cons11_10(x))
gen_c7:c8:c912_10(0) <=> c7
gen_c7:c8:c912_10(+(x, 1)) <=> c9(gen_c7:c8:c912_10(x))
gen_c5:c613_10(0) <=> c5
gen_c5:c613_10(+(x, 1)) <=> c6(gen_c5:c613_10(x))


The following defined symbols remain to be analysed:
LENGTH, NTHTAIL, nthtail

They will be analysed ascendingly in the following order:
LENGTH < NTHTAIL
nthtail < NTHTAIL

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

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LENGTH(gen_nil:cons11_10(n1273_10)) -> gen_c5:c613_10(n1273_10), rt in Omega(1 + n1273_10)

Induction Base:
LENGTH(gen_nil:cons11_10(0)) ->_R^Omega(1)
c5

Induction Step:
LENGTH(gen_nil:cons11_10(+(n1273_10, 1))) ->_R^Omega(1)
c6(LENGTH(gen_nil:cons11_10(n1273_10))) ->_IH
c6(gen_c5:c613_10(c1274_10))

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

(42)
Obligation:
Innermost TRS:
Rules:
NTHTAIL(z0, z1) -> c(COND(ge(z0, length(z1)), z0, z1), GE(z0, length(z1)), LENGTH(z1))
COND(true, z0, z1) -> c1
COND(false, z0, z1) -> c2(TAIL(nthtail(s(z0), z1)), NTHTAIL(s(z0), z1))
TAIL(nil) -> c3
TAIL(cons(z0, z1)) -> c4
LENGTH(nil) -> c5
LENGTH(cons(z0, z1)) -> c6(LENGTH(z1))
GE(z0, 0') -> c7
GE(0', s(z0)) -> c8
GE(s(z0), s(z1)) -> c9(GE(z0, z1))
nthtail(z0, z1) -> cond(ge(z0, length(z1)), z0, z1)
cond(true, z0, z1) -> z1
cond(false, z0, z1) -> tail(nthtail(s(z0), z1))
tail(nil) -> nil
tail(cons(z0, z1)) -> z1
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)

Types:
NTHTAIL :: s:0' -> nil:cons -> c
c :: c1:c2 -> c7:c8:c9 -> c5:c6 -> c
COND :: true:false -> s:0' -> nil:cons -> c1:c2
ge :: s:0' -> s:0' -> true:false
length :: nil:cons -> s:0'
GE :: s:0' -> s:0' -> c7:c8:c9
LENGTH :: nil:cons -> c5:c6
true :: true:false
c1 :: c1:c2
false :: true:false
c2 :: c3:c4 -> c -> c1:c2
TAIL :: nil:cons -> c3:c4
nthtail :: s:0' -> nil:cons -> nil:cons
s :: s:0' -> s:0'
nil :: nil:cons
c3 :: c3:c4
cons :: a -> nil:cons -> nil:cons
c4 :: c3:c4
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
0' :: s:0'
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
cond :: true:false -> s:0' -> nil:cons -> nil:cons
tail :: nil:cons -> nil:cons
hole_c1_10 :: c
hole_s:0'2_10 :: s:0'
hole_nil:cons3_10 :: nil:cons
hole_c1:c24_10 :: c1:c2
hole_c7:c8:c95_10 :: c7:c8:c9
hole_c5:c66_10 :: c5:c6
hole_true:false7_10 :: true:false
hole_c3:c48_10 :: c3:c4
hole_a9_10 :: a
gen_s:0'10_10 :: Nat -> s:0'
gen_nil:cons11_10 :: Nat -> nil:cons
gen_c7:c8:c912_10 :: Nat -> c7:c8:c9
gen_c5:c613_10 :: Nat -> c5:c6


Lemmas:
ge(gen_s:0'10_10(n15_10), gen_s:0'10_10(n15_10)) -> true, rt in Omega(0)
length(gen_nil:cons11_10(n345_10)) -> gen_s:0'10_10(n345_10), rt in Omega(0)
GE(gen_s:0'10_10(n634_10), gen_s:0'10_10(n634_10)) -> gen_c7:c8:c912_10(n634_10), rt in Omega(1 + n634_10)
LENGTH(gen_nil:cons11_10(n1273_10)) -> gen_c5:c613_10(n1273_10), rt in Omega(1 + n1273_10)


Generator Equations:
gen_s:0'10_10(0) <=> 0'
gen_s:0'10_10(+(x, 1)) <=> s(gen_s:0'10_10(x))
gen_nil:cons11_10(0) <=> nil
gen_nil:cons11_10(+(x, 1)) <=> cons(hole_a9_10, gen_nil:cons11_10(x))
gen_c7:c8:c912_10(0) <=> c7
gen_c7:c8:c912_10(+(x, 1)) <=> c9(gen_c7:c8:c912_10(x))
gen_c5:c613_10(0) <=> c5
gen_c5:c613_10(+(x, 1)) <=> c6(gen_c5:c613_10(x))


The following defined symbols remain to be analysed:
nthtail, NTHTAIL

They will be analysed ascendingly in the following order:
nthtail < NTHTAIL