WORST_CASE(Omega(n^1),O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 1040 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 517 ms]
    (12) BOUNDS(1, n^2)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxRelTRS
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) typed CpxTrs
    (17) OrderProof [LOWER BOUND(ID), 0 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 370 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 92 ms]
            (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 606 ms]
            (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 36 ms]
            (30) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   SUB(0, 0) -> c
   SUB(s(z0), 0) -> c1
   SUB(0, s(z0)) -> c2
   SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
   ZERO(nil) -> c4(ZERO2(0, nil))
   ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
   ZERO2(0, nil) -> c6
   ZERO2(0, cons(z0, z1)) -> c7(SUB(z0, z0))
   ZERO2(0, cons(z0, z1)) -> c8(ZERO(z1))
   ZERO2(s(z0), nil) -> c9(ZERO(nil))
   ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))

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

   sub(0, 0) -> 0
   sub(s(z0), 0) -> s(z0)
   sub(0, s(z0)) -> 0
   sub(s(z0), s(z1)) -> sub(z0, z1)
   zero(nil) -> zero2(0, nil)
   zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
   zero2(0, nil) -> nil
   zero2(0, cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
   zero2(s(z0), nil) -> zero(nil)
   zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))

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

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

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


The TRS R consists of the following rules:

   SUB(0, 0) -> c
   SUB(s(z0), 0) -> c1
   SUB(0, s(z0)) -> c2
   SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
   ZERO(nil) -> c4(ZERO2(0, nil))
   ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
   ZERO2(0, nil) -> c6
   ZERO2(0, cons(z0, z1)) -> c7(SUB(z0, z0))
   ZERO2(0, cons(z0, z1)) -> c8(ZERO(z1))
   ZERO2(s(z0), nil) -> c9(ZERO(nil))
   ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))

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

   sub(0, 0) -> 0
   sub(s(z0), 0) -> s(z0)
   sub(0, s(z0)) -> 0
   sub(s(z0), s(z1)) -> sub(z0, z1)
   zero(nil) -> zero2(0, nil)
   zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
   zero2(0, nil) -> nil
   zero2(0, cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
   zero2(s(z0), nil) -> zero(nil)
   zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))

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

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

(4)
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:

   SUB(0, 0) -> c [1]
   SUB(s(z0), 0) -> c1 [1]
   SUB(0, s(z0)) -> c2 [1]
   SUB(s(z0), s(z1)) -> c3(SUB(z0, z1)) [1]
   ZERO(nil) -> c4(ZERO2(0, nil)) [1]
   ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0)) [1]
   ZERO2(0, nil) -> c6 [1]
   ZERO2(0, cons(z0, z1)) -> c7(SUB(z0, z0)) [1]
   ZERO2(0, cons(z0, z1)) -> c8(ZERO(z1)) [1]
   ZERO2(s(z0), nil) -> c9(ZERO(nil)) [1]
   ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2))) [1]
   sub(0, 0) -> 0 [0]
   sub(s(z0), 0) -> s(z0) [0]
   sub(0, s(z0)) -> 0 [0]
   sub(s(z0), s(z1)) -> sub(z0, z1) [0]
   zero(nil) -> zero2(0, nil) [0]
   zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1)) [0]
   zero2(0, nil) -> nil [0]
   zero2(0, cons(z0, z1)) -> cons(sub(z0, z0), zero(z1)) [0]
   zero2(s(z0), nil) -> zero(nil) [0]
   zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2)) [0]

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

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   SUB(0, 0) -> c [1]
   SUB(s(z0), 0) -> c1 [1]
   SUB(0, s(z0)) -> c2 [1]
   SUB(s(z0), s(z1)) -> c3(SUB(z0, z1)) [1]
   ZERO(nil) -> c4(ZERO2(0, nil)) [1]
   ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0)) [1]
   ZERO2(0, nil) -> c6 [1]
   ZERO2(0, cons(z0, z1)) -> c7(SUB(z0, z0)) [1]
   ZERO2(0, cons(z0, z1)) -> c8(ZERO(z1)) [1]
   ZERO2(s(z0), nil) -> c9(ZERO(nil)) [1]
   ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2))) [1]
   sub(0, 0) -> 0 [0]
   sub(s(z0), 0) -> s(z0) [0]
   sub(0, s(z0)) -> 0 [0]
   sub(s(z0), s(z1)) -> sub(z0, z1) [0]
   zero(nil) -> zero2(0, nil) [0]
   zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1)) [0]
   zero2(0, nil) -> nil [0]
   zero2(0, cons(z0, z1)) -> cons(sub(z0, z0), zero(z1)) [0]
   zero2(s(z0), nil) -> zero(nil) [0]
   zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2)) [0]

The TRS has the following type information:
SUB :: 0:s -> 0:s -> c:c1:c2:c3
0 :: 0:s
c :: c:c1:c2:c3
s :: 0:s -> 0:s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
ZERO :: nil:cons -> c4:c5
nil :: nil:cons
c4 :: c6:c7:c8:c9:c10 -> c4:c5
ZERO2 :: 0:s -> nil:cons -> c6:c7:c8:c9:c10
cons :: 0:s -> nil:cons -> nil:cons
c5 :: c6:c7:c8:c9:c10 -> c:c1:c2:c3 -> c4:c5
sub :: 0:s -> 0:s -> 0:s
c6 :: c6:c7:c8:c9:c10
c7 :: c:c1:c2:c3 -> c6:c7:c8:c9:c10
c8 :: c4:c5 -> c6:c7:c8:c9:c10
c9 :: c4:c5 -> c6:c7:c8:c9:c10
c10 :: c4:c5 -> c6:c7:c8:c9:c10
zero :: nil:cons -> nil:cons
zero2 :: 0:s -> nil:cons -> nil:cons

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

(7) 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:

   sub(v0, v1) -> null_sub [0]
   zero(v0) -> null_zero [0]
   zero2(v0, v1) -> null_zero2 [0]
   SUB(v0, v1) -> null_SUB [0]
   ZERO(v0) -> null_ZERO [0]
   ZERO2(v0, v1) -> null_ZERO2 [0]

And the following fresh constants:    null_sub, null_zero, null_zero2, null_SUB, null_ZERO, null_ZERO2

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

(8)
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:

   SUB(0, 0) -> c [1]
   SUB(s(z0), 0) -> c1 [1]
   SUB(0, s(z0)) -> c2 [1]
   SUB(s(z0), s(z1)) -> c3(SUB(z0, z1)) [1]
   ZERO(nil) -> c4(ZERO2(0, nil)) [1]
   ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0)) [1]
   ZERO2(0, nil) -> c6 [1]
   ZERO2(0, cons(z0, z1)) -> c7(SUB(z0, z0)) [1]
   ZERO2(0, cons(z0, z1)) -> c8(ZERO(z1)) [1]
   ZERO2(s(z0), nil) -> c9(ZERO(nil)) [1]
   ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2))) [1]
   sub(0, 0) -> 0 [0]
   sub(s(z0), 0) -> s(z0) [0]
   sub(0, s(z0)) -> 0 [0]
   sub(s(z0), s(z1)) -> sub(z0, z1) [0]
   zero(nil) -> zero2(0, nil) [0]
   zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1)) [0]
   zero2(0, nil) -> nil [0]
   zero2(0, cons(z0, z1)) -> cons(sub(z0, z0), zero(z1)) [0]
   zero2(s(z0), nil) -> zero(nil) [0]
   zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2)) [0]
   sub(v0, v1) -> null_sub [0]
   zero(v0) -> null_zero [0]
   zero2(v0, v1) -> null_zero2 [0]
   SUB(v0, v1) -> null_SUB [0]
   ZERO(v0) -> null_ZERO [0]
   ZERO2(v0, v1) -> null_ZERO2 [0]

The TRS has the following type information:
SUB :: 0:s:null_sub -> 0:s:null_sub -> c:c1:c2:c3:null_SUB
0 :: 0:s:null_sub
c :: c:c1:c2:c3:null_SUB
s :: 0:s:null_sub -> 0:s:null_sub
c1 :: c:c1:c2:c3:null_SUB
c2 :: c:c1:c2:c3:null_SUB
c3 :: c:c1:c2:c3:null_SUB -> c:c1:c2:c3:null_SUB
ZERO :: nil:cons:null_zero:null_zero2 -> c4:c5:null_ZERO
nil :: nil:cons:null_zero:null_zero2
c4 :: c6:c7:c8:c9:c10:null_ZERO2 -> c4:c5:null_ZERO
ZERO2 :: 0:s:null_sub -> nil:cons:null_zero:null_zero2 -> c6:c7:c8:c9:c10:null_ZERO2
cons :: 0:s:null_sub -> nil:cons:null_zero:null_zero2 -> nil:cons:null_zero:null_zero2
c5 :: c6:c7:c8:c9:c10:null_ZERO2 -> c:c1:c2:c3:null_SUB -> c4:c5:null_ZERO
sub :: 0:s:null_sub -> 0:s:null_sub -> 0:s:null_sub
c6 :: c6:c7:c8:c9:c10:null_ZERO2
c7 :: c:c1:c2:c3:null_SUB -> c6:c7:c8:c9:c10:null_ZERO2
c8 :: c4:c5:null_ZERO -> c6:c7:c8:c9:c10:null_ZERO2
c9 :: c4:c5:null_ZERO -> c6:c7:c8:c9:c10:null_ZERO2
c10 :: c4:c5:null_ZERO -> c6:c7:c8:c9:c10:null_ZERO2
zero :: nil:cons:null_zero:null_zero2 -> nil:cons:null_zero:null_zero2
zero2 :: 0:s:null_sub -> nil:cons:null_zero:null_zero2 -> nil:cons:null_zero:null_zero2
null_sub :: 0:s:null_sub
null_zero :: nil:cons:null_zero:null_zero2
null_zero2 :: nil:cons:null_zero:null_zero2
null_SUB :: c:c1:c2:c3:null_SUB
null_ZERO :: c4:c5:null_ZERO
null_ZERO2 :: c6:c7:c8:c9:c10:null_ZERO2

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

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

   0 => 0
   c => 0
   c1 => 1
   c2 => 2
   nil => 0
   c6 => 0
   null_sub => 0
   null_zero => 0
   null_zero2 => 0
   null_SUB => 0
   null_ZERO => 0
   null_ZERO2 => 0

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   SUB(z, z') -{ 1 }-> 2 :|: z0 >= 0, z' = 1 + z0, z = 0
   SUB(z, z') -{ 1 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0
   SUB(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   SUB(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   SUB(z, z') -{ 1 }-> 1 + SUB(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   ZERO(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   ZERO(z) -{ 1 }-> 1 + ZERO2(0, 0) :|: z = 0
   ZERO(z) -{ 1 }-> 1 + ZERO2(sub(z0, z0), 1 + z0 + z1) + SUB(z0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ZERO2(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   ZERO2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   ZERO2(z, z') -{ 1 }-> 1 + ZERO(z1) :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 0
   ZERO2(z, z') -{ 1 }-> 1 + ZERO(0) :|: z = 1 + z0, z0 >= 0, z' = 0
   ZERO2(z, z') -{ 1 }-> 1 + ZERO(1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 1 + z0, z0 >= 0, z2 >= 0
   ZERO2(z, z') -{ 1 }-> 1 + SUB(z0, z0) :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 0
   sub(z, z') -{ 0 }-> sub(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   sub(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
   sub(z, z') -{ 0 }-> 0 :|: z0 >= 0, z' = 1 + z0, z = 0
   sub(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   sub(z, z') -{ 0 }-> 1 + z0 :|: z = 1 + z0, z0 >= 0, z' = 0
   zero(z) -{ 0 }-> zero2(sub(z0, z0), 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   zero(z) -{ 0 }-> zero2(0, 0) :|: z = 0
   zero(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   zero2(z, z') -{ 0 }-> zero(0) :|: z = 1 + z0, z0 >= 0, z' = 0
   zero2(z, z') -{ 0 }-> zero(1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 1 + z0, z0 >= 0, z2 >= 0
   zero2(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
   zero2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   zero2(z, z') -{ 0 }-> 1 + sub(z0, z0) + zero(z1) :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 0

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[fun1(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[fun2(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[sub(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[zero(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[zero2(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(fun(V1, V, Out),1,[],[Out = 0,V1 = 0,V = 0]).
eq(fun(V1, V, Out),1,[],[Out = 1,V1 = 1 + V2,V2 >= 0,V = 0]).
eq(fun(V1, V, Out),1,[],[Out = 2,V3 >= 0,V = 1 + V3,V1 = 0]).
eq(fun(V1, V, Out),1,[fun(V5, V4, Ret1)],[Out = 1 + Ret1,V4 >= 0,V1 = 1 + V5,V5 >= 0,V = 1 + V4]).
eq(fun1(V1, Out),1,[fun2(0, 0, Ret11)],[Out = 1 + Ret11,V1 = 0]).
eq(fun1(V1, Out),1,[sub(V6, V6, Ret010),fun2(Ret010, 1 + V6 + V7, Ret01),fun(V6, V6, Ret12)],[Out = 1 + Ret01 + Ret12,V7 >= 0,V6 >= 0,V1 = 1 + V6 + V7]).
eq(fun2(V1, V, Out),1,[],[Out = 0,V1 = 0,V = 0]).
eq(fun2(V1, V, Out),1,[fun(V8, V8, Ret13)],[Out = 1 + Ret13,V = 1 + V8 + V9,V9 >= 0,V8 >= 0,V1 = 0]).
eq(fun2(V1, V, Out),1,[fun1(V11, Ret14)],[Out = 1 + Ret14,V = 1 + V10 + V11,V11 >= 0,V10 >= 0,V1 = 0]).
eq(fun2(V1, V, Out),1,[fun1(0, Ret15)],[Out = 1 + Ret15,V1 = 1 + V12,V12 >= 0,V = 0]).
eq(fun2(V1, V, Out),1,[fun1(1 + V13 + V14, Ret16)],[Out = 1 + Ret16,V13 >= 0,V = 1 + V13 + V14,V1 = 1 + V15,V15 >= 0,V14 >= 0]).
eq(sub(V1, V, Out),0,[],[Out = 0,V1 = 0,V = 0]).
eq(sub(V1, V, Out),0,[],[Out = 1 + V16,V1 = 1 + V16,V16 >= 0,V = 0]).
eq(sub(V1, V, Out),0,[],[Out = 0,V17 >= 0,V = 1 + V17,V1 = 0]).
eq(sub(V1, V, Out),0,[sub(V19, V18, Ret)],[Out = Ret,V18 >= 0,V1 = 1 + V19,V19 >= 0,V = 1 + V18]).
eq(zero(V1, Out),0,[zero2(0, 0, Ret2)],[Out = Ret2,V1 = 0]).
eq(zero(V1, Out),0,[sub(V21, V21, Ret0),zero2(Ret0, 1 + V21 + V20, Ret3)],[Out = Ret3,V20 >= 0,V21 >= 0,V1 = 1 + V20 + V21]).
eq(zero2(V1, V, Out),0,[],[Out = 0,V1 = 0,V = 0]).
eq(zero2(V1, V, Out),0,[sub(V23, V23, Ret011),zero(V22, Ret17)],[Out = 1 + Ret011 + Ret17,V = 1 + V22 + V23,V22 >= 0,V23 >= 0,V1 = 0]).
eq(zero2(V1, V, Out),0,[zero(0, Ret4)],[Out = Ret4,V1 = 1 + V24,V24 >= 0,V = 0]).
eq(zero2(V1, V, Out),0,[zero(1 + V27 + V25, Ret5)],[Out = Ret5,V27 >= 0,V = 1 + V25 + V27,V1 = 1 + V26,V26 >= 0,V25 >= 0]).
eq(sub(V1, V, Out),0,[],[Out = 0,V29 >= 0,V28 >= 0,V1 = V29,V = V28]).
eq(zero(V1, Out),0,[],[Out = 0,V30 >= 0,V1 = V30]).
eq(zero2(V1, V, Out),0,[],[Out = 0,V32 >= 0,V31 >= 0,V1 = V32,V = V31]).
eq(fun(V1, V, Out),0,[],[Out = 0,V33 >= 0,V34 >= 0,V1 = V33,V = V34]).
eq(fun1(V1, Out),0,[],[Out = 0,V35 >= 0,V1 = V35]).
eq(fun2(V1, V, Out),0,[],[Out = 0,V36 >= 0,V37 >= 0,V1 = V36,V = V37]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,Out),[V1],[Out]).
input_output_vars(fun2(V1,V,Out),[V1,V],[Out]).
input_output_vars(sub(V1,V,Out),[V1,V],[Out]).
input_output_vars(zero(V1,Out),[V1],[Out]).
input_output_vars(zero2(V1,V,Out),[V1,V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [fun/3]
1. recursive  : [sub/3]
2. recursive [non_tail] : [fun1/2,fun2/3]
3. recursive  : [zero/2,zero2/3]
4. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into fun/3
1. SCC is partially evaluated into sub/3
2. SCC is partially evaluated into fun1/2
3. SCC is partially evaluated into zero/2
4. SCC is partially evaluated into start/2

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

### Specialization of cost equations fun/3 
* CE 30 is refined into CE [34] 
* CE 31 is refined into CE [35] 
* CE 29 is refined into CE [36] 
* CE 33 is refined into CE [37] 
* CE 32 is refined into CE [38] 


### Cost equations --> "Loop" of fun/3 
* CEs [38] --> Loop 19 
* CEs [34] --> Loop 20 
* CEs [35] --> Loop 21 
* CEs [36,37] --> Loop 22 

### Ranking functions of CR fun(V1,V,Out) 
* RF of phase [19]: [V,V1]

#### Partial ranking functions of CR fun(V1,V,Out) 
* Partial RF of phase [19]:
  - RF of loop [19:1]:
    V
    V1


### Specialization of cost equations sub/3 
* CE 19 is refined into CE [39] 
* CE 18 is refined into CE [40] 
* CE 20 is refined into CE [41] 
* CE 21 is refined into CE [42] 


### Cost equations --> "Loop" of sub/3 
* CEs [42] --> Loop 23 
* CEs [39] --> Loop 24 
* CEs [40,41] --> Loop 25 

### Ranking functions of CR sub(V1,V,Out) 
* RF of phase [23]: [V,V1]

#### Partial ranking functions of CR sub(V1,V,Out) 
* Partial RF of phase [23]:
  - RF of loop [23:1]:
    V
    V1


### Specialization of cost equations fun1/2 
* CE 22 is refined into CE [43,44] 
* CE 26 is refined into CE [45,46,47,48] 
* CE 28 is refined into CE [49] 
* CE 23 is refined into CE [50] 
* CE 27 is refined into CE [51] 
* CE 25 is refined into CE [52,53] 
* CE 24 is discarded (unfeasible) 


### Cost equations --> "Loop" of fun1/2 
* CEs [53] --> Loop 26 
* CEs [52] --> Loop 27 
* CEs [48] --> Loop 28 
* CEs [46,47] --> Loop 29 
* CEs [44] --> Loop 30 
* CEs [45] --> Loop 31 
* CEs [43] --> Loop 32 
* CEs [49] --> Loop 33 
* CEs [50,51] --> Loop 34 

### Ranking functions of CR fun1(V1,Out) 
* RF of phase [26,27]: [V1]

#### Partial ranking functions of CR fun1(V1,Out) 
* Partial RF of phase [26,27]:
  - RF of loop [26:1]:
    V1-1
  - RF of loop [27:1]:
    V1


### Specialization of cost equations zero/2 
* CE 14 is refined into CE [54] 
* CE 15 is refined into CE [55] 
* CE 17 is refined into CE [56] 
* CE 16 is discarded (unfeasible) 


### Cost equations --> "Loop" of zero/2 
* CEs [56] --> Loop 35 
* CEs [54,55] --> Loop 36 

### Ranking functions of CR zero(V1,Out) 
* RF of phase [35]: [V1]

#### Partial ranking functions of CR zero(V1,Out) 
* Partial RF of phase [35]:
  - RF of loop [35:1]:
    V1


### Specialization of cost equations start/2 
* CE 3 is refined into CE [57] 
* CE 6 is refined into CE [58,59] 
* CE 4 is refined into CE [60,61] 
* CE 7 is refined into CE [62,63,64,65] 
* CE 8 is refined into CE [66,67] 
* CE 1 is refined into CE [68] 
* CE 2 is refined into CE [69,70] 
* CE 5 is refined into CE [71,72,73] 
* CE 9 is refined into CE [74] 
* CE 10 is refined into CE [75,76,77,78,79,80] 
* CE 11 is refined into CE [81,82,83,84] 
* CE 12 is refined into CE [85,86,87] 
* CE 13 is refined into CE [88,89] 


### Cost equations --> "Loop" of start/2 
* CEs [57,58,59,76,85] --> Loop 37 
* CEs [60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,77,78,79,80,81,82,83,84,86,87,88,89] --> Loop 38 

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

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


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

#### Cost of chains of fun(V1,V,Out):
* Chain [[19],22]: 1*it(19)+1
  Such that:it(19) =< Out

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

* Chain [[19],21]: 1*it(19)+1
  Such that:it(19) =< Out

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

* Chain [[19],20]: 1*it(19)+1
  Such that:it(19) =< Out

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

* Chain [22]: 1
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [21]: 1
  with precondition: [V1=0,Out=2,V>=1] 

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


#### Cost of chains of sub(V1,V,Out):
* Chain [[23],25]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

* Chain [[23],24]: 0
  with precondition: [V1=Out+V,V>=1,V1>=V+1] 

* Chain [25]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [24]: 0
  with precondition: [V=0,V1=Out,V1>=1] 


#### Cost of chains of fun1(V1,Out):
* Chain [[26,27],34]: 6*it(26)+1*s(3)+2
  Such that:aux(4) =< V1
it(26) =< aux(4)
s(3) =< it(26)*aux(4)

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

* Chain [[26,27],33]: 6*it(26)+1*s(3)+0
  Such that:aux(5) =< V1
it(26) =< aux(5)
s(3) =< it(26)*aux(5)

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

* Chain [[26,27],32]: 6*it(26)+1*s(3)+2
  Such that:aux(6) =< V1
it(26) =< aux(6)
s(3) =< it(26)*aux(6)

  with precondition: [V1>=2,Out>=3] 

* Chain [[26,27],31]: 6*it(26)+1*s(3)+4
  Such that:aux(7) =< V1
it(26) =< aux(7)
s(3) =< it(26)*aux(7)

  with precondition: [V1>=2,Out>=4] 

* Chain [[26,27],30]: 7*it(26)+1*s(3)+2
  Such that:aux(8) =< V1
it(26) =< aux(8)
s(3) =< it(26)*aux(8)

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

* Chain [[26,27],29]: 8*it(26)+1*s(3)+4
  Such that:aux(10) =< V1
it(26) =< aux(10)
s(3) =< it(26)*aux(10)

  with precondition: [V1>=3,Out>=5] 

* Chain [[26,27],28]: 8*it(26)+1*s(3)+4
  Such that:aux(12) =< V1
it(26) =< aux(12)
s(3) =< it(26)*aux(12)

  with precondition: [V1>=3,Out>=6] 

* Chain [34]: 2
  with precondition: [V1=0,Out=1] 

* Chain [33]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [32]: 2
  with precondition: [Out=1,V1>=1] 

* Chain [31]: 4
  with precondition: [Out=2,V1>=1] 

* Chain [30]: 1*s(4)+2
  Such that:s(4) =< V1

  with precondition: [Out>=2,V1>=Out] 

* Chain [29]: 2*s(5)+4
  Such that:aux(9) =< V1
s(5) =< aux(9)

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

* Chain [28]: 2*s(7)+4
  Such that:aux(11) =< V1
s(7) =< aux(11)

  with precondition: [Out>=4,2*V1>=Out] 


#### Cost of chains of zero(V1,Out):
* Chain [[35],36]: 0
  with precondition: [Out>=1,V1>=Out] 

* Chain [36]: 0
  with precondition: [Out=0,V1>=0] 


#### Cost of chains of start(V1,V):
* Chain [38]: 106*s(36)+14*s(37)+1*s(42)+1*s(43)+52*s(46)+7*s(47)+5
  Such that:s(45) =< V1
s(42) =< V1+2
s(43) =< V+1
aux(14) =< V
s(36) =< aux(14)
s(46) =< s(45)
s(47) =< s(46)*s(45)
s(37) =< s(36)*aux(14)

  with precondition: [V1>=0] 

* Chain [37]: 3
  with precondition: [V=0,V1>=1] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [38] with precondition: [V1>=0] 
    - Upper bound: 52*V1+5+7*V1*V1+nat(V)*106+nat(V)*14*nat(V)+(V1+2)+nat(V+1) 
    - Complexity: n^2 
* Chain [37] with precondition: [V=0,V1>=1] 
    - Upper bound: 3 
    - Complexity: constant 

### Maximum cost of start(V1,V): 52*V1+2+7*V1*V1+nat(V)*106+nat(V)*14*nat(V)+(V1+2)+nat(V+1)+3 
Asymptotic class: n^2 
* Total analysis performed in 417 ms.


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

(12)
BOUNDS(1, n^2)

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

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

(14)
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:

   SUB(0', 0') -> c
   SUB(s(z0), 0') -> c1
   SUB(0', s(z0)) -> c2
   SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
   ZERO(nil) -> c4(ZERO2(0', nil))
   ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
   ZERO2(0', nil) -> c6
   ZERO2(0', cons(z0, z1)) -> c7(SUB(z0, z0))
   ZERO2(0', cons(z0, z1)) -> c8(ZERO(z1))
   ZERO2(s(z0), nil) -> c9(ZERO(nil))
   ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))

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

   sub(0', 0') -> 0'
   sub(s(z0), 0') -> s(z0)
   sub(0', s(z0)) -> 0'
   sub(s(z0), s(z1)) -> sub(z0, z1)
   zero(nil) -> zero2(0', nil)
   zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
   zero2(0', nil) -> nil
   zero2(0', cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
   zero2(s(z0), nil) -> zero(nil)
   zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))

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

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

(16)
Obligation:
Innermost TRS:
Rules:
SUB(0', 0') -> c
SUB(s(z0), 0') -> c1
SUB(0', s(z0)) -> c2
SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
ZERO(nil) -> c4(ZERO2(0', nil))
ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
ZERO2(0', nil) -> c6
ZERO2(0', cons(z0, z1)) -> c7(SUB(z0, z0))
ZERO2(0', cons(z0, z1)) -> c8(ZERO(z1))
ZERO2(s(z0), nil) -> c9(ZERO(nil))
ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))
sub(0', 0') -> 0'
sub(s(z0), 0') -> s(z0)
sub(0', s(z0)) -> 0'
sub(s(z0), s(z1)) -> sub(z0, z1)
zero(nil) -> zero2(0', nil)
zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
zero2(0', nil) -> nil
zero2(0', cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
zero2(s(z0), nil) -> zero(nil)
zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))

Types:
SUB :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
ZERO :: nil:cons -> c4:c5
nil :: nil:cons
c4 :: c6:c7:c8:c9:c10 -> c4:c5
ZERO2 :: 0':s -> nil:cons -> c6:c7:c8:c9:c10
cons :: 0':s -> nil:cons -> nil:cons
c5 :: c6:c7:c8:c9:c10 -> c:c1:c2:c3 -> c4:c5
sub :: 0':s -> 0':s -> 0':s
c6 :: c6:c7:c8:c9:c10
c7 :: c:c1:c2:c3 -> c6:c7:c8:c9:c10
c8 :: c4:c5 -> c6:c7:c8:c9:c10
c9 :: c4:c5 -> c6:c7:c8:c9:c10
c10 :: c4:c5 -> c6:c7:c8:c9:c10
zero :: nil:cons -> nil:cons
zero2 :: 0':s -> nil:cons -> nil:cons
hole_c:c1:c2:c31_11 :: c:c1:c2:c3
hole_0':s2_11 :: 0':s
hole_c4:c53_11 :: c4:c5
hole_nil:cons4_11 :: nil:cons
hole_c6:c7:c8:c9:c105_11 :: c6:c7:c8:c9:c10
gen_c:c1:c2:c36_11 :: Nat -> c:c1:c2:c3
gen_0':s7_11 :: Nat -> 0':s
gen_nil:cons8_11 :: Nat -> nil:cons

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

(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
SUB, ZERO, sub, zero

They will be analysed ascendingly in the following order:
SUB < ZERO
sub < ZERO
sub < zero

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

(18)
Obligation:
Innermost TRS:
Rules:
SUB(0', 0') -> c
SUB(s(z0), 0') -> c1
SUB(0', s(z0)) -> c2
SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
ZERO(nil) -> c4(ZERO2(0', nil))
ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
ZERO2(0', nil) -> c6
ZERO2(0', cons(z0, z1)) -> c7(SUB(z0, z0))
ZERO2(0', cons(z0, z1)) -> c8(ZERO(z1))
ZERO2(s(z0), nil) -> c9(ZERO(nil))
ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))
sub(0', 0') -> 0'
sub(s(z0), 0') -> s(z0)
sub(0', s(z0)) -> 0'
sub(s(z0), s(z1)) -> sub(z0, z1)
zero(nil) -> zero2(0', nil)
zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
zero2(0', nil) -> nil
zero2(0', cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
zero2(s(z0), nil) -> zero(nil)
zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))

Types:
SUB :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
ZERO :: nil:cons -> c4:c5
nil :: nil:cons
c4 :: c6:c7:c8:c9:c10 -> c4:c5
ZERO2 :: 0':s -> nil:cons -> c6:c7:c8:c9:c10
cons :: 0':s -> nil:cons -> nil:cons
c5 :: c6:c7:c8:c9:c10 -> c:c1:c2:c3 -> c4:c5
sub :: 0':s -> 0':s -> 0':s
c6 :: c6:c7:c8:c9:c10
c7 :: c:c1:c2:c3 -> c6:c7:c8:c9:c10
c8 :: c4:c5 -> c6:c7:c8:c9:c10
c9 :: c4:c5 -> c6:c7:c8:c9:c10
c10 :: c4:c5 -> c6:c7:c8:c9:c10
zero :: nil:cons -> nil:cons
zero2 :: 0':s -> nil:cons -> nil:cons
hole_c:c1:c2:c31_11 :: c:c1:c2:c3
hole_0':s2_11 :: 0':s
hole_c4:c53_11 :: c4:c5
hole_nil:cons4_11 :: nil:cons
hole_c6:c7:c8:c9:c105_11 :: c6:c7:c8:c9:c10
gen_c:c1:c2:c36_11 :: Nat -> c:c1:c2:c3
gen_0':s7_11 :: Nat -> 0':s
gen_nil:cons8_11 :: Nat -> nil:cons


Generator Equations:
gen_c:c1:c2:c36_11(0) <=> c
gen_c:c1:c2:c36_11(+(x, 1)) <=> c3(gen_c:c1:c2:c36_11(x))
gen_0':s7_11(0) <=> 0'
gen_0':s7_11(+(x, 1)) <=> s(gen_0':s7_11(x))
gen_nil:cons8_11(0) <=> nil
gen_nil:cons8_11(+(x, 1)) <=> cons(0', gen_nil:cons8_11(x))


The following defined symbols remain to be analysed:
SUB, ZERO, sub, zero

They will be analysed ascendingly in the following order:
SUB < ZERO
sub < ZERO
sub < zero

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
SUB(gen_0':s7_11(n10_11), gen_0':s7_11(n10_11)) -> gen_c:c1:c2:c36_11(n10_11), rt in Omega(1 + n10_11)

Induction Base:
SUB(gen_0':s7_11(0), gen_0':s7_11(0)) ->_R^Omega(1)
c

Induction Step:
SUB(gen_0':s7_11(+(n10_11, 1)), gen_0':s7_11(+(n10_11, 1))) ->_R^Omega(1)
c3(SUB(gen_0':s7_11(n10_11), gen_0':s7_11(n10_11))) ->_IH
c3(gen_c:c1:c2:c36_11(c11_11))

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

(20)
Complex Obligation (BEST)

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

(21)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
SUB(0', 0') -> c
SUB(s(z0), 0') -> c1
SUB(0', s(z0)) -> c2
SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
ZERO(nil) -> c4(ZERO2(0', nil))
ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
ZERO2(0', nil) -> c6
ZERO2(0', cons(z0, z1)) -> c7(SUB(z0, z0))
ZERO2(0', cons(z0, z1)) -> c8(ZERO(z1))
ZERO2(s(z0), nil) -> c9(ZERO(nil))
ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))
sub(0', 0') -> 0'
sub(s(z0), 0') -> s(z0)
sub(0', s(z0)) -> 0'
sub(s(z0), s(z1)) -> sub(z0, z1)
zero(nil) -> zero2(0', nil)
zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
zero2(0', nil) -> nil
zero2(0', cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
zero2(s(z0), nil) -> zero(nil)
zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))

Types:
SUB :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
ZERO :: nil:cons -> c4:c5
nil :: nil:cons
c4 :: c6:c7:c8:c9:c10 -> c4:c5
ZERO2 :: 0':s -> nil:cons -> c6:c7:c8:c9:c10
cons :: 0':s -> nil:cons -> nil:cons
c5 :: c6:c7:c8:c9:c10 -> c:c1:c2:c3 -> c4:c5
sub :: 0':s -> 0':s -> 0':s
c6 :: c6:c7:c8:c9:c10
c7 :: c:c1:c2:c3 -> c6:c7:c8:c9:c10
c8 :: c4:c5 -> c6:c7:c8:c9:c10
c9 :: c4:c5 -> c6:c7:c8:c9:c10
c10 :: c4:c5 -> c6:c7:c8:c9:c10
zero :: nil:cons -> nil:cons
zero2 :: 0':s -> nil:cons -> nil:cons
hole_c:c1:c2:c31_11 :: c:c1:c2:c3
hole_0':s2_11 :: 0':s
hole_c4:c53_11 :: c4:c5
hole_nil:cons4_11 :: nil:cons
hole_c6:c7:c8:c9:c105_11 :: c6:c7:c8:c9:c10
gen_c:c1:c2:c36_11 :: Nat -> c:c1:c2:c3
gen_0':s7_11 :: Nat -> 0':s
gen_nil:cons8_11 :: Nat -> nil:cons


Generator Equations:
gen_c:c1:c2:c36_11(0) <=> c
gen_c:c1:c2:c36_11(+(x, 1)) <=> c3(gen_c:c1:c2:c36_11(x))
gen_0':s7_11(0) <=> 0'
gen_0':s7_11(+(x, 1)) <=> s(gen_0':s7_11(x))
gen_nil:cons8_11(0) <=> nil
gen_nil:cons8_11(+(x, 1)) <=> cons(0', gen_nil:cons8_11(x))


The following defined symbols remain to be analysed:
SUB, ZERO, sub, zero

They will be analysed ascendingly in the following order:
SUB < ZERO
sub < ZERO
sub < zero

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
Innermost TRS:
Rules:
SUB(0', 0') -> c
SUB(s(z0), 0') -> c1
SUB(0', s(z0)) -> c2
SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
ZERO(nil) -> c4(ZERO2(0', nil))
ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
ZERO2(0', nil) -> c6
ZERO2(0', cons(z0, z1)) -> c7(SUB(z0, z0))
ZERO2(0', cons(z0, z1)) -> c8(ZERO(z1))
ZERO2(s(z0), nil) -> c9(ZERO(nil))
ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))
sub(0', 0') -> 0'
sub(s(z0), 0') -> s(z0)
sub(0', s(z0)) -> 0'
sub(s(z0), s(z1)) -> sub(z0, z1)
zero(nil) -> zero2(0', nil)
zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
zero2(0', nil) -> nil
zero2(0', cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
zero2(s(z0), nil) -> zero(nil)
zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))

Types:
SUB :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
ZERO :: nil:cons -> c4:c5
nil :: nil:cons
c4 :: c6:c7:c8:c9:c10 -> c4:c5
ZERO2 :: 0':s -> nil:cons -> c6:c7:c8:c9:c10
cons :: 0':s -> nil:cons -> nil:cons
c5 :: c6:c7:c8:c9:c10 -> c:c1:c2:c3 -> c4:c5
sub :: 0':s -> 0':s -> 0':s
c6 :: c6:c7:c8:c9:c10
c7 :: c:c1:c2:c3 -> c6:c7:c8:c9:c10
c8 :: c4:c5 -> c6:c7:c8:c9:c10
c9 :: c4:c5 -> c6:c7:c8:c9:c10
c10 :: c4:c5 -> c6:c7:c8:c9:c10
zero :: nil:cons -> nil:cons
zero2 :: 0':s -> nil:cons -> nil:cons
hole_c:c1:c2:c31_11 :: c:c1:c2:c3
hole_0':s2_11 :: 0':s
hole_c4:c53_11 :: c4:c5
hole_nil:cons4_11 :: nil:cons
hole_c6:c7:c8:c9:c105_11 :: c6:c7:c8:c9:c10
gen_c:c1:c2:c36_11 :: Nat -> c:c1:c2:c3
gen_0':s7_11 :: Nat -> 0':s
gen_nil:cons8_11 :: Nat -> nil:cons


Lemmas:
SUB(gen_0':s7_11(n10_11), gen_0':s7_11(n10_11)) -> gen_c:c1:c2:c36_11(n10_11), rt in Omega(1 + n10_11)


Generator Equations:
gen_c:c1:c2:c36_11(0) <=> c
gen_c:c1:c2:c36_11(+(x, 1)) <=> c3(gen_c:c1:c2:c36_11(x))
gen_0':s7_11(0) <=> 0'
gen_0':s7_11(+(x, 1)) <=> s(gen_0':s7_11(x))
gen_nil:cons8_11(0) <=> nil
gen_nil:cons8_11(+(x, 1)) <=> cons(0', gen_nil:cons8_11(x))


The following defined symbols remain to be analysed:
sub, ZERO, zero

They will be analysed ascendingly in the following order:
sub < ZERO
sub < zero

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sub(gen_0':s7_11(n878_11), gen_0':s7_11(n878_11)) -> gen_0':s7_11(0), rt in Omega(0)

Induction Base:
sub(gen_0':s7_11(0), gen_0':s7_11(0)) ->_R^Omega(0)
0'

Induction Step:
sub(gen_0':s7_11(+(n878_11, 1)), gen_0':s7_11(+(n878_11, 1))) ->_R^Omega(0)
sub(gen_0':s7_11(n878_11), gen_0':s7_11(n878_11)) ->_IH
gen_0':s7_11(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:
SUB(0', 0') -> c
SUB(s(z0), 0') -> c1
SUB(0', s(z0)) -> c2
SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
ZERO(nil) -> c4(ZERO2(0', nil))
ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
ZERO2(0', nil) -> c6
ZERO2(0', cons(z0, z1)) -> c7(SUB(z0, z0))
ZERO2(0', cons(z0, z1)) -> c8(ZERO(z1))
ZERO2(s(z0), nil) -> c9(ZERO(nil))
ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))
sub(0', 0') -> 0'
sub(s(z0), 0') -> s(z0)
sub(0', s(z0)) -> 0'
sub(s(z0), s(z1)) -> sub(z0, z1)
zero(nil) -> zero2(0', nil)
zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
zero2(0', nil) -> nil
zero2(0', cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
zero2(s(z0), nil) -> zero(nil)
zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))

Types:
SUB :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
ZERO :: nil:cons -> c4:c5
nil :: nil:cons
c4 :: c6:c7:c8:c9:c10 -> c4:c5
ZERO2 :: 0':s -> nil:cons -> c6:c7:c8:c9:c10
cons :: 0':s -> nil:cons -> nil:cons
c5 :: c6:c7:c8:c9:c10 -> c:c1:c2:c3 -> c4:c5
sub :: 0':s -> 0':s -> 0':s
c6 :: c6:c7:c8:c9:c10
c7 :: c:c1:c2:c3 -> c6:c7:c8:c9:c10
c8 :: c4:c5 -> c6:c7:c8:c9:c10
c9 :: c4:c5 -> c6:c7:c8:c9:c10
c10 :: c4:c5 -> c6:c7:c8:c9:c10
zero :: nil:cons -> nil:cons
zero2 :: 0':s -> nil:cons -> nil:cons
hole_c:c1:c2:c31_11 :: c:c1:c2:c3
hole_0':s2_11 :: 0':s
hole_c4:c53_11 :: c4:c5
hole_nil:cons4_11 :: nil:cons
hole_c6:c7:c8:c9:c105_11 :: c6:c7:c8:c9:c10
gen_c:c1:c2:c36_11 :: Nat -> c:c1:c2:c3
gen_0':s7_11 :: Nat -> 0':s
gen_nil:cons8_11 :: Nat -> nil:cons


Lemmas:
SUB(gen_0':s7_11(n10_11), gen_0':s7_11(n10_11)) -> gen_c:c1:c2:c36_11(n10_11), rt in Omega(1 + n10_11)
sub(gen_0':s7_11(n878_11), gen_0':s7_11(n878_11)) -> gen_0':s7_11(0), rt in Omega(0)


Generator Equations:
gen_c:c1:c2:c36_11(0) <=> c
gen_c:c1:c2:c36_11(+(x, 1)) <=> c3(gen_c:c1:c2:c36_11(x))
gen_0':s7_11(0) <=> 0'
gen_0':s7_11(+(x, 1)) <=> s(gen_0':s7_11(x))
gen_nil:cons8_11(0) <=> nil
gen_nil:cons8_11(+(x, 1)) <=> cons(0', gen_nil:cons8_11(x))


The following defined symbols remain to be analysed:
ZERO, zero
----------------------------------------

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ZERO(gen_nil:cons8_11(n1828_11)) -> *9_11, rt in Omega(n1828_11)

Induction Base:
ZERO(gen_nil:cons8_11(0))

Induction Step:
ZERO(gen_nil:cons8_11(+(n1828_11, 1))) ->_R^Omega(1)
c5(ZERO2(sub(0', 0'), cons(0', gen_nil:cons8_11(n1828_11))), SUB(0', 0')) ->_L^Omega(0)
c5(ZERO2(gen_0':s7_11(0), cons(0', gen_nil:cons8_11(n1828_11))), SUB(0', 0')) ->_R^Omega(1)
c5(c8(ZERO(gen_nil:cons8_11(n1828_11))), SUB(0', 0')) ->_IH
c5(c8(*9_11), SUB(0', 0')) ->_L^Omega(1)
c5(c8(*9_11), gen_c:c1:c2:c36_11(0))

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

(28)
Obligation:
Innermost TRS:
Rules:
SUB(0', 0') -> c
SUB(s(z0), 0') -> c1
SUB(0', s(z0)) -> c2
SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
ZERO(nil) -> c4(ZERO2(0', nil))
ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
ZERO2(0', nil) -> c6
ZERO2(0', cons(z0, z1)) -> c7(SUB(z0, z0))
ZERO2(0', cons(z0, z1)) -> c8(ZERO(z1))
ZERO2(s(z0), nil) -> c9(ZERO(nil))
ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))
sub(0', 0') -> 0'
sub(s(z0), 0') -> s(z0)
sub(0', s(z0)) -> 0'
sub(s(z0), s(z1)) -> sub(z0, z1)
zero(nil) -> zero2(0', nil)
zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
zero2(0', nil) -> nil
zero2(0', cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
zero2(s(z0), nil) -> zero(nil)
zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))

Types:
SUB :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
ZERO :: nil:cons -> c4:c5
nil :: nil:cons
c4 :: c6:c7:c8:c9:c10 -> c4:c5
ZERO2 :: 0':s -> nil:cons -> c6:c7:c8:c9:c10
cons :: 0':s -> nil:cons -> nil:cons
c5 :: c6:c7:c8:c9:c10 -> c:c1:c2:c3 -> c4:c5
sub :: 0':s -> 0':s -> 0':s
c6 :: c6:c7:c8:c9:c10
c7 :: c:c1:c2:c3 -> c6:c7:c8:c9:c10
c8 :: c4:c5 -> c6:c7:c8:c9:c10
c9 :: c4:c5 -> c6:c7:c8:c9:c10
c10 :: c4:c5 -> c6:c7:c8:c9:c10
zero :: nil:cons -> nil:cons
zero2 :: 0':s -> nil:cons -> nil:cons
hole_c:c1:c2:c31_11 :: c:c1:c2:c3
hole_0':s2_11 :: 0':s
hole_c4:c53_11 :: c4:c5
hole_nil:cons4_11 :: nil:cons
hole_c6:c7:c8:c9:c105_11 :: c6:c7:c8:c9:c10
gen_c:c1:c2:c36_11 :: Nat -> c:c1:c2:c3
gen_0':s7_11 :: Nat -> 0':s
gen_nil:cons8_11 :: Nat -> nil:cons


Lemmas:
SUB(gen_0':s7_11(n10_11), gen_0':s7_11(n10_11)) -> gen_c:c1:c2:c36_11(n10_11), rt in Omega(1 + n10_11)
sub(gen_0':s7_11(n878_11), gen_0':s7_11(n878_11)) -> gen_0':s7_11(0), rt in Omega(0)
ZERO(gen_nil:cons8_11(n1828_11)) -> *9_11, rt in Omega(n1828_11)


Generator Equations:
gen_c:c1:c2:c36_11(0) <=> c
gen_c:c1:c2:c36_11(+(x, 1)) <=> c3(gen_c:c1:c2:c36_11(x))
gen_0':s7_11(0) <=> 0'
gen_0':s7_11(+(x, 1)) <=> s(gen_0':s7_11(x))
gen_nil:cons8_11(0) <=> nil
gen_nil:cons8_11(+(x, 1)) <=> cons(0', gen_nil:cons8_11(x))


The following defined symbols remain to be analysed:
zero
----------------------------------------

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
zero(gen_nil:cons8_11(n7888_11)) -> gen_nil:cons8_11(n7888_11), rt in Omega(0)

Induction Base:
zero(gen_nil:cons8_11(0)) ->_R^Omega(0)
zero2(0', nil) ->_R^Omega(0)
nil

Induction Step:
zero(gen_nil:cons8_11(+(n7888_11, 1))) ->_R^Omega(0)
zero2(sub(0', 0'), cons(0', gen_nil:cons8_11(n7888_11))) ->_L^Omega(0)
zero2(gen_0':s7_11(0), cons(0', gen_nil:cons8_11(n7888_11))) ->_R^Omega(0)
cons(sub(0', 0'), zero(gen_nil:cons8_11(n7888_11))) ->_L^Omega(0)
cons(gen_0':s7_11(0), zero(gen_nil:cons8_11(n7888_11))) ->_IH
cons(gen_0':s7_11(0), gen_nil:cons8_11(c7889_11))

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

(30)
BOUNDS(1, INF)