WORST_CASE(Omega(n^1),O(n^1))
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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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^1).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 583 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxWeightedTrs
    (5) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) CompleteCoflocoProof [FINISHED, 2433 ms]
    (14) BOUNDS(1, n^1)
(15) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(16) TRS for Loop Detection
    (17) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   -'(z0, 0) -> c
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   P(s(z0)) -> c2
   F(s(z0), z1) -> c3(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), P(-(s(z0), z1)), -'(s(z0), z1))
   F(s(z0), z1) -> c4(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), P(-(z1, s(z0))), -'(z1, s(z0)))
   F(z0, s(z1)) -> c5(F(p(-(z0, s(z1))), p(-(s(z1), z0))), P(-(z0, s(z1))), -'(z0, s(z1)))
   F(z0, s(z1)) -> c6(F(p(-(z0, s(z1))), p(-(s(z1), z0))), P(-(s(z1), z0)), -'(s(z1), z0))

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

   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   p(s(z0)) -> z0
   f(s(z0), z1) -> f(p(-(s(z0), z1)), p(-(z1, s(z0))))
   f(z0, s(z1)) -> f(p(-(z0, s(z1))), p(-(s(z1), z0)))

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^1).


The TRS R consists of the following rules:

   -'(z0, 0) -> c
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   P(s(z0)) -> c2
   F(s(z0), z1) -> c3(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), P(-(s(z0), z1)), -'(s(z0), z1))
   F(s(z0), z1) -> c4(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), P(-(z1, s(z0))), -'(z1, s(z0)))
   F(z0, s(z1)) -> c5(F(p(-(z0, s(z1))), p(-(s(z1), z0))), P(-(z0, s(z1))), -'(z0, s(z1)))
   F(z0, s(z1)) -> c6(F(p(-(z0, s(z1))), p(-(s(z1), z0))), P(-(s(z1), z0)), -'(s(z1), z0))

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

   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   p(s(z0)) -> z0
   f(s(z0), z1) -> f(p(-(s(z0), z1)), p(-(z1, s(z0))))
   f(z0, s(z1)) -> f(p(-(z0, s(z1))), p(-(s(z1), z0)))

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^1).


The TRS R consists of the following rules:

   -'(z0, 0) -> c [1]
   -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1]
   P(s(z0)) -> c2 [1]
   F(s(z0), z1) -> c3(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), P(-(s(z0), z1)), -'(s(z0), z1)) [1]
   F(s(z0), z1) -> c4(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), P(-(z1, s(z0))), -'(z1, s(z0))) [1]
   F(z0, s(z1)) -> c5(F(p(-(z0, s(z1))), p(-(s(z1), z0))), P(-(z0, s(z1))), -'(z0, s(z1))) [1]
   F(z0, s(z1)) -> c6(F(p(-(z0, s(z1))), p(-(s(z1), z0))), P(-(s(z1), z0)), -'(s(z1), z0)) [1]
   -(z0, 0) -> z0 [0]
   -(s(z0), s(z1)) -> -(z0, z1) [0]
   p(s(z0)) -> z0 [0]
   f(s(z0), z1) -> f(p(-(s(z0), z1)), p(-(z1, s(z0)))) [0]
   f(z0, s(z1)) -> f(p(-(z0, s(z1))), p(-(s(z1), z0))) [0]

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

(5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

- => minus

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

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


The TRS R consists of the following rules:

   -'(z0, 0) -> c [1]
   -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1]
   P(s(z0)) -> c2 [1]
   F(s(z0), z1) -> c3(F(p(minus(s(z0), z1)), p(minus(z1, s(z0)))), P(minus(s(z0), z1)), -'(s(z0), z1)) [1]
   F(s(z0), z1) -> c4(F(p(minus(s(z0), z1)), p(minus(z1, s(z0)))), P(minus(z1, s(z0))), -'(z1, s(z0))) [1]
   F(z0, s(z1)) -> c5(F(p(minus(z0, s(z1))), p(minus(s(z1), z0))), P(minus(z0, s(z1))), -'(z0, s(z1))) [1]
   F(z0, s(z1)) -> c6(F(p(minus(z0, s(z1))), p(minus(s(z1), z0))), P(minus(s(z1), z0)), -'(s(z1), z0)) [1]
   minus(z0, 0) -> z0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   p(s(z0)) -> z0 [0]
   f(s(z0), z1) -> f(p(minus(s(z0), z1)), p(minus(z1, s(z0)))) [0]
   f(z0, s(z1)) -> f(p(minus(z0, s(z1))), p(minus(s(z1), z0))) [0]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   -'(z0, 0) -> c [1]
   -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1]
   P(s(z0)) -> c2 [1]
   F(s(z0), z1) -> c3(F(p(minus(s(z0), z1)), p(minus(z1, s(z0)))), P(minus(s(z0), z1)), -'(s(z0), z1)) [1]
   F(s(z0), z1) -> c4(F(p(minus(s(z0), z1)), p(minus(z1, s(z0)))), P(minus(z1, s(z0))), -'(z1, s(z0))) [1]
   F(z0, s(z1)) -> c5(F(p(minus(z0, s(z1))), p(minus(s(z1), z0))), P(minus(z0, s(z1))), -'(z0, s(z1))) [1]
   F(z0, s(z1)) -> c6(F(p(minus(z0, s(z1))), p(minus(s(z1), z0))), P(minus(s(z1), z0)), -'(s(z1), z0)) [1]
   minus(z0, 0) -> z0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   p(s(z0)) -> z0 [0]
   f(s(z0), z1) -> f(p(minus(s(z0), z1)), p(minus(z1, s(z0)))) [0]
   f(z0, s(z1)) -> f(p(minus(z0, s(z1))), p(minus(s(z1), z0))) [0]

The TRS has the following type information:
-' :: 0:s -> 0:s -> c:c1
0 :: 0:s
c :: c:c1
s :: 0:s -> 0:s
c1 :: c:c1 -> c:c1
P :: 0:s -> c2
c2 :: c2
F :: 0:s -> 0:s -> c3:c4:c5:c6
c3 :: c3:c4:c5:c6 -> c2 -> c:c1 -> c3:c4:c5:c6
p :: 0:s -> 0:s
minus :: 0:s -> 0:s -> 0:s
c4 :: c3:c4:c5:c6 -> c2 -> c:c1 -> c3:c4:c5:c6
c5 :: c3:c4:c5:c6 -> c2 -> c:c1 -> c3:c4:c5:c6
c6 :: c3:c4:c5:c6 -> c2 -> c:c1 -> c3:c4:c5:c6
f :: 0:s -> 0:s -> f

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

(9) 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:

   minus(v0, v1) -> null_minus [0]
   p(v0) -> null_p [0]
   f(v0, v1) -> null_f [0]
   -'(v0, v1) -> null_-' [0]
   P(v0) -> null_P [0]
   F(v0, v1) -> null_F [0]

And the following fresh constants:    null_minus, null_p, null_f, null_-', null_P, null_F

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

(10)
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:

   -'(z0, 0) -> c [1]
   -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1]
   P(s(z0)) -> c2 [1]
   F(s(z0), z1) -> c3(F(p(minus(s(z0), z1)), p(minus(z1, s(z0)))), P(minus(s(z0), z1)), -'(s(z0), z1)) [1]
   F(s(z0), z1) -> c4(F(p(minus(s(z0), z1)), p(minus(z1, s(z0)))), P(minus(z1, s(z0))), -'(z1, s(z0))) [1]
   F(z0, s(z1)) -> c5(F(p(minus(z0, s(z1))), p(minus(s(z1), z0))), P(minus(z0, s(z1))), -'(z0, s(z1))) [1]
   F(z0, s(z1)) -> c6(F(p(minus(z0, s(z1))), p(minus(s(z1), z0))), P(minus(s(z1), z0)), -'(s(z1), z0)) [1]
   minus(z0, 0) -> z0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   p(s(z0)) -> z0 [0]
   f(s(z0), z1) -> f(p(minus(s(z0), z1)), p(minus(z1, s(z0)))) [0]
   f(z0, s(z1)) -> f(p(minus(z0, s(z1))), p(minus(s(z1), z0))) [0]
   minus(v0, v1) -> null_minus [0]
   p(v0) -> null_p [0]
   f(v0, v1) -> null_f [0]
   -'(v0, v1) -> null_-' [0]
   P(v0) -> null_P [0]
   F(v0, v1) -> null_F [0]

The TRS has the following type information:
-' :: 0:s:null_minus:null_p -> 0:s:null_minus:null_p -> c:c1:null_-'
0 :: 0:s:null_minus:null_p
c :: c:c1:null_-'
s :: 0:s:null_minus:null_p -> 0:s:null_minus:null_p
c1 :: c:c1:null_-' -> c:c1:null_-'
P :: 0:s:null_minus:null_p -> c2:null_P
c2 :: c2:null_P
F :: 0:s:null_minus:null_p -> 0:s:null_minus:null_p -> c3:c4:c5:c6:null_F
c3 :: c3:c4:c5:c6:null_F -> c2:null_P -> c:c1:null_-' -> c3:c4:c5:c6:null_F
p :: 0:s:null_minus:null_p -> 0:s:null_minus:null_p
minus :: 0:s:null_minus:null_p -> 0:s:null_minus:null_p -> 0:s:null_minus:null_p
c4 :: c3:c4:c5:c6:null_F -> c2:null_P -> c:c1:null_-' -> c3:c4:c5:c6:null_F
c5 :: c3:c4:c5:c6:null_F -> c2:null_P -> c:c1:null_-' -> c3:c4:c5:c6:null_F
c6 :: c3:c4:c5:c6:null_F -> c2:null_P -> c:c1:null_-' -> c3:c4:c5:c6:null_F
f :: 0:s:null_minus:null_p -> 0:s:null_minus:null_p -> null_f
null_minus :: 0:s:null_minus:null_p
null_p :: 0:s:null_minus:null_p
null_f :: null_f
null_-' :: c:c1:null_-'
null_P :: c2:null_P
null_F :: c3:c4:c5:c6:null_F

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

(11) 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
   c2 => 1
   null_minus => 0
   null_p => 0
   null_f => 0
   null_-' => 0
   null_P => 0
   null_F => 0

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ 1 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   -'(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   -'(z, z') -{ 1 }-> 1 + -'(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   F(z, z') -{ 1 }-> 1 + F(p(minus(z0, 1 + z1)), p(minus(1 + z1, z0))) + P(minus(z0, 1 + z1)) + -'(z0, 1 + z1) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + F(p(minus(z0, 1 + z1)), p(minus(1 + z1, z0))) + P(minus(1 + z1, z0)) + -'(1 + z1, z0) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + F(p(minus(1 + z0, z1)), p(minus(z1, 1 + z0))) + P(minus(z1, 1 + z0)) + -'(z1, 1 + z0) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   F(z, z') -{ 1 }-> 1 + F(p(minus(1 + z0, z1)), p(minus(z1, 1 + z0))) + P(minus(1 + z0, z1)) + -'(1 + z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   P(z) -{ 1 }-> 1 :|: z = 1 + z0, z0 >= 0
   P(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   f(z, z') -{ 0 }-> f(p(minus(z0, 1 + z1)), p(minus(1 + z1, z0))) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   f(z, z') -{ 0 }-> f(p(minus(1 + z0, z1)), p(minus(z1, 1 + z0))) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   minus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   p(z) -{ 0 }-> z0 :|: z = 1 + z0, z0 >= 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

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

(13) 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,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[p(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[f(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(fun(V1, V, Out),1,[],[Out = 0,V1 = V2,V2 >= 0,V = 0]).
eq(fun(V1, V, Out),1,[fun(V4, V3, Ret1)],[Out = 1 + Ret1,V3 >= 0,V1 = 1 + V4,V4 >= 0,V = 1 + V3]).
eq(fun1(V1, Out),1,[],[Out = 1,V1 = 1 + V5,V5 >= 0]).
eq(fun2(V1, V, Out),1,[minus(1 + V6, V7, Ret00100),p(Ret00100, Ret0010),minus(V7, 1 + V6, Ret00110),p(Ret00110, Ret0011),fun2(Ret0010, Ret0011, Ret001),minus(1 + V6, V7, Ret010),fun1(Ret010, Ret01),fun(1 + V6, V7, Ret11)],[Out = 1 + Ret001 + Ret01 + Ret11,V7 >= 0,V1 = 1 + V6,V = V7,V6 >= 0]).
eq(fun2(V1, V, Out),1,[minus(1 + V8, V9, Ret001001),p(Ret001001, Ret00101),minus(V9, 1 + V8, Ret001101),p(Ret001101, Ret00111),fun2(Ret00101, Ret00111, Ret0012),minus(V9, 1 + V8, Ret0101),fun1(Ret0101, Ret011),fun(V9, 1 + V8, Ret12)],[Out = 1 + Ret0012 + Ret011 + Ret12,V9 >= 0,V1 = 1 + V8,V = V9,V8 >= 0]).
eq(fun2(V1, V, Out),1,[minus(V10, 1 + V11, Ret001002),p(Ret001002, Ret00102),minus(1 + V11, V10, Ret001102),p(Ret001102, Ret00112),fun2(Ret00102, Ret00112, Ret0013),minus(V10, 1 + V11, Ret0102),fun1(Ret0102, Ret012),fun(V10, 1 + V11, Ret13)],[Out = 1 + Ret0013 + Ret012 + Ret13,V1 = V10,V11 >= 0,V10 >= 0,V = 1 + V11]).
eq(fun2(V1, V, Out),1,[minus(V13, 1 + V12, Ret001003),p(Ret001003, Ret00103),minus(1 + V12, V13, Ret001103),p(Ret001103, Ret00113),fun2(Ret00103, Ret00113, Ret0014),minus(1 + V12, V13, Ret0103),fun1(Ret0103, Ret013),fun(1 + V12, V13, Ret14)],[Out = 1 + Ret0014 + Ret013 + Ret14,V1 = V13,V12 >= 0,V13 >= 0,V = 1 + V12]).
eq(minus(V1, V, Out),0,[],[Out = V14,V1 = V14,V14 >= 0,V = 0]).
eq(minus(V1, V, Out),0,[minus(V16, V15, Ret)],[Out = Ret,V15 >= 0,V1 = 1 + V16,V16 >= 0,V = 1 + V15]).
eq(p(V1, Out),0,[],[Out = V17,V1 = 1 + V17,V17 >= 0]).
eq(f(V1, V, Out),0,[minus(1 + V19, V18, Ret00),p(Ret00, Ret0),minus(V18, 1 + V19, Ret10),p(Ret10, Ret15),f(Ret0, Ret15, Ret2)],[Out = Ret2,V18 >= 0,V1 = 1 + V19,V = V18,V19 >= 0]).
eq(f(V1, V, Out),0,[minus(V21, 1 + V20, Ret002),p(Ret002, Ret02),minus(1 + V20, V21, Ret101),p(Ret101, Ret16),f(Ret02, Ret16, Ret3)],[Out = Ret3,V1 = V21,V20 >= 0,V21 >= 0,V = 1 + V20]).
eq(minus(V1, V, Out),0,[],[Out = 0,V23 >= 0,V22 >= 0,V1 = V23,V = V22]).
eq(p(V1, Out),0,[],[Out = 0,V24 >= 0,V1 = V24]).
eq(f(V1, V, Out),0,[],[Out = 0,V26 >= 0,V25 >= 0,V1 = V26,V = V25]).
eq(fun(V1, V, Out),0,[],[Out = 0,V27 >= 0,V28 >= 0,V1 = V27,V = V28]).
eq(fun1(V1, Out),0,[],[Out = 0,V29 >= 0,V1 = V29]).
eq(fun2(V1, V, Out),0,[],[Out = 0,V30 >= 0,V31 >= 0,V1 = V30,V = V31]).
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(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(p(V1,Out),[V1],[Out]).
input_output_vars(f(V1,V,Out),[V1,V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [minus/3]
1. non_recursive  : [p/2]
2. recursive  : [f/3]
3. recursive  : [fun/3]
4. non_recursive  : [fun1/2]
5. recursive [non_tail] : [fun2/3]
6. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into minus/3
1. SCC is partially evaluated into p/2
2. SCC is partially evaluated into f/3
3. SCC is partially evaluated into fun/3
4. SCC is partially evaluated into fun1/2
5. SCC is partially evaluated into fun2/3
6. SCC is partially evaluated into start/2

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

### Specialization of cost equations minus/3 
* CE 19 is refined into CE [25] 
* CE 17 is refined into CE [26] 
* CE 18 is refined into CE [27] 


### Cost equations --> "Loop" of minus/3 
* CEs [27] --> Loop 17 
* CEs [25] --> Loop 18 
* CEs [26] --> Loop 19 

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [17]: [V,V1]

#### Partial ranking functions of CR minus(V1,V,Out) 
* Partial RF of phase [17]:
  - RF of loop [17:1]:
    V
    V1


### Specialization of cost equations p/2 
* CE 20 is refined into CE [28] 
* CE 21 is refined into CE [29] 


### Cost equations --> "Loop" of p/2 
* CEs [28] --> Loop 20 
* CEs [29] --> Loop 21 

### Ranking functions of CR p(V1,Out) 

#### Partial ranking functions of CR p(V1,Out) 


### Specialization of cost equations f/3 
* CE 24 is refined into CE [30] 
* CE 22 is refined into CE [31,32,33,34,35,36,37,38] 
* CE 23 is refined into CE [39,40,41,42,43,44,45,46] 


### Cost equations --> "Loop" of f/3 
* CEs [38,46] --> Loop 22 
* CEs [35,43] --> Loop 23 
* CEs [37,45] --> Loop 24 
* CEs [32] --> Loop 25 
* CEs [31,33] --> Loop 26 
* CEs [40] --> Loop 27 
* CEs [34,36,39,41,42,44] --> Loop 28 
* CEs [30] --> Loop 29 

### Ranking functions of CR f(V1,V,Out) 
* RF of phase [25]: [V1]
* RF of phase [27]: [V]

#### Partial ranking functions of CR f(V1,V,Out) 
* Partial RF of phase [25]:
  - RF of loop [25:1]:
    V1
* Partial RF of phase [27]:
  - RF of loop [27:1]:
    V


### Specialization of cost equations fun/3 
* CE 7 is refined into CE [47] 
* CE 9 is refined into CE [48] 
* CE 8 is refined into CE [49] 


### Cost equations --> "Loop" of fun/3 
* CEs [49] --> Loop 30 
* CEs [47,48] --> Loop 31 

### Ranking functions of CR fun(V1,V,Out) 
* RF of phase [30]: [V,V1]

#### Partial ranking functions of CR fun(V1,V,Out) 
* Partial RF of phase [30]:
  - RF of loop [30:1]:
    V
    V1


### Specialization of cost equations fun1/2 
* CE 10 is refined into CE [50] 
* CE 11 is refined into CE [51] 


### Cost equations --> "Loop" of fun1/2 
* CEs [50] --> Loop 32 
* CEs [51] --> Loop 33 

### Ranking functions of CR fun1(V1,Out) 

#### Partial ranking functions of CR fun1(V1,Out) 


### Specialization of cost equations fun2/3 
* CE 16 is refined into CE [52] 
* CE 12 is refined into CE [53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88] 
* CE 13 is refined into CE [89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118] 
* CE 14 is refined into CE [119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148] 
* CE 15 is refined into CE [149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184] 


### Cost equations --> "Loop" of fun2/3 
* CEs [88,148] --> Loop 34 
* CEs [84,86,87,118,144,146,147,184] --> Loop 35 
* CEs [83,85,117,143,145,183] --> Loop 36 
* CEs [108,174] --> Loop 37 
* CEs [72,104,106,107,132,170,172,173] --> Loop 38 
* CEs [71,103,105,131,169,171] --> Loop 39 
* CEs [66,78,126,138] --> Loop 40 
* CEs [96,102,162,168] --> Loop 41 
* CEs [64,65,74,76,77,110,124,125,134,136,137,176] --> Loop 42 
* CEs [95,101,161,167] --> Loop 43 
* CEs [62,68,70,80,82,92,94,98,100,112,114,116,122,128,130,140,142,158,160,164,166,178,180,182] --> Loop 44 
* CEs [69,79,81,111,113,115,129,139,141,177,179,181] --> Loop 45 
* CEs [57] --> Loop 46 
* CEs [56,58,90] --> Loop 47 
* CEs [54,60] --> Loop 48 
* CEs [53,55,59,61,89,91] --> Loop 49 
* CEs [153] --> Loop 50 
* CEs [120,152,154] --> Loop 51 
* CEs [150,156] --> Loop 52 
* CEs [63,67,73,75,93,97,99,109,119,121,123,127,133,135,149,151,155,157,159,163,165,175] --> Loop 53 
* CEs [52] --> Loop 54 

### Ranking functions of CR fun2(V1,V,Out) 
* RF of phase [46,47]: [V1]
* RF of phase [50,51]: [V]

#### Partial ranking functions of CR fun2(V1,V,Out) 
* Partial RF of phase [46,47]:
  - RF of loop [46:1,47:1]:
    V1
* Partial RF of phase [50,51]:
  - RF of loop [50:1,51:1]:
    V


### Specialization of cost equations start/2 
* CE 1 is refined into CE [185,186] 
* CE 2 is refined into CE [187,188] 
* CE 3 is refined into CE [189,190,191,192,193,194,195,196,197,198,199,200,201,202,203] 
* CE 4 is refined into CE [204,205,206] 
* CE 5 is refined into CE [207,208] 
* CE 6 is refined into CE [209,210,211] 


### Cost equations --> "Loop" of start/2 
* CEs [211] --> Loop 55 
* CEs [190,204,210] --> Loop 56 
* CEs [185,186,187,188,189,191,192,193,194,195,196,197,198,199,200,201,202,203,205,206,207,208,209] --> Loop 57 

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

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


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

#### Cost of chains of minus(V1,V,Out):
* Chain [[17],19]: 0
  with precondition: [V1=Out+V,V>=1,V1>=V] 

* Chain [[17],18]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

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

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


#### Cost of chains of p(V1,Out):
* Chain [21]: 0
  with precondition: [Out=0,V1>=0] 

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


#### Cost of chains of f(V1,V,Out):
* Chain [[27],29]: 0
  with precondition: [V1=0,Out=0,V>=1] 

* Chain [[27],28,29]: 0
  with precondition: [V1=0,Out=0,V>=2] 

* Chain [[25],29]: 0
  with precondition: [V=0,Out=0,V1>=1] 

* Chain [[25],26,29]: 0
  with precondition: [V=0,Out=0,V1>=2] 

* Chain [29]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [28,29]: 0
  with precondition: [Out=0,V1>=0,V>=1] 

* Chain [26,29]: 0
  with precondition: [Out=0,V1>=1,V>=0] 

* Chain [24,29]: 0
  with precondition: [Out=0,V1=V,V1>=1] 

* Chain [23,[27],29]: 0
  with precondition: [Out=0,V1>=1,V>=V1+2] 

* Chain [23,[27],28,29]: 0
  with precondition: [Out=0,V1>=1,V>=V1+3] 

* Chain [23,29]: 0
  with precondition: [Out=0,V1>=1,V>=V1+1] 

* Chain [23,28,29]: 0
  with precondition: [Out=0,V1>=1,V>=V1+2] 

* Chain [22,[25],29]: 0
  with precondition: [Out=0,V>=1,V1>=V+2] 

* Chain [22,[25],26,29]: 0
  with precondition: [Out=0,V>=1,V1>=V+3] 

* Chain [22,29]: 0
  with precondition: [Out=0,V>=1,V1>=V+1] 

* Chain [22,26,29]: 0
  with precondition: [Out=0,V>=1,V1>=V+2] 


#### Cost of chains of fun(V1,V,Out):
* Chain [[30],31]: 1*it(30)+1
  Such that:it(30) =< Out

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

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


#### Cost of chains of fun1(V1,Out):
* Chain [33]: 0
  with precondition: [Out=0,V1>=0] 

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


#### Cost of chains of fun2(V1,V,Out):
* Chain [[50,51],54]: 5*it(50)+0
  Such that:aux(3) =< V
it(50) =< aux(3)

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

* Chain [[50,51],53,54]: 5*it(50)+2
  Such that:aux(4) =< V
it(50) =< aux(4)

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

* Chain [[50,51],52,54]: 5*it(50)+3
  Such that:aux(5) =< V
it(50) =< aux(5)

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

* Chain [[46,47],54]: 5*it(46)+0
  Such that:aux(8) =< V1
it(46) =< aux(8)

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

* Chain [[46,47],49,54]: 5*it(46)+2
  Such that:aux(9) =< V1
it(46) =< aux(9)

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

* Chain [[46,47],48,54]: 5*it(46)+3
  Such that:aux(10) =< V1
it(46) =< aux(10)

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

* Chain [54]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [53,54]: 2
  with precondition: [Out=1,V1>=0,V>=1] 

* Chain [52,54]: 3
  with precondition: [V1=0,Out=2,V>=1] 

* Chain [49,54]: 2
  with precondition: [Out=1,V1>=1,V>=0] 

* Chain [48,54]: 3
  with precondition: [V=0,Out=2,V1>=1] 

* Chain [45,54]: 2
  with precondition: [Out=1,V1=V,V1>=1] 

* Chain [44,54]: 4*s(1)+20*s(2)+2
  Such that:aux(11) =< V1
aux(12) =< V
s(2) =< aux(11)
s(1) =< aux(12)

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

* Chain [43,54]: 3
  with precondition: [Out=2,V1>=1,V>=V1+1] 

* Chain [42,54]: 8*s(25)+3
  Such that:aux(13) =< V
s(25) =< aux(13)

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

* Chain [41,54]: 4*s(33)+3
  Such that:aux(14) =< V1
s(33) =< aux(14)

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

* Chain [40,54]: 4*s(37)+3
  Such that:aux(15) =< V
s(37) =< aux(15)

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

* Chain [39,[50,51],54]: 5*it(50)+2
  Such that:aux(3) =< -V1+V
it(50) =< aux(3)

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

* Chain [39,[50,51],53,54]: 5*it(50)+4
  Such that:aux(4) =< -V1+V
it(50) =< aux(4)

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

* Chain [39,[50,51],52,54]: 5*it(50)+5
  Such that:aux(5) =< -V1+V
it(50) =< aux(5)

  with precondition: [V1>=1,Out>=4,V>=V1+3,2*V>=2*V1+Out+1] 

* Chain [39,54]: 2
  with precondition: [Out=1,V1>=1,V>=V1+1] 

* Chain [39,53,54]: 4
  with precondition: [Out=2,V1>=1,V>=V1+2] 

* Chain [39,52,54]: 5
  with precondition: [Out=3,V1>=1,V>=V1+2] 

* Chain [38,[50,51],54]: 5*it(50)+6*s(41)+3
  Such that:aux(3) =< -V1+V
aux(16) =< V1
it(50) =< aux(3)
s(41) =< aux(16)

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

* Chain [38,[50,51],53,54]: 5*it(50)+6*s(41)+5
  Such that:aux(4) =< -V1+V
aux(16) =< V1
it(50) =< aux(4)
s(41) =< aux(16)

  with precondition: [V1>=1,Out>=4,V>=V1+3,2*V>=Out+V1+2] 

* Chain [38,[50,51],52,54]: 5*it(50)+6*s(41)+6
  Such that:aux(5) =< -V1+V
aux(16) =< V1
it(50) =< aux(5)
s(41) =< aux(16)

  with precondition: [V1>=1,Out>=5,V>=V1+3,2*V>=Out+V1+1] 

* Chain [38,54]: 6*s(41)+3
  Such that:aux(16) =< V1
s(41) =< aux(16)

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

* Chain [38,53,54]: 6*s(41)+5
  Such that:aux(16) =< V1
s(41) =< aux(16)

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

* Chain [38,52,54]: 6*s(41)+6
  Such that:aux(16) =< V1
s(41) =< aux(16)

  with precondition: [Out>=4,V>=V1+2,V1+3>=Out] 

* Chain [37,[50,51],54]: 5*it(50)+2*s(47)+3
  Such that:aux(3) =< -V1+V
aux(17) =< V1
it(50) =< aux(3)
s(47) =< aux(17)

  with precondition: [V1>=1,Out>=4,V>=V1+2,2*V>=Out+V1] 

* Chain [37,[50,51],53,54]: 5*it(50)+2*s(47)+5
  Such that:aux(4) =< -V1+V
aux(17) =< V1
it(50) =< aux(4)
s(47) =< aux(17)

  with precondition: [V1>=1,Out>=5,V>=V1+3,2*V>=Out+V1+1] 

* Chain [37,[50,51],52,54]: 5*it(50)+2*s(47)+6
  Such that:aux(5) =< -V1+V
aux(17) =< V1
it(50) =< aux(5)
s(47) =< aux(17)

  with precondition: [V1>=1,Out>=6,V>=V1+3,2*V>=Out+V1] 

* Chain [37,54]: 2*s(47)+3
  Such that:aux(17) =< V1
s(47) =< aux(17)

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

* Chain [37,53,54]: 2*s(47)+5
  Such that:aux(17) =< V1
s(47) =< aux(17)

  with precondition: [Out>=4,V>=V1+2,V1+3>=Out] 

* Chain [37,52,54]: 2*s(47)+6
  Such that:aux(17) =< V1
s(47) =< aux(17)

  with precondition: [Out>=5,V>=V1+2,V1+4>=Out] 

* Chain [36,[46,47],54]: 5*it(46)+2
  Such that:aux(8) =< V1-V
it(46) =< aux(8)

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

* Chain [36,[46,47],49,54]: 5*it(46)+4
  Such that:aux(9) =< V1-V
it(46) =< aux(9)

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

* Chain [36,[46,47],48,54]: 5*it(46)+5
  Such that:aux(10) =< V1-V
it(46) =< aux(10)

  with precondition: [V>=1,Out>=4,V1>=V+3,2*V1>=2*V+Out+1] 

* Chain [36,54]: 2
  with precondition: [Out=1,V>=1,V1>=V+1] 

* Chain [36,49,54]: 4
  with precondition: [Out=2,V>=1,V1>=V+2] 

* Chain [36,48,54]: 5
  with precondition: [Out=3,V>=1,V1>=V+2] 

* Chain [35,[46,47],54]: 5*it(46)+6*s(49)+3
  Such that:aux(8) =< V1-V
aux(18) =< V+1
it(46) =< aux(8)
s(49) =< aux(18)

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

* Chain [35,[46,47],49,54]: 5*it(46)+6*s(49)+5
  Such that:aux(9) =< V1-V
aux(18) =< V+1
it(46) =< aux(9)
s(49) =< aux(18)

  with precondition: [V>=1,Out>=4,V1>=V+3,2*V1>=Out+V+2] 

* Chain [35,[46,47],48,54]: 5*it(46)+6*s(49)+6
  Such that:aux(10) =< V1-V
aux(18) =< V+1
it(46) =< aux(10)
s(49) =< aux(18)

  with precondition: [V>=1,Out>=5,V1>=V+3,2*V1>=Out+V+1] 

* Chain [35,54]: 6*s(49)+3
  Such that:aux(18) =< V+1
s(49) =< aux(18)

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

* Chain [35,49,54]: 6*s(49)+5
  Such that:aux(18) =< V+1
s(49) =< aux(18)

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

* Chain [35,48,54]: 6*s(49)+6
  Such that:aux(18) =< V+1
s(49) =< aux(18)

  with precondition: [Out>=4,V1>=V+2,V+3>=Out] 

* Chain [34,[46,47],54]: 5*it(46)+2*s(55)+3
  Such that:aux(8) =< V1-V
aux(19) =< V+1
it(46) =< aux(8)
s(55) =< aux(19)

  with precondition: [V>=1,Out>=4,V1>=V+2,2*V1>=Out+V] 

* Chain [34,[46,47],49,54]: 5*it(46)+2*s(55)+5
  Such that:aux(9) =< V1-V
aux(19) =< V+1
it(46) =< aux(9)
s(55) =< aux(19)

  with precondition: [V>=1,Out>=5,V1>=V+3,2*V1>=Out+V+1] 

* Chain [34,[46,47],48,54]: 5*it(46)+2*s(55)+6
  Such that:aux(10) =< V1-V
aux(19) =< V+1
it(46) =< aux(10)
s(55) =< aux(19)

  with precondition: [V>=1,Out>=6,V1>=V+3,2*V1>=Out+V] 

* Chain [34,54]: 2*s(55)+3
  Such that:aux(19) =< V+1
s(55) =< aux(19)

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

* Chain [34,49,54]: 2*s(55)+5
  Such that:aux(19) =< V+1
s(55) =< aux(19)

  with precondition: [Out>=4,V1>=V+2,V+3>=Out] 

* Chain [34,48,54]: 2*s(55)+6
  Such that:aux(19) =< V+1
s(55) =< aux(19)

  with precondition: [Out>=5,V1>=V+2,V+4>=Out] 


#### Cost of chains of start(V1,V):
* Chain [57]: 32*s(163)+72*s(168)+45*s(171)+45*s(173)+48*s(177)+6
  Such that:aux(34) =< -V1+V
aux(35) =< V1
aux(36) =< V1-V
aux(37) =< V
aux(38) =< V+1
s(163) =< aux(37)
s(173) =< aux(34)
s(168) =< aux(35)
s(171) =< aux(36)
s(177) =< aux(38)

  with precondition: [V1>=0] 

* Chain [56]: 15*s(201)+3
  Such that:s(200) =< V1
s(201) =< s(200)

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

* Chain [55]: 0
  with precondition: [V1=V,V1>=1] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [57] with precondition: [V1>=0] 
    - Upper bound: 72*V1+6+nat(V)*32+nat(V+1)*48+nat(-V1+V)*45+nat(V1-V)*45 
    - Complexity: n 
* Chain [56] with precondition: [V=0,V1>=0] 
    - Upper bound: 15*V1+3 
    - Complexity: n 
* Chain [55] with precondition: [V1=V,V1>=1] 
    - Upper bound: 0 
    - Complexity: constant 

### Maximum cost of start(V1,V): 57*V1+3+nat(V)*32+nat(V+1)*48+nat(-V1+V)*45+nat(V1-V)*45+(15*V1+3) 
Asymptotic class: n 
* Total analysis performed in 2304 ms.


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

(14)
BOUNDS(1, n^1)

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

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

(16)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   -'(z0, 0) -> c
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   P(s(z0)) -> c2
   F(s(z0), z1) -> c3(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), P(-(s(z0), z1)), -'(s(z0), z1))
   F(s(z0), z1) -> c4(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), P(-(z1, s(z0))), -'(z1, s(z0)))
   F(z0, s(z1)) -> c5(F(p(-(z0, s(z1))), p(-(s(z1), z0))), P(-(z0, s(z1))), -'(z0, s(z1)))
   F(z0, s(z1)) -> c6(F(p(-(z0, s(z1))), p(-(s(z1), z0))), P(-(s(z1), z0)), -'(s(z1), z0))

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

   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   p(s(z0)) -> z0
   f(s(z0), z1) -> f(p(-(s(z0), z1)), p(-(z1, s(z0))))
   f(z0, s(z1)) -> f(p(-(z0, s(z1))), p(-(s(z1), z0)))

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

(17) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

-'(s(z0), s(z1)) ->^+ c1(-'(z0, z1))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [z0 / s(z0), z1 / s(z1)].

The result substitution is [ ].




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

(18)
Complex Obligation (BEST)

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

(19)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   -'(z0, 0) -> c
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   P(s(z0)) -> c2
   F(s(z0), z1) -> c3(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), P(-(s(z0), z1)), -'(s(z0), z1))
   F(s(z0), z1) -> c4(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), P(-(z1, s(z0))), -'(z1, s(z0)))
   F(z0, s(z1)) -> c5(F(p(-(z0, s(z1))), p(-(s(z1), z0))), P(-(z0, s(z1))), -'(z0, s(z1)))
   F(z0, s(z1)) -> c6(F(p(-(z0, s(z1))), p(-(s(z1), z0))), P(-(s(z1), z0)), -'(s(z1), z0))

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

   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   p(s(z0)) -> z0
   f(s(z0), z1) -> f(p(-(s(z0), z1)), p(-(z1, s(z0))))
   f(z0, s(z1)) -> f(p(-(z0, s(z1))), p(-(s(z1), z0)))

Rewrite Strategy: INNERMOST
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(20) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(21)
BOUNDS(n^1, INF)

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(22)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   -'(z0, 0) -> c
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   P(s(z0)) -> c2
   F(s(z0), z1) -> c3(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), P(-(s(z0), z1)), -'(s(z0), z1))
   F(s(z0), z1) -> c4(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), P(-(z1, s(z0))), -'(z1, s(z0)))
   F(z0, s(z1)) -> c5(F(p(-(z0, s(z1))), p(-(s(z1), z0))), P(-(z0, s(z1))), -'(z0, s(z1)))
   F(z0, s(z1)) -> c6(F(p(-(z0, s(z1))), p(-(s(z1), z0))), P(-(s(z1), z0)), -'(s(z1), z0))

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

   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   p(s(z0)) -> z0
   f(s(z0), z1) -> f(p(-(s(z0), z1)), p(-(z1, s(z0))))
   f(z0, s(z1)) -> f(p(-(z0, s(z1))), p(-(s(z1), z0)))

Rewrite Strategy: INNERMOST