WORST_CASE(?,O(n^1))
proof of input_MXJtuWY7cW.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(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) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxWeightedTrs
(15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxTypedWeightedTrs
(17) CompletionProof [UPPER BOUND(ID), 0 ms]
(18) CpxTypedWeightedCompleteTrs
(19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(20) CpxRNTS
(21) CompleteCoflocoProof [FINISHED, 827 ms]
(22) BOUNDS(1, n^1)


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

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


The TRS R consists of the following rules:

   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   p(s(x)) -> x
   f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
   f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))

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:
   -(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)))
Tuples:
   -'(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))
S tuples:
   -'(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))
K tuples:none
Defined Rule Symbols:   -_2, p_1, f_2

Defined Pair Symbols:   -'_2, P_1, F_2

Compound Symbols:   c, c1_1, c2, c3_3, c4_3, c5_3, c6_3


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   P(s(z0)) -> c2
   -'(z0, 0) -> c

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

(4)
Obligation:
Complexity Dependency Tuples Problem

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)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   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))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   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))
K tuples:none
Defined Rule Symbols:   -_2, p_1, f_2

Defined Pair Symbols:   -'_2, F_2

Compound Symbols:   c1_1, c3_3, c4_3, c5_3, c6_3


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

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

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)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   F(s(z0), z1) -> c3(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), -'(s(z0), z1))
   F(s(z0), z1) -> c4(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), -'(z1, s(z0)))
   F(z0, s(z1)) -> c5(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)))
   F(z0, s(z1)) -> c6(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(s(z1), z0))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   F(s(z0), z1) -> c3(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), -'(s(z0), z1))
   F(s(z0), z1) -> c4(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), -'(z1, s(z0)))
   F(z0, s(z1)) -> c5(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)))
   F(z0, s(z1)) -> c6(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(s(z1), z0))
K tuples:none
Defined Rule Symbols:   -_2, p_1, f_2

Defined Pair Symbols:   -'_2, F_2

Compound Symbols:   c1_1, c3_2, c4_2, c5_2, c6_2


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   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)))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(z0)) -> z0
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   F(s(z0), z1) -> c3(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), -'(s(z0), z1))
   F(s(z0), z1) -> c4(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), -'(z1, s(z0)))
   F(z0, s(z1)) -> c5(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)))
   F(z0, s(z1)) -> c6(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(s(z1), z0))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   F(s(z0), z1) -> c3(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), -'(s(z0), z1))
   F(s(z0), z1) -> c4(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), -'(z1, s(z0)))
   F(z0, s(z1)) -> c5(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)))
   F(z0, s(z1)) -> c6(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(s(z1), z0))
K tuples:none
Defined Rule Symbols:   p_1, -_2

Defined Pair Symbols:   -'_2, F_2

Compound Symbols:   c1_1, c3_2, c4_2, c5_2, c6_2


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

(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^1).


The TRS R consists of the following rules:

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

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

   p(s(z0)) -> z0
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, 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^1).


The TRS R consists of the following rules:

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

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

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

- => minus

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

(14)
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:

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

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

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

(16)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

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

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

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

(17) 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:

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

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

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

(18)
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:

   -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1]
   F(s(z0), z1) -> c3(F(p(minus(s(z0), z1)), p(minus(z1, s(z0)))), -'(s(z0), z1)) [1]
   F(s(z0), z1) -> c4(F(p(minus(s(z0), z1)), 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))), -'(z0, s(z1))) [1]
   F(z0, s(z1)) -> c6(F(p(minus(z0, s(z1))), p(minus(s(z1), z0))), -'(s(z1), z0)) [1]
   p(s(z0)) -> z0 [0]
   minus(z0, 0) -> z0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   p(v0) -> null_p [0]
   minus(v0, v1) -> null_minus [0]
   -'(v0, v1) -> null_-' [0]
   F(v0, v1) -> null_F [0]

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

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

(19) 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
   null_p => 0
   null_minus => 0
   null_-' => 0
   null_F => 0

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

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(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))) + -'(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))) + -'(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))) + -'(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))) + -'(1 + z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   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.

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

(21) 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, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[p(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(fun(V1, V, Out),1,[fun(V3, V2, Ret1)],[Out = 1 + Ret1,V2 >= 0,V1 = 1 + V3,V3 >= 0,V = 1 + V2]).
eq(fun1(V1, V, Out),1,[minus(1 + V5, V4, Ret0100),p(Ret0100, Ret010),minus(V4, 1 + V5, Ret0110),p(Ret0110, Ret011),fun1(Ret010, Ret011, Ret01),fun(1 + V5, V4, Ret11)],[Out = 1 + Ret01 + Ret11,V4 >= 0,V1 = 1 + V5,V = V4,V5 >= 0]).
eq(fun1(V1, V, Out),1,[minus(1 + V7, V6, Ret01001),p(Ret01001, Ret0101),minus(V6, 1 + V7, Ret01101),p(Ret01101, Ret0111),fun1(Ret0101, Ret0111, Ret012),fun(V6, 1 + V7, Ret12)],[Out = 1 + Ret012 + Ret12,V6 >= 0,V1 = 1 + V7,V = V6,V7 >= 0]).
eq(fun1(V1, V, Out),1,[minus(V8, 1 + V9, Ret01002),p(Ret01002, Ret0102),minus(1 + V9, V8, Ret01102),p(Ret01102, Ret0112),fun1(Ret0102, Ret0112, Ret013),fun(V8, 1 + V9, Ret13)],[Out = 1 + Ret013 + Ret13,V1 = V8,V9 >= 0,V8 >= 0,V = 1 + V9]).
eq(fun1(V1, V, Out),1,[minus(V11, 1 + V10, Ret01003),p(Ret01003, Ret0103),minus(1 + V10, V11, Ret01103),p(Ret01103, Ret0113),fun1(Ret0103, Ret0113, Ret014),fun(1 + V10, V11, Ret14)],[Out = 1 + Ret014 + Ret14,V1 = V11,V10 >= 0,V11 >= 0,V = 1 + V10]).
eq(p(V1, Out),0,[],[Out = V12,V1 = 1 + V12,V12 >= 0]).
eq(minus(V1, V, Out),0,[],[Out = V13,V1 = V13,V13 >= 0,V = 0]).
eq(minus(V1, V, Out),0,[minus(V15, V14, Ret)],[Out = Ret,V14 >= 0,V1 = 1 + V15,V15 >= 0,V = 1 + V14]).
eq(p(V1, Out),0,[],[Out = 0,V16 >= 0,V1 = V16]).
eq(minus(V1, V, Out),0,[],[Out = 0,V18 >= 0,V17 >= 0,V1 = V18,V = V17]).
eq(fun(V1, V, Out),0,[],[Out = 0,V20 >= 0,V19 >= 0,V1 = V20,V = V19]).
eq(fun1(V1, V, Out),0,[],[Out = 0,V21 >= 0,V22 >= 0,V1 = V21,V = V22]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,Out),[V1,V],[Out]).
input_output_vars(p(V1,Out),[V1],[Out]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [fun/3]
1. recursive  : [minus/3]
2. non_recursive  : [p/2]
3. recursive [non_tail] : [fun1/3]
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 minus/3
2. SCC is partially evaluated into p/2
3. SCC is partially evaluated into fun1/3
4. SCC is partially evaluated into start/2

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

### Specialization of cost equations fun/3 
* CE 6 is refined into CE [17] 
* CE 5 is refined into CE [18] 


### Cost equations --> "Loop" of fun/3 
* CEs [18] --> Loop 12 
* CEs [17] --> Loop 13 

### Ranking functions of CR fun(V1,V,Out) 
* RF of phase [12]: [V,V1]

#### Partial ranking functions of CR fun(V1,V,Out) 
* Partial RF of phase [12]:
  - RF of loop [12:1]:
    V
    V1


### Specialization of cost equations minus/3 
* CE 16 is refined into CE [19] 
* CE 14 is refined into CE [20] 
* CE 15 is refined into CE [21] 


### Cost equations --> "Loop" of minus/3 
* CEs [21] --> Loop 14 
* CEs [19] --> Loop 15 
* CEs [20] --> Loop 16 

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

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


### Specialization of cost equations p/2 
* CE 12 is refined into CE [22] 
* CE 13 is refined into CE [23] 


### Cost equations --> "Loop" of p/2 
* CEs [22] --> Loop 17 
* CEs [23] --> Loop 18 

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

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


### Specialization of cost equations fun1/3 
* CE 11 is refined into CE [24] 
* CE 7 is refined into CE [25,26,27,28,29,30,31,32,33,34,35,36,37,38] 
* CE 8 is refined into CE [39,40,41,42,43,44,45,46,47,48,49,50,51,52] 
* CE 9 is refined into CE [53,54,55,56,57,58,59,60,61,62,63,64,65,66] 
* CE 10 is refined into CE [67,68,69,70,71,72,73,74,75,76,77,78,79,80] 


### Cost equations --> "Loop" of fun1/3 
* CEs [38,52,66,80] --> Loop 19 
* CEs [37,51,65,79] --> Loop 20 
* CEs [32,46,60,74] --> Loop 21 
* CEs [31,45,59,73] --> Loop 22 
* CEs [34,48,62,76] --> Loop 23 
* CEs [28,30,36,42,44,50,56,58,64,70,72,78] --> Loop 24 
* CEs [35,49,63,77] --> Loop 25 
* CEs [26,40] --> Loop 26 
* CEs [25,27,39,41] --> Loop 27 
* CEs [54,68] --> Loop 28 
* CEs [29,33,43,47,53,55,57,61,67,69,71,75] --> Loop 29 
* CEs [24] --> Loop 30 

### Ranking functions of CR fun1(V1,V,Out) 
* RF of phase [26]: [V1]
* RF of phase [28]: [V]

#### Partial ranking functions of CR fun1(V1,V,Out) 
* Partial RF of phase [26]:
  - RF of loop [26:1]:
    V1
* Partial RF of phase [28]:
  - RF of loop [28:1]:
    V


### Specialization of cost equations start/2 
* CE 1 is refined into CE [81,82] 
* CE 2 is refined into CE [83,84,85,86,87,88,89,90,91,92,93] 
* CE 3 is refined into CE [94,95] 
* CE 4 is refined into CE [96,97,98] 


### Cost equations --> "Loop" of start/2 
* CEs [84,96] --> Loop 31 
* CEs [81,82,83,85,86,87,88,89,90,91,92,93,94,95,97,98] --> Loop 32 

### 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 [[12],13]: 1*it(12)+0
  Such that:it(12) =< Out

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

* Chain [13]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of minus(V1,V,Out):
* Chain [[14],16]: 0
  with precondition: [V1=Out+V,V>=1,V1>=V] 

* Chain [[14],15]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

* Chain [16]: 0
  with precondition: [V=0,V1=Out,V1>=0] 

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


#### Cost of chains of p(V1,Out):
* Chain [18]: 0
  with precondition: [Out=0,V1>=0] 

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


#### Cost of chains of fun1(V1,V,Out):
* Chain [[28],30]: 1*it(28)+0
  Such that:it(28) =< Out

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

* Chain [[28],29,30]: 1*it(28)+1
  Such that:it(28) =< Out

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

* Chain [[26],30]: 1*it(26)+0
  Such that:it(26) =< Out

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

* Chain [[26],27,30]: 1*it(26)+1
  Such that:it(26) =< Out

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

* Chain [30]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [29,30]: 1
  with precondition: [Out=1,V1>=0,V>=1] 

* Chain [27,30]: 1
  with precondition: [Out=1,V1>=1,V>=0] 

* Chain [25,30]: 1
  with precondition: [Out=1,V1=V,V1>=1] 

* Chain [24,30]: 4*s(1)+8*s(2)+1
  Such that:aux(1) =< V1
aux(2) =< V
s(2) =< aux(1)
s(1) =< aux(2)

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

* Chain [23,30]: 4*s(13)+1
  Such that:aux(3) =< V
s(13) =< aux(3)

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

* Chain [22,[28],30]: 1*it(28)+1
  Such that:it(28) =< Out

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

* Chain [22,[28],29,30]: 1*it(28)+2
  Such that:it(28) =< Out

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

* Chain [22,30]: 1
  with precondition: [Out=1,V1>=1,V>=V1+1] 

* Chain [22,29,30]: 2
  with precondition: [Out=2,V1>=1,V>=V1+2] 

* Chain [21,[28],30]: 1*it(28)+4*s(17)+1
  Such that:it(28) =< -V1+V
aux(4) =< V1
s(17) =< aux(4)

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

* Chain [21,[28],29,30]: 1*it(28)+4*s(17)+2
  Such that:it(28) =< -V1+V
aux(4) =< V1
s(17) =< aux(4)

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

* Chain [21,30]: 4*s(17)+1
  Such that:aux(4) =< V1
s(17) =< aux(4)

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

* Chain [21,29,30]: 4*s(17)+2
  Such that:aux(4) =< V1
s(17) =< aux(4)

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

* Chain [20,[26],30]: 1*it(26)+1
  Such that:it(26) =< Out

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

* Chain [20,[26],27,30]: 1*it(26)+2
  Such that:it(26) =< Out

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

* Chain [20,30]: 1
  with precondition: [Out=1,V>=1,V1>=V+1] 

* Chain [20,27,30]: 2
  with precondition: [Out=2,V>=1,V1>=V+2] 

* Chain [19,[26],30]: 1*it(26)+4*s(21)+1
  Such that:it(26) =< V1-V
aux(5) =< V+1
s(21) =< aux(5)

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

* Chain [19,[26],27,30]: 1*it(26)+4*s(21)+2
  Such that:it(26) =< V1-V
aux(5) =< V+1
s(21) =< aux(5)

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

* Chain [19,30]: 4*s(21)+1
  Such that:aux(5) =< V+1
s(21) =< aux(5)

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

* Chain [19,27,30]: 4*s(21)+2
  Such that:aux(5) =< V+1
s(21) =< aux(5)

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


#### Cost of chains of start(V1,V):
* Chain [32]: 11*s(59)+4*s(63)+4*s(65)+24*s(69)+16*s(73)+2
  Such that:aux(15) =< -V1+V
aux(16) =< V1
aux(17) =< V1-V
aux(18) =< V
aux(19) =< V+1
s(59) =< aux(18)
s(63) =< aux(15)
s(69) =< aux(16)
s(65) =< aux(17)
s(73) =< aux(19)

  with precondition: [V1>=0] 

* Chain [31]: 2*s(83)+1
  Such that:s(82) =< V1
s(83) =< s(82)

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


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [32] with precondition: [V1>=0] 
    - Upper bound: 24*V1+2+nat(V)*11+nat(V+1)*16+nat(-V1+V)*4+nat(V1-V)*4 
    - Complexity: n 
* Chain [31] with precondition: [V=0,V1>=0] 
    - Upper bound: 2*V1+1 
    - Complexity: n 

### Maximum cost of start(V1,V): 22*V1+1+nat(V)*11+nat(V+1)*16+nat(-V1+V)*4+nat(V1-V)*4+(2*V1+1) 
Asymptotic class: n 
* Total analysis performed in 850 ms.


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

(22)
BOUNDS(1, n^1)