WORST_CASE(?,O(n^1))
proof of input_NxuqtFv0iF.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTypedWeightedTrs
(15) CompletionProof [UPPER BOUND(ID), 0 ms]
(16) CpxTypedWeightedCompleteTrs
(17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(18) CpxRNTS
(19) CompleteCoflocoProof [FINISHED, 1119 ms]
(20) 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:

   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   id_inc(x) -> x
   id_inc(x) -> s(x)
   div(x, y) -> if(ge(y, s(0)), ge(x, y), x, y)
   if(false, b, x, y) -> div_by_zero
   if(true, false, x, y) -> 0
   if(true, true, x, y) -> id_inc(div(minus(x, y), y))

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:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   minus(z0, 0) -> z0
   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   id_inc(z0) -> z0
   id_inc(z0) -> s(z0)
   div(z0, z1) -> if(ge(z1, s(0)), ge(z0, z1), z0, z1)
   if(false, z0, z1, z2) -> div_by_zero
   if(true, false, z0, z1) -> 0
   if(true, true, z0, z1) -> id_inc(div(minus(z0, z1), z1))
Tuples:
   GE(z0, 0) -> c
   GE(0, s(z0)) -> c1
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   MINUS(z0, 0) -> c3
   MINUS(0, z0) -> c4
   MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
   ID_INC(z0) -> c6
   ID_INC(z0) -> c7
   DIV(z0, z1) -> c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0)))
   DIV(z0, z1) -> c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z0, z1))
   IF(false, z0, z1, z2) -> c10
   IF(true, false, z0, z1) -> c11
   IF(true, true, z0, z1) -> c12(ID_INC(div(minus(z0, z1), z1)), DIV(minus(z0, z1), z1), MINUS(z0, z1))
S tuples:
   GE(z0, 0) -> c
   GE(0, s(z0)) -> c1
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   MINUS(z0, 0) -> c3
   MINUS(0, z0) -> c4
   MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
   ID_INC(z0) -> c6
   ID_INC(z0) -> c7
   DIV(z0, z1) -> c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0)))
   DIV(z0, z1) -> c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z0, z1))
   IF(false, z0, z1, z2) -> c10
   IF(true, false, z0, z1) -> c11
   IF(true, true, z0, z1) -> c12(ID_INC(div(minus(z0, z1), z1)), DIV(minus(z0, z1), z1), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:   ge_2, minus_2, id_inc_1, div_2, if_4

Defined Pair Symbols:   GE_2, MINUS_2, ID_INC_1, DIV_2, IF_4

Compound Symbols:   c, c1, c2_1, c3, c4, c5_1, c6, c7, c8_2, c9_2, c10, c11, c12_3


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 8 trailing nodes:
   MINUS(0, z0) -> c4
   MINUS(z0, 0) -> c3
   IF(false, z0, z1, z2) -> c10
   GE(z0, 0) -> c
   IF(true, false, z0, z1) -> c11
   ID_INC(z0) -> c7
   ID_INC(z0) -> c6
   GE(0, s(z0)) -> c1

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   minus(z0, 0) -> z0
   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   id_inc(z0) -> z0
   id_inc(z0) -> s(z0)
   div(z0, z1) -> if(ge(z1, s(0)), ge(z0, z1), z0, z1)
   if(false, z0, z1, z2) -> div_by_zero
   if(true, false, z0, z1) -> 0
   if(true, true, z0, z1) -> id_inc(div(minus(z0, z1), z1))
Tuples:
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
   DIV(z0, z1) -> c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0)))
   DIV(z0, z1) -> c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z0, z1))
   IF(true, true, z0, z1) -> c12(ID_INC(div(minus(z0, z1), z1)), DIV(minus(z0, z1), z1), MINUS(z0, z1))
S tuples:
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
   DIV(z0, z1) -> c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0)))
   DIV(z0, z1) -> c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z0, z1))
   IF(true, true, z0, z1) -> c12(ID_INC(div(minus(z0, z1), z1)), DIV(minus(z0, z1), z1), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:   ge_2, minus_2, id_inc_1, div_2, if_4

Defined Pair Symbols:   GE_2, MINUS_2, DIV_2, IF_4

Compound Symbols:   c2_1, c5_1, c8_2, c9_2, c12_3


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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   minus(z0, 0) -> z0
   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   id_inc(z0) -> z0
   id_inc(z0) -> s(z0)
   div(z0, z1) -> if(ge(z1, s(0)), ge(z0, z1), z0, z1)
   if(false, z0, z1, z2) -> div_by_zero
   if(true, false, z0, z1) -> 0
   if(true, true, z0, z1) -> id_inc(div(minus(z0, z1), z1))
Tuples:
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
   DIV(z0, z1) -> c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0)))
   DIV(z0, z1) -> c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z0, z1))
   IF(true, true, z0, z1) -> c12(DIV(minus(z0, z1), z1), MINUS(z0, z1))
S tuples:
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
   DIV(z0, z1) -> c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0)))
   DIV(z0, z1) -> c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z0, z1))
   IF(true, true, z0, z1) -> c12(DIV(minus(z0, z1), z1), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:   ge_2, minus_2, id_inc_1, div_2, if_4

Defined Pair Symbols:   GE_2, MINUS_2, DIV_2, IF_4

Compound Symbols:   c2_1, c5_1, c8_2, c9_2, c12_2


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   id_inc(z0) -> z0
   id_inc(z0) -> s(z0)
   div(z0, z1) -> if(ge(z1, s(0)), ge(z0, z1), z0, z1)
   if(false, z0, z1, z2) -> div_by_zero
   if(true, false, z0, z1) -> 0
   if(true, true, z0, z1) -> id_inc(div(minus(z0, z1), z1))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   ge(z0, 0) -> true
   minus(z0, 0) -> z0
   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
Tuples:
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
   DIV(z0, z1) -> c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0)))
   DIV(z0, z1) -> c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z0, z1))
   IF(true, true, z0, z1) -> c12(DIV(minus(z0, z1), z1), MINUS(z0, z1))
S tuples:
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
   DIV(z0, z1) -> c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0)))
   DIV(z0, z1) -> c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z0, z1))
   IF(true, true, z0, z1) -> c12(DIV(minus(z0, z1), z1), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:   ge_2, minus_2

Defined Pair Symbols:   GE_2, MINUS_2, DIV_2, IF_4

Compound Symbols:   c2_1, c5_1, c8_2, c9_2, c12_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:

   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
   DIV(z0, z1) -> c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0)))
   DIV(z0, z1) -> c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z0, z1))
   IF(true, true, z0, z1) -> c12(DIV(minus(z0, z1), z1), MINUS(z0, z1))

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

   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   ge(z0, 0) -> true
   minus(z0, 0) -> z0
   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(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:

   GE(s(z0), s(z1)) -> c2(GE(z0, z1)) [1]
   MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) [1]
   DIV(z0, z1) -> c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0))) [1]
   DIV(z0, z1) -> c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z0, z1)) [1]
   IF(true, true, z0, z1) -> c12(DIV(minus(z0, z1), z1), MINUS(z0, z1)) [1]
   ge(0, s(z0)) -> false [0]
   ge(s(z0), s(z1)) -> ge(z0, z1) [0]
   ge(z0, 0) -> true [0]
   minus(z0, 0) -> z0 [0]
   minus(0, z0) -> 0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]

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

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

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

   GE(s(z0), s(z1)) -> c2(GE(z0, z1)) [1]
   MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) [1]
   DIV(z0, z1) -> c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0))) [1]
   DIV(z0, z1) -> c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z0, z1)) [1]
   IF(true, true, z0, z1) -> c12(DIV(minus(z0, z1), z1), MINUS(z0, z1)) [1]
   ge(0, s(z0)) -> false [0]
   ge(s(z0), s(z1)) -> ge(z0, z1) [0]
   ge(z0, 0) -> true [0]
   minus(z0, 0) -> z0 [0]
   minus(0, z0) -> 0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]

The TRS has the following type information:
GE :: s:0 -> s:0 -> c2
s :: s:0 -> s:0
c2 :: c2 -> c2
MINUS :: s:0 -> s:0 -> c5
c5 :: c5 -> c5
DIV :: s:0 -> s:0 -> c8:c9
c8 :: c12 -> c2 -> c8:c9
IF :: true:false -> true:false -> s:0 -> s:0 -> c12
ge :: s:0 -> s:0 -> true:false
0 :: s:0
c9 :: c12 -> c2 -> c8:c9
true :: true:false
c12 :: c8:c9 -> c5 -> c12
minus :: s:0 -> s:0 -> s:0
false :: true:false

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

(15) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   ge(v0, v1) -> null_ge [0]
   minus(v0, v1) -> null_minus [0]
   GE(v0, v1) -> null_GE [0]
   MINUS(v0, v1) -> null_MINUS [0]
   IF(v0, v1, v2, v3) -> null_IF [0]

And the following fresh constants:    null_ge, null_minus, null_GE, null_MINUS, null_IF, const

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

(16)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   GE(s(z0), s(z1)) -> c2(GE(z0, z1)) [1]
   MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) [1]
   DIV(z0, z1) -> c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0))) [1]
   DIV(z0, z1) -> c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z0, z1)) [1]
   IF(true, true, z0, z1) -> c12(DIV(minus(z0, z1), z1), MINUS(z0, z1)) [1]
   ge(0, s(z0)) -> false [0]
   ge(s(z0), s(z1)) -> ge(z0, z1) [0]
   ge(z0, 0) -> true [0]
   minus(z0, 0) -> z0 [0]
   minus(0, z0) -> 0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   ge(v0, v1) -> null_ge [0]
   minus(v0, v1) -> null_minus [0]
   GE(v0, v1) -> null_GE [0]
   MINUS(v0, v1) -> null_MINUS [0]
   IF(v0, v1, v2, v3) -> null_IF [0]

The TRS has the following type information:
GE :: s:0:null_minus -> s:0:null_minus -> c2:null_GE
s :: s:0:null_minus -> s:0:null_minus
c2 :: c2:null_GE -> c2:null_GE
MINUS :: s:0:null_minus -> s:0:null_minus -> c5:null_MINUS
c5 :: c5:null_MINUS -> c5:null_MINUS
DIV :: s:0:null_minus -> s:0:null_minus -> c8:c9
c8 :: c12:null_IF -> c2:null_GE -> c8:c9
IF :: true:false:null_ge -> true:false:null_ge -> s:0:null_minus -> s:0:null_minus -> c12:null_IF
ge :: s:0:null_minus -> s:0:null_minus -> true:false:null_ge
0 :: s:0:null_minus
c9 :: c12:null_IF -> c2:null_GE -> c8:c9
true :: true:false:null_ge
c12 :: c8:c9 -> c5:null_MINUS -> c12:null_IF
minus :: s:0:null_minus -> s:0:null_minus -> s:0:null_minus
false :: true:false:null_ge
null_ge :: true:false:null_ge
null_minus :: s:0:null_minus
null_GE :: c2:null_GE
null_MINUS :: c5:null_MINUS
null_IF :: c12:null_IF
const :: c8:c9

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

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

   0 => 0
   true => 2
   false => 1
   null_ge => 0
   null_minus => 0
   null_GE => 0
   null_MINUS => 0
   null_IF => 0
   const => 0

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

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

   DIV(z, z') -{ 1 }-> 1 + IF(ge(z1, 1 + 0), ge(z0, z1), z0, z1) + GE(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   DIV(z, z') -{ 1 }-> 1 + IF(ge(z1, 1 + 0), ge(z0, z1), z0, z1) + GE(z1, 1 + 0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   GE(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   GE(z, z') -{ 1 }-> 1 + GE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   IF(z, z', z'', z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   IF(z, z', z'', z2) -{ 1 }-> 1 + DIV(minus(z0, z1), z1) + MINUS(z0, z1) :|: z = 2, z1 >= 0, z' = 2, z2 = z1, z'' = z0, z0 >= 0
   MINUS(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   ge(z, z') -{ 0 }-> ge(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   ge(z, z') -{ 0 }-> 2 :|: z = z0, z0 >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z0 >= 0, z' = 1 + z0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   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 :|: z0 >= 0, z = 0, z' = z0
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V, V12, V13),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V12, V13),0,[fun1(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V12, V13),0,[fun2(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V12, V13),0,[fun3(V1, V, V12, V13, Out)],[V1 >= 0,V >= 0,V12 >= 0,V13 >= 0]).
eq(start(V1, V, V12, V13),0,[ge(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V12, V13),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,[fun1(V5, V4, Ret11)],[Out = 1 + Ret11,V4 >= 0,V1 = 1 + V5,V5 >= 0,V = 1 + V4]).
eq(fun2(V1, V, Out),1,[ge(V6, 1 + 0, Ret010),ge(V7, V6, Ret011),fun3(Ret010, Ret011, V7, V6, Ret01),fun(V6, 1 + 0, Ret12)],[Out = 1 + Ret01 + Ret12,V1 = V7,V6 >= 0,V = V6,V7 >= 0]).
eq(fun2(V1, V, Out),1,[ge(V9, 1 + 0, Ret0101),ge(V8, V9, Ret0111),fun3(Ret0101, Ret0111, V8, V9, Ret012),fun(V8, V9, Ret13)],[Out = 1 + Ret012 + Ret13,V1 = V8,V9 >= 0,V = V9,V8 >= 0]).
eq(fun3(V1, V, V12, V13, Out),1,[minus(V11, V10, Ret0102),fun2(Ret0102, V10, Ret013),fun1(V11, V10, Ret14)],[Out = 1 + Ret013 + Ret14,V1 = 2,V10 >= 0,V = 2,V13 = V10,V12 = V11,V11 >= 0]).
eq(ge(V1, V, Out),0,[],[Out = 1,V14 >= 0,V = 1 + V14,V1 = 0]).
eq(ge(V1, V, Out),0,[ge(V16, V15, Ret)],[Out = Ret,V15 >= 0,V1 = 1 + V16,V16 >= 0,V = 1 + V15]).
eq(ge(V1, V, Out),0,[],[Out = 2,V1 = V17,V17 >= 0,V = 0]).
eq(minus(V1, V, Out),0,[],[Out = V18,V1 = V18,V18 >= 0,V = 0]).
eq(minus(V1, V, Out),0,[],[Out = 0,V19 >= 0,V1 = 0,V = V19]).
eq(minus(V1, V, Out),0,[minus(V21, V20, Ret2)],[Out = Ret2,V20 >= 0,V1 = 1 + V21,V21 >= 0,V = 1 + V20]).
eq(ge(V1, V, Out),0,[],[Out = 0,V23 >= 0,V22 >= 0,V1 = V23,V = V22]).
eq(minus(V1, V, Out),0,[],[Out = 0,V25 >= 0,V24 >= 0,V1 = V25,V = V24]).
eq(fun(V1, V, Out),0,[],[Out = 0,V27 >= 0,V26 >= 0,V1 = V27,V = V26]).
eq(fun1(V1, V, Out),0,[],[Out = 0,V28 >= 0,V29 >= 0,V1 = V28,V = V29]).
eq(fun3(V1, V, V12, V13, Out),0,[],[Out = 0,V13 = V32,V31 >= 0,V12 = V33,V30 >= 0,V1 = V31,V = V30,V33 >= 0,V32 >= 0]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun2(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun3(V1,V,V12,V13,Out),[V1,V,V12,V13],[Out]).
input_output_vars(ge(V1,V,Out),[V1,V],[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  : [fun1/3]
2. recursive  : [minus/3]
3. recursive  : [ge/3]
4. recursive [non_tail] : [fun2/3,fun3/5]
5. non_recursive  : [start/4]

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

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

### Specialization of cost equations fun/3 
* CE 18 is refined into CE [23] 
* CE 17 is refined into CE [24] 


### Cost equations --> "Loop" of fun/3 
* CEs [24] --> Loop 15 
* CEs [23] --> Loop 16 

### Ranking functions of CR fun(V1,V,Out) 
* RF of phase [15]: [V,V1]

#### Partial ranking functions of CR fun(V1,V,Out) 
* Partial RF of phase [15]:
  - RF of loop [15:1]:
    V
    V1


### Specialization of cost equations fun1/3 
* CE 16 is refined into CE [25] 
* CE 15 is refined into CE [26] 


### Cost equations --> "Loop" of fun1/3 
* CEs [26] --> Loop 17 
* CEs [25] --> Loop 18 

### Ranking functions of CR fun1(V1,V,Out) 
* RF of phase [17]: [V,V1]

#### Partial ranking functions of CR fun1(V1,V,Out) 
* Partial RF of phase [17]:
  - RF of loop [17:1]:
    V
    V1


### Specialization of cost equations minus/3 
* CE 9 is refined into CE [27] 
* CE 8 is refined into CE [28] 
* CE 10 is refined into CE [29] 


### Cost equations --> "Loop" of minus/3 
* CEs [29] --> Loop 19 
* CEs [27] --> Loop 20 
* CEs [28] --> Loop 21 

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

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


### Specialization of cost equations ge/3 
* CE 22 is refined into CE [30] 
* CE 21 is refined into CE [31] 
* CE 19 is refined into CE [32] 
* CE 20 is refined into CE [33] 


### Cost equations --> "Loop" of ge/3 
* CEs [33] --> Loop 22 
* CEs [30] --> Loop 23 
* CEs [31] --> Loop 24 
* CEs [32] --> Loop 25 

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

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


### Specialization of cost equations fun2/3 
* CE 13 is refined into CE [34,35,36,37,38,39,40,41] 
* CE 14 is refined into CE [42,43,44,45,46,47,48,49] 
* CE 11 is refined into CE [50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66] 
* CE 12 is refined into CE [67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85] 


### Cost equations --> "Loop" of fun2/3 
* CEs [59,66] --> Loop 26 
* CEs [55,57,62,64] --> Loop 27 
* CEs [50,51,53,67,68,71] --> Loop 28 
* CEs [70,73,75,77,79,81,83,85] --> Loop 29 
* CEs [52,54,56,58,60,61,63,65,69,72,74,76,78,80,82,84] --> Loop 30 
* CEs [41,49] --> Loop 31 
* CEs [39,40,47,48] --> Loop 32 
* CEs [38,46] --> Loop 33 
* CEs [37,45] --> Loop 34 
* CEs [35,36,43,44] --> Loop 35 
* CEs [34,42] --> Loop 36 

### Ranking functions of CR fun2(V1,V,Out) 
* RF of phase [31,32,33]: [V1,V1-V+1]

#### Partial ranking functions of CR fun2(V1,V,Out) 
* Partial RF of phase [31,32,33]:
  - RF of loop [31:1,32:1,33:1]:
    V1
    V1-V+1


### Specialization of cost equations start/4 
* CE 1 is refined into CE [86] 
* CE 2 is refined into CE [87,88,89,90,91,92,93,94,95,96,97,98,99,100,101] 
* CE 3 is refined into CE [102,103] 
* CE 4 is refined into CE [104,105] 
* CE 5 is refined into CE [106,107,108,109,110] 
* CE 6 is refined into CE [111,112,113,114,115] 
* CE 7 is refined into CE [116,117,118] 


### Cost equations --> "Loop" of start/4 
* CEs [112,116] --> Loop 37 
* CEs [87,88,89,90,91,92,93,94,95,96,97,98,99,100,101] --> Loop 38 
* CEs [86,102,103,104,105,106,107,108,109,110,111,113,114,115,117,118] --> Loop 39 

### Ranking functions of CR start(V1,V,V12,V13) 

#### Partial ranking functions of CR start(V1,V,V12,V13) 


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

#### Cost of chains of fun(V1,V,Out):
* Chain [[15],16]: 1*it(15)+0
  Such that:it(15) =< Out

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

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


#### Cost of chains of fun1(V1,V,Out):
* Chain [[17],18]: 1*it(17)+0
  Such that:it(17) =< Out

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

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


#### Cost of chains of minus(V1,V,Out):
* Chain [[19],21]: 0
  with precondition: [V1=Out+V,V>=1,V1>=V] 

* Chain [[19],20]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

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

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


#### Cost of chains of ge(V1,V,Out):
* Chain [[22],25]: 0
  with precondition: [Out=1,V1>=1,V>=V1+1] 

* Chain [[22],24]: 0
  with precondition: [Out=2,V>=1,V1>=V] 

* Chain [[22],23]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

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

* Chain [24]: 0
  with precondition: [V=0,Out=2,V1>=0] 

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


#### Cost of chains of fun2(V1,V,Out):
* Chain [[31,32,33],36,30]: 6*it(31)+6*s(15)+2*s(16)+3
  Such that:aux(6) =< V1
aux(7) =< V1-V+1
aux(10) =< V1-V
aux(4) =< aux(6)
it(31) =< aux(6)
s(17) =< aux(6)
aux(4) =< aux(7)
it(31) =< aux(7)
aux(4) =< aux(10)
it(31) =< aux(10)
s(17) =< aux(10)
s(16) =< aux(4)
s(15) =< s(17)

  with precondition: [V>=1,Out>=5,V1>=2*V,4*V1+5>=6*V+Out] 

* Chain [[31,32,33],36,29]: 6*it(31)+6*s(15)+2*s(16)+11
  Such that:aux(6) =< V1
aux(7) =< V1-V+1
aux(12) =< V1-V
aux(4) =< aux(6)
it(31) =< aux(6)
s(17) =< aux(6)
aux(4) =< aux(7)
it(31) =< aux(7)
aux(4) =< aux(12)
it(31) =< aux(12)
s(17) =< aux(12)
s(16) =< aux(4)
s(15) =< s(17)

  with precondition: [V>=1,Out>=6,V1>=2*V,4*V1+6>=6*V+Out] 

* Chain [[31,32,33],35,30]: 6*it(31)+6*s(15)+2*s(16)+3*s(29)+1*s(31)+3
  Such that:s(31) =< 1
aux(6) =< V1
aux(7) =< V1-V+1
aux(13) =< V
aux(14) =< V1-V
s(29) =< aux(13)
aux(4) =< aux(6)
it(31) =< aux(6)
s(17) =< aux(6)
aux(4) =< aux(7)
it(31) =< aux(7)
aux(4) =< aux(14)
it(31) =< aux(14)
s(17) =< aux(14)
s(16) =< aux(4)
s(15) =< s(17)

  with precondition: [V>=1,Out>=6,V1>=2*V,4*V1+5>=5*V+Out] 

* Chain [[31,32,33],35,29]: 6*it(31)+6*s(15)+2*s(16)+3*s(29)+1*s(31)+11
  Such that:s(31) =< 1
aux(6) =< V1
aux(7) =< V1-V+1
aux(13) =< V
aux(15) =< V1-V
s(29) =< aux(13)
aux(4) =< aux(6)
it(31) =< aux(6)
s(17) =< aux(6)
aux(4) =< aux(7)
it(31) =< aux(7)
aux(4) =< aux(15)
it(31) =< aux(15)
s(17) =< aux(15)
s(16) =< aux(4)
s(15) =< s(17)

  with precondition: [V>=1,Out>=7,V1>=2*V,4*V1+6>=5*V+Out] 

* Chain [[31,32,33],34,30]: 6*it(31)+6*s(15)+2*s(16)+3*s(33)+1*s(36)+3
  Such that:s(36) =< 1
aux(6) =< V1
aux(7) =< V1-V+1
aux(17) =< V
aux(18) =< V1-V
s(33) =< aux(17)
aux(4) =< aux(6)
it(31) =< aux(6)
s(17) =< aux(6)
aux(4) =< aux(7)
it(31) =< aux(7)
aux(4) =< aux(18)
it(31) =< aux(18)
s(17) =< aux(18)
s(16) =< aux(4)
s(15) =< s(17)

  with precondition: [V>=1,Out>=7,V1>=2*V,4*V1+5>=4*V+Out] 

* Chain [[31,32,33],34,29]: 6*it(31)+6*s(15)+2*s(16)+3*s(33)+1*s(36)+11
  Such that:s(36) =< 1
aux(6) =< V1
aux(7) =< V1-V+1
aux(17) =< V
aux(19) =< V1-V
s(33) =< aux(17)
aux(4) =< aux(6)
it(31) =< aux(6)
s(17) =< aux(6)
aux(4) =< aux(7)
it(31) =< aux(7)
aux(4) =< aux(19)
it(31) =< aux(19)
s(17) =< aux(19)
s(16) =< aux(4)
s(15) =< s(17)

  with precondition: [V>=1,Out>=8,V1>=2*V,4*V1+6>=4*V+Out] 

* Chain [[31,32,33],30]: 6*it(31)+6*s(15)+2*s(16)+1
  Such that:aux(7) =< V1-V+1
aux(20) =< V1
aux(4) =< aux(20)
it(31) =< aux(20)
aux(4) =< aux(7)
it(31) =< aux(7)
s(16) =< aux(4)
s(15) =< aux(20)

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

* Chain [[31,32,33],29]: 6*it(31)+6*s(15)+2*s(16)+9
  Such that:aux(7) =< V1-V+1
aux(21) =< V1
aux(4) =< aux(21)
it(31) =< aux(21)
aux(4) =< aux(7)
it(31) =< aux(7)
s(16) =< aux(4)
s(15) =< aux(21)

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

* Chain [[31,32,33],27]: 6*it(31)+8*s(15)+2*s(16)+2*s(37)+1
  Such that:aux(7) =< V1-V+1
aux(23) =< V
aux(24) =< V1
s(15) =< aux(24)
s(37) =< aux(23)
aux(4) =< aux(24)
it(31) =< aux(24)
aux(4) =< aux(7)
it(31) =< aux(7)
s(16) =< aux(4)

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

* Chain [[31,32,33],26]: 6*it(31)+6*s(15)+2*s(16)+2*s(41)+1
  Such that:aux(6) =< V1
aux(7) =< V1-V+1
aux(25) =< V
aux(26) =< V1-V
s(41) =< aux(25)
aux(4) =< aux(6)
it(31) =< aux(6)
s(17) =< aux(6)
aux(4) =< aux(7)
it(31) =< aux(7)
aux(4) =< aux(26)
it(31) =< aux(26)
s(17) =< aux(26)
s(16) =< aux(4)
s(15) =< s(17)

  with precondition: [V>=1,Out>=4,V1>=2*V,4*V1+3>=5*V+Out] 

* Chain [36,30]: 3
  with precondition: [Out=3,V>=1,V1>=V] 

* Chain [36,29]: 11
  with precondition: [Out=4,V>=1,V1>=V] 

* Chain [35,30]: 3*s(29)+1*s(31)+3
  Such that:s(31) =< 1
aux(13) =< V
s(29) =< aux(13)

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

* Chain [35,29]: 3*s(29)+1*s(31)+11
  Such that:s(31) =< 1
aux(13) =< V
s(29) =< aux(13)

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

* Chain [34,30]: 3*s(33)+1*s(36)+3
  Such that:s(36) =< 1
aux(17) =< V
s(33) =< aux(17)

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

* Chain [34,29]: 3*s(33)+1*s(36)+11
  Such that:s(36) =< 1
aux(17) =< V
s(33) =< aux(17)

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

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

* Chain [29]: 9
  with precondition: [Out=2,V1>=0,V>=1] 

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

* Chain [27]: 2*s(37)+2*s(38)+1
  Such that:aux(22) =< V1
aux(23) =< V
s(38) =< aux(22)
s(37) =< aux(23)

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

* Chain [26]: 2*s(41)+1
  Such that:aux(25) =< V
s(41) =< aux(25)

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


#### Cost of chains of start(V1,V,V12,V13):
* Chain [39]: 34*s(151)+8*s(158)+42*s(161)+14*s(163)+42*s(164)+22*s(165)+18*s(167)+6*s(168)+11
  Such that:aux(37) =< 1
aux(38) =< V1
aux(39) =< V1-V
aux(40) =< V1-V+1
aux(41) =< V
s(151) =< aux(41)
s(158) =< aux(37)
s(160) =< aux(38)
s(161) =< aux(38)
s(162) =< aux(38)
s(160) =< aux(40)
s(161) =< aux(40)
s(160) =< aux(39)
s(161) =< aux(39)
s(162) =< aux(39)
s(163) =< s(160)
s(164) =< s(162)
s(165) =< aux(38)
s(166) =< aux(38)
s(167) =< aux(38)
s(166) =< aux(40)
s(167) =< aux(40)
s(168) =< s(166)

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

* Chain [38]: 71*s(189)+16*s(198)+84*s(201)+28*s(203)+84*s(204)+44*s(205)+36*s(207)+12*s(208)+12
  Such that:aux(45) =< 1
aux(46) =< V12-2*V13
aux(47) =< V12-2*V13+1
aux(48) =< V12-V13
aux(49) =< V13
s(189) =< aux(49)
s(198) =< aux(45)
s(200) =< aux(48)
s(201) =< aux(48)
s(202) =< aux(48)
s(200) =< aux(47)
s(201) =< aux(47)
s(200) =< aux(46)
s(201) =< aux(46)
s(202) =< aux(46)
s(203) =< s(200)
s(204) =< s(202)
s(205) =< aux(48)
s(206) =< aux(48)
s(207) =< aux(48)
s(206) =< aux(47)
s(207) =< aux(47)
s(208) =< s(206)

  with precondition: [V1=2,V=2,V12>=0,V13>=0] 

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


Closed-form bounds of start(V1,V,V12,V13): 
-------------------------------------
* Chain [39] with precondition: [V1>=0,V>=0] 
    - Upper bound: 144*V1+34*V+19 
    - Complexity: n 
* Chain [38] with precondition: [V1=2,V=2,V12>=0,V13>=0] 
    - Upper bound: 71*V13+28+nat(V12-V13)*288 
    - Complexity: n 
* Chain [37] with precondition: [V=0,V1>=0] 
    - Upper bound: 0 
    - Complexity: constant 

### Maximum cost of start(V1,V,V12,V13): max([144*V1+34*V+19,nat(V13)*71+28+nat(V12-V13)*288]) 
Asymptotic class: n 
* Total analysis performed in 1007 ms.


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

(20)
BOUNDS(1, n^1)