WORST_CASE(?,O(n^2))
proof of input_8Mm4N6IHRD.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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^2).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
(3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CdtProblem
(5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
(7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxRelTRS
(9) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxWeightedTrs
(11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTypedWeightedTrs
(13) CompletionProof [UPPER BOUND(ID), 0 ms]
(14) CpxTypedWeightedCompleteTrs
(15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(16) CpxRNTS
(17) CompleteCoflocoProof [FINISHED, 858 ms]
(18) BOUNDS(1, n^2)


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

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


The TRS R consists of the following rules:

   minus(x, y) -> cond(equal(min(x, y), y), x, y)
   cond(true, x, y) -> s(minus(x, s(y)))
   min(0, v) -> 0
   min(u, 0) -> 0
   min(s(u), s(v)) -> s(min(u, v))
   equal(0, 0) -> true
   equal(s(x), 0) -> false
   equal(0, s(y)) -> false
   equal(s(x), s(y)) -> equal(x, 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:
   minus(z0, z1) -> cond(equal(min(z0, z1), z1), z0, z1)
   cond(true, z0, z1) -> s(minus(z0, s(z1)))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   equal(0, 0) -> true
   equal(s(z0), 0) -> false
   equal(0, s(z0)) -> false
   equal(s(z0), s(z1)) -> equal(z0, z1)
Tuples:
   MINUS(z0, z1) -> c(COND(equal(min(z0, z1), z1), z0, z1), EQUAL(min(z0, z1), z1), MIN(z0, z1))
   COND(true, z0, z1) -> c1(MINUS(z0, s(z1)))
   MIN(0, z0) -> c2
   MIN(z0, 0) -> c3
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   EQUAL(0, 0) -> c5
   EQUAL(s(z0), 0) -> c6
   EQUAL(0, s(z0)) -> c7
   EQUAL(s(z0), s(z1)) -> c8(EQUAL(z0, z1))
S tuples:
   MINUS(z0, z1) -> c(COND(equal(min(z0, z1), z1), z0, z1), EQUAL(min(z0, z1), z1), MIN(z0, z1))
   COND(true, z0, z1) -> c1(MINUS(z0, s(z1)))
   MIN(0, z0) -> c2
   MIN(z0, 0) -> c3
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   EQUAL(0, 0) -> c5
   EQUAL(s(z0), 0) -> c6
   EQUAL(0, s(z0)) -> c7
   EQUAL(s(z0), s(z1)) -> c8(EQUAL(z0, z1))
K tuples:none
Defined Rule Symbols:   minus_2, cond_3, min_2, equal_2

Defined Pair Symbols:   MINUS_2, COND_3, MIN_2, EQUAL_2

Compound Symbols:   c_3, c1_1, c2, c3, c4_1, c5, c6, c7, c8_1


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing nodes:
   EQUAL(0, 0) -> c5
   EQUAL(s(z0), 0) -> c6
   MIN(0, z0) -> c2
   EQUAL(0, s(z0)) -> c7
   MIN(z0, 0) -> c3

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, z1) -> cond(equal(min(z0, z1), z1), z0, z1)
   cond(true, z0, z1) -> s(minus(z0, s(z1)))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   equal(0, 0) -> true
   equal(s(z0), 0) -> false
   equal(0, s(z0)) -> false
   equal(s(z0), s(z1)) -> equal(z0, z1)
Tuples:
   MINUS(z0, z1) -> c(COND(equal(min(z0, z1), z1), z0, z1), EQUAL(min(z0, z1), z1), MIN(z0, z1))
   COND(true, z0, z1) -> c1(MINUS(z0, s(z1)))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   EQUAL(s(z0), s(z1)) -> c8(EQUAL(z0, z1))
S tuples:
   MINUS(z0, z1) -> c(COND(equal(min(z0, z1), z1), z0, z1), EQUAL(min(z0, z1), z1), MIN(z0, z1))
   COND(true, z0, z1) -> c1(MINUS(z0, s(z1)))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   EQUAL(s(z0), s(z1)) -> c8(EQUAL(z0, z1))
K tuples:none
Defined Rule Symbols:   minus_2, cond_3, min_2, equal_2

Defined Pair Symbols:   MINUS_2, COND_3, MIN_2, EQUAL_2

Compound Symbols:   c_3, c1_1, c4_1, c8_1


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   minus(z0, z1) -> cond(equal(min(z0, z1), z1), z0, z1)
   cond(true, z0, z1) -> s(minus(z0, s(z1)))

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   equal(0, 0) -> true
   equal(s(z0), 0) -> false
   equal(0, s(z0)) -> false
   equal(s(z0), s(z1)) -> equal(z0, z1)
   min(0, z0) -> 0
   min(z0, 0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
Tuples:
   MINUS(z0, z1) -> c(COND(equal(min(z0, z1), z1), z0, z1), EQUAL(min(z0, z1), z1), MIN(z0, z1))
   COND(true, z0, z1) -> c1(MINUS(z0, s(z1)))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   EQUAL(s(z0), s(z1)) -> c8(EQUAL(z0, z1))
S tuples:
   MINUS(z0, z1) -> c(COND(equal(min(z0, z1), z1), z0, z1), EQUAL(min(z0, z1), z1), MIN(z0, z1))
   COND(true, z0, z1) -> c1(MINUS(z0, s(z1)))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   EQUAL(s(z0), s(z1)) -> c8(EQUAL(z0, z1))
K tuples:none
Defined Rule Symbols:   equal_2, min_2

Defined Pair Symbols:   MINUS_2, COND_3, MIN_2, EQUAL_2

Compound Symbols:   c_3, c1_1, c4_1, c8_1


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

(7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   MINUS(z0, z1) -> c(COND(equal(min(z0, z1), z1), z0, z1), EQUAL(min(z0, z1), z1), MIN(z0, z1))
   COND(true, z0, z1) -> c1(MINUS(z0, s(z1)))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   EQUAL(s(z0), s(z1)) -> c8(EQUAL(z0, z1))

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

   equal(0, 0) -> true
   equal(s(z0), 0) -> false
   equal(0, s(z0)) -> false
   equal(s(z0), s(z1)) -> equal(z0, z1)
   min(0, z0) -> 0
   min(z0, 0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))

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

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

(10)
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:

   MINUS(z0, z1) -> c(COND(equal(min(z0, z1), z1), z0, z1), EQUAL(min(z0, z1), z1), MIN(z0, z1)) [1]
   COND(true, z0, z1) -> c1(MINUS(z0, s(z1))) [1]
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) [1]
   EQUAL(s(z0), s(z1)) -> c8(EQUAL(z0, z1)) [1]
   equal(0, 0) -> true [0]
   equal(s(z0), 0) -> false [0]
   equal(0, s(z0)) -> false [0]
   equal(s(z0), s(z1)) -> equal(z0, z1) [0]
   min(0, z0) -> 0 [0]
   min(z0, 0) -> 0 [0]
   min(s(z0), s(z1)) -> s(min(z0, z1)) [0]

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

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

(12)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   MINUS(z0, z1) -> c(COND(equal(min(z0, z1), z1), z0, z1), EQUAL(min(z0, z1), z1), MIN(z0, z1)) [1]
   COND(true, z0, z1) -> c1(MINUS(z0, s(z1))) [1]
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) [1]
   EQUAL(s(z0), s(z1)) -> c8(EQUAL(z0, z1)) [1]
   equal(0, 0) -> true [0]
   equal(s(z0), 0) -> false [0]
   equal(0, s(z0)) -> false [0]
   equal(s(z0), s(z1)) -> equal(z0, z1) [0]
   min(0, z0) -> 0 [0]
   min(z0, 0) -> 0 [0]
   min(s(z0), s(z1)) -> s(min(z0, z1)) [0]

The TRS has the following type information:
MINUS :: s:0 -> s:0 -> c
c :: c1 -> c8 -> c4 -> c
COND :: true:false -> s:0 -> s:0 -> c1
equal :: s:0 -> s:0 -> true:false
min :: s:0 -> s:0 -> s:0
EQUAL :: s:0 -> s:0 -> c8
MIN :: s:0 -> s:0 -> c4
true :: true:false
c1 :: c -> c1
s :: s:0 -> s:0
c4 :: c4 -> c4
c8 :: c8 -> c8
0 :: s:0
false :: true:false

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

(13) 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:

   equal(v0, v1) -> null_equal [0]
   min(v0, v1) -> null_min [0]
   COND(v0, v1, v2) -> null_COND [0]
   MIN(v0, v1) -> null_MIN [0]
   EQUAL(v0, v1) -> null_EQUAL [0]

And the following fresh constants:    null_equal, null_min, null_COND, null_MIN, null_EQUAL, const

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

(14)
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:

   MINUS(z0, z1) -> c(COND(equal(min(z0, z1), z1), z0, z1), EQUAL(min(z0, z1), z1), MIN(z0, z1)) [1]
   COND(true, z0, z1) -> c1(MINUS(z0, s(z1))) [1]
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) [1]
   EQUAL(s(z0), s(z1)) -> c8(EQUAL(z0, z1)) [1]
   equal(0, 0) -> true [0]
   equal(s(z0), 0) -> false [0]
   equal(0, s(z0)) -> false [0]
   equal(s(z0), s(z1)) -> equal(z0, z1) [0]
   min(0, z0) -> 0 [0]
   min(z0, 0) -> 0 [0]
   min(s(z0), s(z1)) -> s(min(z0, z1)) [0]
   equal(v0, v1) -> null_equal [0]
   min(v0, v1) -> null_min [0]
   COND(v0, v1, v2) -> null_COND [0]
   MIN(v0, v1) -> null_MIN [0]
   EQUAL(v0, v1) -> null_EQUAL [0]

The TRS has the following type information:
MINUS :: s:0:null_min -> s:0:null_min -> c
c :: c1:null_COND -> c8:null_EQUAL -> c4:null_MIN -> c
COND :: true:false:null_equal -> s:0:null_min -> s:0:null_min -> c1:null_COND
equal :: s:0:null_min -> s:0:null_min -> true:false:null_equal
min :: s:0:null_min -> s:0:null_min -> s:0:null_min
EQUAL :: s:0:null_min -> s:0:null_min -> c8:null_EQUAL
MIN :: s:0:null_min -> s:0:null_min -> c4:null_MIN
true :: true:false:null_equal
c1 :: c -> c1:null_COND
s :: s:0:null_min -> s:0:null_min
c4 :: c4:null_MIN -> c4:null_MIN
c8 :: c8:null_EQUAL -> c8:null_EQUAL
0 :: s:0:null_min
false :: true:false:null_equal
null_equal :: true:false:null_equal
null_min :: s:0:null_min
null_COND :: c1:null_COND
null_MIN :: c4:null_MIN
null_EQUAL :: c8:null_EQUAL
const :: c

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

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

   true => 2
   0 => 0
   false => 1
   null_equal => 0
   null_min => 0
   null_COND => 0
   null_MIN => 0
   null_EQUAL => 0
   const => 0

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

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   COND(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   COND(z, z', z'') -{ 1 }-> 1 + MINUS(z0, 1 + z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   EQUAL(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   EQUAL(z, z') -{ 1 }-> 1 + EQUAL(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MIN(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   MIN(z, z') -{ 1 }-> 1 + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MINUS(z, z') -{ 1 }-> 1 + COND(equal(min(z0, z1), z1), z0, z1) + EQUAL(min(z0, z1), z1) + MIN(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   equal(z, z') -{ 0 }-> equal(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   equal(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   equal(z, z') -{ 0 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0
   equal(z, z') -{ 0 }-> 1 :|: z0 >= 0, z' = 1 + z0, z = 0
   equal(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   min(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   min(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   min(z, z') -{ 0 }-> 1 + min(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V, V4),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V4),0,[fun1(V1, V, V4, Out)],[V1 >= 0,V >= 0,V4 >= 0]).
eq(start(V1, V, V4),0,[fun3(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V4),0,[fun2(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V4),0,[equal(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V4),0,[min(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(fun(V1, V, Out),1,[min(V3, V2, Ret00100),equal(Ret00100, V2, Ret0010),fun1(Ret0010, V3, V2, Ret001),min(V3, V2, Ret010),fun2(Ret010, V2, Ret01),fun3(V3, V2, Ret1)],[Out = 1 + Ret001 + Ret01 + Ret1,V1 = V3,V2 >= 0,V = V2,V3 >= 0]).
eq(fun1(V1, V, V4, Out),1,[fun(V6, 1 + V5, Ret11)],[Out = 1 + Ret11,V1 = 2,V5 >= 0,V6 >= 0,V = V6,V4 = V5]).
eq(fun3(V1, V, Out),1,[fun3(V8, V7, Ret12)],[Out = 1 + Ret12,V7 >= 0,V1 = 1 + V8,V8 >= 0,V = 1 + V7]).
eq(fun2(V1, V, Out),1,[fun2(V9, V10, Ret13)],[Out = 1 + Ret13,V10 >= 0,V1 = 1 + V9,V9 >= 0,V = 1 + V10]).
eq(equal(V1, V, Out),0,[],[Out = 2,V1 = 0,V = 0]).
eq(equal(V1, V, Out),0,[],[Out = 1,V1 = 1 + V11,V11 >= 0,V = 0]).
eq(equal(V1, V, Out),0,[],[Out = 1,V12 >= 0,V = 1 + V12,V1 = 0]).
eq(equal(V1, V, Out),0,[equal(V14, V13, Ret)],[Out = Ret,V13 >= 0,V1 = 1 + V14,V14 >= 0,V = 1 + V13]).
eq(min(V1, V, Out),0,[],[Out = 0,V15 >= 0,V1 = 0,V = V15]).
eq(min(V1, V, Out),0,[],[Out = 0,V1 = V16,V16 >= 0,V = 0]).
eq(min(V1, V, Out),0,[min(V17, V18, Ret14)],[Out = 1 + Ret14,V18 >= 0,V1 = 1 + V17,V17 >= 0,V = 1 + V18]).
eq(equal(V1, V, Out),0,[],[Out = 0,V20 >= 0,V19 >= 0,V1 = V20,V = V19]).
eq(min(V1, V, Out),0,[],[Out = 0,V22 >= 0,V21 >= 0,V1 = V22,V = V21]).
eq(fun1(V1, V, V4, Out),0,[],[Out = 0,V24 >= 0,V4 = V25,V23 >= 0,V1 = V24,V = V23,V25 >= 0]).
eq(fun3(V1, V, Out),0,[],[Out = 0,V26 >= 0,V27 >= 0,V1 = V26,V = V27]).
eq(fun2(V1, V, Out),0,[],[Out = 0,V29 >= 0,V28 >= 0,V1 = V29,V = V28]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,V4,Out),[V1,V,V4],[Out]).
input_output_vars(fun3(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun2(V1,V,Out),[V1,V],[Out]).
input_output_vars(equal(V1,V,Out),[V1,V],[Out]).
input_output_vars(min(V1,V,Out),[V1,V],[Out]).


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

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

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

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

### Specialization of cost equations equal/3 
* CE 18 is refined into CE [22] 
* CE 15 is refined into CE [23] 
* CE 16 is refined into CE [24] 
* CE 14 is refined into CE [25] 
* CE 17 is refined into CE [26] 


### Cost equations --> "Loop" of equal/3 
* CEs [26] --> Loop 15 
* CEs [22] --> Loop 16 
* CEs [23] --> Loop 17 
* CEs [24] --> Loop 18 
* CEs [25] --> Loop 19 

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

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


### Specialization of cost equations fun2/3 
* CE 13 is refined into CE [27] 
* CE 12 is refined into CE [28] 


### Cost equations --> "Loop" of fun2/3 
* CEs [28] --> Loop 20 
* CEs [27] --> Loop 21 

### Ranking functions of CR fun2(V1,V,Out) 
* RF of phase [20]: [V,V1]

#### Partial ranking functions of CR fun2(V1,V,Out) 
* Partial RF of phase [20]:
  - RF of loop [20:1]:
    V
    V1


### Specialization of cost equations fun3/3 
* CE 11 is refined into CE [29] 
* CE 10 is refined into CE [30] 


### Cost equations --> "Loop" of fun3/3 
* CEs [30] --> Loop 22 
* CEs [29] --> Loop 23 

### Ranking functions of CR fun3(V1,V,Out) 
* RF of phase [22]: [V,V1]

#### Partial ranking functions of CR fun3(V1,V,Out) 
* Partial RF of phase [22]:
  - RF of loop [22:1]:
    V
    V1


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


### Cost equations --> "Loop" of min/3 
* CEs [33] --> Loop 24 
* CEs [31,32] --> Loop 25 

### Ranking functions of CR min(V1,V,Out) 
* RF of phase [24]: [V,V1]

#### Partial ranking functions of CR min(V1,V,Out) 
* Partial RF of phase [24]:
  - RF of loop [24:1]:
    V
    V1


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


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

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

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


### Specialization of cost equations start/3 
* CE 1 is refined into CE [72] 
* CE 2 is refined into CE [73,74,75,76] 
* CE 3 is refined into CE [77,78,79,80,81,82,83] 
* CE 4 is refined into CE [84,85] 
* CE 5 is refined into CE [86,87] 
* CE 6 is refined into CE [88,89,90,91,92,93,94] 
* CE 7 is refined into CE [95,96] 


### Cost equations --> "Loop" of start/3 
* CEs [94] --> Loop 35 
* CEs [73,74,75,76] --> Loop 36 
* CEs [89] --> Loop 37 
* CEs [72,77,78,79,80,81,82,83,84,85,86,87,88,90,91,92,93,95,96] --> Loop 38 

### Ranking functions of CR start(V1,V,V4) 

#### Partial ranking functions of CR start(V1,V,V4) 


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

#### Cost of chains of equal(V1,V,Out):
* Chain [[15],19]: 0
  with precondition: [Out=2,V1=V,V1>=1] 

* Chain [[15],18]: 0
  with precondition: [Out=1,V1>=1,V>=V1+1] 

* Chain [[15],17]: 0
  with precondition: [Out=1,V>=1,V1>=V+1] 

* Chain [[15],16]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

* Chain [19]: 0
  with precondition: [V1=0,V=0,Out=2] 

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

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

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


#### Cost of chains of fun2(V1,V,Out):
* Chain [[20],21]: 1*it(20)+0
  Such that:it(20) =< Out

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

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


#### Cost of chains of fun3(V1,V,Out):
* Chain [[22],23]: 1*it(22)+0
  Such that:it(22) =< Out

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

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


#### Cost of chains of min(V1,V,Out):
* Chain [[24],25]: 0
  with precondition: [Out>=1,V1>=Out,V>=Out] 

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


#### Cost of chains of fun(V1,V,Out):
* Chain [[31,32,33],30]: 6*it(31)+2*s(10)+3*s(12)+1
  Such that:aux(3) =< V1+1
aux(7) =< V1-V+1
it(31) =< aux(7)
aux(4) =< aux(3)
s(11) =< it(31)*aux(3)
s(13) =< it(31)*aux(4)
s(12) =< s(13)
s(10) =< s(11)

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

* Chain [[31,32,33],29]: 6*it(31)+2*s(10)+3*s(12)+9*s(14)+3*s(23)+1
  Such that:aux(8) =< V1
aux(10) =< V1+1
aux(11) =< V1-V+1
s(23) =< aux(8)
s(14) =< aux(10)
it(31) =< aux(11)
aux(4) =< aux(10)
s(11) =< it(31)*aux(10)
s(13) =< it(31)*aux(4)
s(12) =< s(13)
s(10) =< s(11)

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

* Chain [[31,32,33],28]: 6*it(31)+2*s(10)+3*s(12)+5*s(26)+3*s(27)+1
  Such that:aux(13) =< V1
aux(15) =< V1+1
aux(16) =< V1-V+1
s(26) =< aux(13)
s(27) =< aux(15)
it(31) =< aux(16)
aux(4) =< aux(15)
s(11) =< it(31)*aux(15)
s(13) =< it(31)*aux(4)
s(12) =< s(13)
s(10) =< s(11)

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

* Chain [[31,32,33],27]: 6*it(31)+2*s(10)+3*s(12)+3*s(34)+1
  Such that:aux(17) =< V1
aux(3) =< V1+1
aux(6) =< V1-V
aux(5) =< V1-V+1
s(34) =< aux(17)
it(31) =< aux(5)
it(31) =< aux(6)
aux(4) =< aux(3)
s(11) =< it(31)*aux(3)
s(13) =< it(31)*aux(4)
s(12) =< s(13)
s(10) =< s(11)

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

* Chain [[31,32,33],26]: 6*it(31)+2*s(10)+3*s(12)+2*s(37)+1
  Such that:aux(18) =< V1
aux(3) =< V1+1
aux(6) =< V1-V
aux(5) =< V1-V+1
s(37) =< aux(18)
it(31) =< aux(5)
it(31) =< aux(6)
aux(4) =< aux(3)
s(11) =< it(31)*aux(3)
s(13) =< it(31)*aux(4)
s(12) =< s(13)
s(10) =< s(11)

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

* Chain [34,[31,32,33],30]: 6*it(31)+2*s(10)+3*s(12)+3
  Such that:aux(7) =< V1
aux(3) =< V1+1
it(31) =< aux(7)
aux(4) =< aux(3)
s(11) =< it(31)*aux(3)
s(13) =< it(31)*aux(4)
s(12) =< s(13)
s(10) =< s(11)

  with precondition: [V=0,V1>=1,Out>=5] 

* Chain [34,[31,32,33],29]: 9*it(31)+2*s(10)+3*s(12)+9*s(14)+3
  Such that:aux(10) =< V1+1
aux(19) =< V1
it(31) =< aux(19)
s(14) =< aux(10)
aux(4) =< aux(10)
s(11) =< it(31)*aux(10)
s(13) =< it(31)*aux(4)
s(12) =< s(13)
s(10) =< s(11)

  with precondition: [V=0,V1>=1,Out>=6] 

* Chain [34,[31,32,33],28]: 11*it(31)+2*s(10)+3*s(12)+3*s(27)+3
  Such that:aux(15) =< V1+1
aux(20) =< V1
it(31) =< aux(20)
s(27) =< aux(15)
aux(4) =< aux(15)
s(11) =< it(31)*aux(15)
s(13) =< it(31)*aux(4)
s(12) =< s(13)
s(10) =< s(11)

  with precondition: [V=0,V1>=1,Out>=7] 

* Chain [34,[31,32,33],27]: 9*it(31)+2*s(10)+3*s(12)+3
  Such that:aux(3) =< V1+1
aux(21) =< V1
it(31) =< aux(21)
aux(4) =< aux(3)
s(11) =< it(31)*aux(3)
s(13) =< it(31)*aux(4)
s(12) =< s(13)
s(10) =< s(11)

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

* Chain [34,[31,32,33],26]: 8*it(31)+2*s(10)+3*s(12)+3
  Such that:aux(3) =< V1+1
aux(22) =< V1
it(31) =< aux(22)
aux(4) =< aux(3)
s(11) =< it(31)*aux(3)
s(13) =< it(31)*aux(4)
s(12) =< s(13)
s(10) =< s(11)

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

* Chain [34,30]: 3
  with precondition: [V=0,Out=3,V1>=0] 

* Chain [34,29]: 9*s(14)+3*s(23)+3
  Such that:aux(9) =< 1
aux(8) =< V1
s(23) =< aux(8)
s(14) =< aux(9)

  with precondition: [V=0,Out=4,V1>=1] 

* Chain [34,28]: 5*s(26)+3*s(27)+3
  Such that:aux(14) =< 1
aux(13) =< V1
s(26) =< aux(13)
s(27) =< aux(14)

  with precondition: [V=0,Out=5,V1>=1] 

* Chain [34,27]: 3*s(34)+3
  Such that:aux(17) =< 1
s(34) =< aux(17)

  with precondition: [V=0,Out=4,V1>=1] 

* Chain [34,26]: 2*s(37)+3
  Such that:aux(18) =< 1
s(37) =< aux(18)

  with precondition: [V=0,Out=5,V1>=1] 

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

* Chain [29]: 9*s(14)+3*s(23)+1
  Such that:aux(8) =< V1
aux(9) =< V
s(23) =< aux(8)
s(14) =< aux(9)

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

* Chain [28]: 5*s(26)+3*s(27)+1
  Such that:aux(13) =< V1
aux(14) =< V
s(26) =< aux(13)
s(27) =< aux(14)

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

* Chain [27]: 3*s(34)+1
  Such that:aux(17) =< V
s(34) =< aux(17)

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

* Chain [26]: 2*s(37)+1
  Such that:aux(18) =< V
s(37) =< aux(18)

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


#### Cost of chains of start(V1,V,V4):
* Chain [38]: 17*s(155)+72*s(156)+15*s(165)+10*s(166)+24*s(167)+12*s(174)+6*s(178)+4*s(179)+18*s(181)+9*s(184)+6*s(185)+19*s(186)+3
  Such that:s(171) =< V1-V
s(172) =< V1-V+1
aux(32) =< 1
aux(33) =< V1
aux(34) =< V1+1
aux(35) =< V
s(186) =< aux(35)
s(155) =< aux(32)
s(156) =< aux(33)
s(162) =< aux(34)
s(163) =< s(156)*aux(34)
s(164) =< s(156)*s(162)
s(165) =< s(164)
s(166) =< s(163)
s(167) =< aux(34)
s(174) =< s(172)
s(174) =< s(171)
s(176) =< s(174)*aux(34)
s(177) =< s(174)*s(162)
s(178) =< s(177)
s(179) =< s(176)
s(181) =< s(172)
s(182) =< s(181)*aux(34)
s(183) =< s(181)*s(162)
s(184) =< s(183)
s(185) =< s(182)

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

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

* Chain [36]: 21*s(202)+30*s(203)+15*s(207)+10*s(208)+12*s(209)+17*s(215)+2
  Such that:s(199) =< V+1
aux(36) =< V-V4
aux(37) =< V
aux(38) =< V4+1
s(202) =< aux(37)
s(203) =< aux(36)
s(204) =< s(199)
s(205) =< s(203)*s(199)
s(206) =< s(203)*s(204)
s(207) =< s(206)
s(208) =< s(205)
s(209) =< s(199)
s(215) =< aux(38)

  with precondition: [V1=2,V>=0,V4>=0] 

* Chain [35]: 0
  with precondition: [V1=V,V1>=1] 


Closed-form bounds of start(V1,V,V4): 
-------------------------------------
* Chain [38] with precondition: [V1>=0,V>=0] 
    - Upper bound: 72*V1+20+(V1+1)*(25*V1)+19*V+(24*V1+24)+(25*V1+25)*nat(V1-V+1)+nat(V1-V+1)*30 
    - Complexity: n^2 
* Chain [37] with precondition: [V1=0,V>=1] 
    - Upper bound: 0 
    - Complexity: constant 
* Chain [36] with precondition: [V1=2,V>=0,V4>=0] 
    - Upper bound: 33*V+14+(25*V+25)*nat(V-V4)+(17*V4+17)+nat(V-V4)*30 
    - Complexity: n^2 
* Chain [35] with precondition: [V1=V,V1>=1] 
    - Upper bound: 0 
    - Complexity: constant 

### Maximum cost of start(V1,V,V4): 19*V+2+max([14*V+12+(25*V+25)*nat(V-V4)+nat(V4+1)*17+nat(V-V4)*30,72*V1+18+(V1+1)*(25*V1)+(24*V1+24)+(25*V1+25)*nat(V1-V+1)+nat(V1-V+1)*30]) 
Asymptotic class: n^2 
* Total analysis performed in 839 ms.


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

(18)
BOUNDS(1, n^2)