/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 949 ms]
    (10) BOUNDS(1, n^1)
(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTRS
    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) typed CpxTrs
    (15) OrderProof [LOWER BOUND(ID), 0 ms]
    (16) typed CpxTrs
    (17) RewriteLemmaProof [LOWER BOUND(ID), 1022 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 24 ms]
            (24) typed CpxTrs


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

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


The TRS R consists of the following rules:

   minus(X, s(Y)) -> pred(minus(X, Y))
   minus(X, 0) -> X
   pred(s(X)) -> X
   le(s(X), s(Y)) -> le(X, Y)
   le(s(X), 0) -> false
   le(0, Y) -> true
   gcd(0, Y) -> 0
   gcd(s(X), 0) -> s(X)
   gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
   if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
   if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

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

(2)
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:

   minus(X, s(Y)) -> pred(minus(X, Y)) [1]
   minus(X, 0) -> X [1]
   pred(s(X)) -> X [1]
   le(s(X), s(Y)) -> le(X, Y) [1]
   le(s(X), 0) -> false [1]
   le(0, Y) -> true [1]
   gcd(0, Y) -> 0 [1]
   gcd(s(X), 0) -> s(X) [1]
   gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y)) [1]
   if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y)) [1]
   if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X)) [1]

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

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

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   minus(X, s(Y)) -> pred(minus(X, Y)) [1]
   minus(X, 0) -> X [1]
   pred(s(X)) -> X [1]
   le(s(X), s(Y)) -> le(X, Y) [1]
   le(s(X), 0) -> false [1]
   le(0, Y) -> true [1]
   gcd(0, Y) -> 0 [1]
   gcd(s(X), 0) -> s(X) [1]
   gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y)) [1]
   if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y)) [1]
   if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X)) [1]

The TRS has the following type information:
minus :: s:0 -> s:0 -> s:0
s :: s:0 -> s:0
pred :: s:0 -> s:0
0 :: s:0
le :: s:0 -> s:0 -> false:true
false :: false:true
true :: false:true
gcd :: s:0 -> s:0 -> s:0
if :: false:true -> s:0 -> s:0 -> s:0

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

(5) 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:

   pred(v0) -> null_pred [0]
   if(v0, v1, v2) -> null_if [0]
   minus(v0, v1) -> null_minus [0]
   le(v0, v1) -> null_le [0]
   gcd(v0, v1) -> null_gcd [0]

And the following fresh constants:    null_pred, null_if, null_minus, null_le, null_gcd

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

(6)
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(X, s(Y)) -> pred(minus(X, Y)) [1]
   minus(X, 0) -> X [1]
   pred(s(X)) -> X [1]
   le(s(X), s(Y)) -> le(X, Y) [1]
   le(s(X), 0) -> false [1]
   le(0, Y) -> true [1]
   gcd(0, Y) -> 0 [1]
   gcd(s(X), 0) -> s(X) [1]
   gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y)) [1]
   if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y)) [1]
   if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X)) [1]
   pred(v0) -> null_pred [0]
   if(v0, v1, v2) -> null_if [0]
   minus(v0, v1) -> null_minus [0]
   le(v0, v1) -> null_le [0]
   gcd(v0, v1) -> null_gcd [0]

The TRS has the following type information:
minus :: s:0:null_pred:null_if:null_minus:null_gcd -> s:0:null_pred:null_if:null_minus:null_gcd -> s:0:null_pred:null_if:null_minus:null_gcd
s :: s:0:null_pred:null_if:null_minus:null_gcd -> s:0:null_pred:null_if:null_minus:null_gcd
pred :: s:0:null_pred:null_if:null_minus:null_gcd -> s:0:null_pred:null_if:null_minus:null_gcd
0 :: s:0:null_pred:null_if:null_minus:null_gcd
le :: s:0:null_pred:null_if:null_minus:null_gcd -> s:0:null_pred:null_if:null_minus:null_gcd -> false:true:null_le
false :: false:true:null_le
true :: false:true:null_le
gcd :: s:0:null_pred:null_if:null_minus:null_gcd -> s:0:null_pred:null_if:null_minus:null_gcd -> s:0:null_pred:null_if:null_minus:null_gcd
if :: false:true:null_le -> s:0:null_pred:null_if:null_minus:null_gcd -> s:0:null_pred:null_if:null_minus:null_gcd -> s:0:null_pred:null_if:null_minus:null_gcd
null_pred :: s:0:null_pred:null_if:null_minus:null_gcd
null_if :: s:0:null_pred:null_if:null_minus:null_gcd
null_minus :: s:0:null_pred:null_if:null_minus:null_gcd
null_le :: false:true:null_le
null_gcd :: s:0:null_pred:null_if:null_minus:null_gcd

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

(7) 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
   false => 1
   true => 2
   null_pred => 0
   null_if => 0
   null_minus => 0
   null_le => 0
   null_gcd => 0

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

(8)
Obligation:
Complexity RNTS consisting of the following rules:

   gcd(z, z') -{ 1 }-> if(le(Y, X), 1 + X, 1 + Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   gcd(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, z = 0
   gcd(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   gcd(z, z') -{ 1 }-> 1 + X :|: z = 1 + X, X >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> gcd(minus(X, Y), 1 + Y) :|: z = 2, z'' = 1 + Y, Y >= 0, z' = 1 + X, X >= 0
   if(z, z', z'') -{ 1 }-> gcd(minus(Y, X), 1 + X) :|: z'' = 1 + Y, Y >= 0, z = 1, z' = 1 + X, X >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   le(z, z') -{ 1 }-> le(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   le(z, z') -{ 1 }-> 2 :|: z' = Y, Y >= 0, z = 0
   le(z, z') -{ 1 }-> 1 :|: z = 1 + X, X >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   minus(z, z') -{ 1 }-> X :|: X >= 0, z = X, z' = 0
   minus(z, z') -{ 1 }-> pred(minus(X, Y)) :|: Y >= 0, z' = 1 + Y, X >= 0, z = X
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   pred(z) -{ 1 }-> X :|: z = 1 + X, X >= 0
   pred(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V, V2),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V2),0,[pred(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V2),0,[le(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V2),0,[gcd(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V2),0,[if(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(minus(V1, V, Out),1,[minus(X1, Y1, Ret0),pred(Ret0, Ret)],[Out = Ret,Y1 >= 0,V = 1 + Y1,X1 >= 0,V1 = X1]).
eq(minus(V1, V, Out),1,[],[Out = X2,X2 >= 0,V1 = X2,V = 0]).
eq(pred(V1, Out),1,[],[Out = X3,V1 = 1 + X3,X3 >= 0]).
eq(le(V1, V, Out),1,[le(X4, Y2, Ret1)],[Out = Ret1,V1 = 1 + X4,Y2 >= 0,V = 1 + Y2,X4 >= 0]).
eq(le(V1, V, Out),1,[],[Out = 1,V1 = 1 + X5,X5 >= 0,V = 0]).
eq(le(V1, V, Out),1,[],[Out = 2,V = Y3,Y3 >= 0,V1 = 0]).
eq(gcd(V1, V, Out),1,[],[Out = 0,V = Y4,Y4 >= 0,V1 = 0]).
eq(gcd(V1, V, Out),1,[],[Out = 1 + X6,V1 = 1 + X6,X6 >= 0,V = 0]).
eq(gcd(V1, V, Out),1,[le(Y5, X7, Ret01),if(Ret01, 1 + X7, 1 + Y5, Ret2)],[Out = Ret2,V1 = 1 + X7,Y5 >= 0,V = 1 + Y5,X7 >= 0]).
eq(if(V1, V, V2, Out),1,[minus(X8, Y6, Ret02),gcd(Ret02, 1 + Y6, Ret3)],[Out = Ret3,V1 = 2,V2 = 1 + Y6,Y6 >= 0,V = 1 + X8,X8 >= 0]).
eq(if(V1, V, V2, Out),1,[minus(Y7, X9, Ret03),gcd(Ret03, 1 + X9, Ret4)],[Out = Ret4,V2 = 1 + Y7,Y7 >= 0,V1 = 1,V = 1 + X9,X9 >= 0]).
eq(pred(V1, Out),0,[],[Out = 0,V3 >= 0,V1 = V3]).
eq(if(V1, V, V2, Out),0,[],[Out = 0,V5 >= 0,V2 = V6,V4 >= 0,V1 = V5,V = V4,V6 >= 0]).
eq(minus(V1, V, Out),0,[],[Out = 0,V8 >= 0,V7 >= 0,V1 = V8,V = V7]).
eq(le(V1, V, Out),0,[],[Out = 0,V9 >= 0,V10 >= 0,V1 = V9,V = V10]).
eq(gcd(V1, V, Out),0,[],[Out = 0,V11 >= 0,V12 >= 0,V1 = V11,V = V12]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(pred(V1,Out),[V1],[Out]).
input_output_vars(le(V1,V,Out),[V1,V],[Out]).
input_output_vars(gcd(V1,V,Out),[V1,V],[Out]).
input_output_vars(if(V1,V,V2,Out),[V1,V,V2],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [pred/2]
1. recursive [non_tail] : [minus/3]
2. recursive  : [le/3]
3. recursive  : [gcd/3,if/4]
4. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into pred/2
1. SCC is partially evaluated into minus/3
2. SCC is partially evaluated into le/3
3. SCC is partially evaluated into gcd/3
4. SCC is partially evaluated into start/3

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

### Specialization of cost equations pred/2 
* CE 17 is refined into CE [23] 
* CE 18 is refined into CE [24] 


### Cost equations --> "Loop" of pred/2 
* CEs [23] --> Loop 16 
* CEs [24] --> Loop 17 

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

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


### Specialization of cost equations minus/3 
* CE 10 is refined into CE [25] 
* CE 9 is refined into CE [26] 
* CE 8 is refined into CE [27,28] 


### Cost equations --> "Loop" of minus/3 
* CEs [28] --> Loop 18 
* CEs [27] --> Loop 19 
* CEs [25] --> Loop 20 
* CEs [26] --> Loop 21 

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [18]: [V]
* RF of phase [19]: [V]

#### Partial ranking functions of CR minus(V1,V,Out) 
* Partial RF of phase [18]:
  - RF of loop [18:1]:
    V
* Partial RF of phase [19]:
  - RF of loop [19:1]:
    V


### Specialization of cost equations le/3 
* CE 22 is refined into CE [29] 
* CE 20 is refined into CE [30] 
* CE 21 is refined into CE [31] 
* CE 19 is refined into CE [32] 


### Cost equations --> "Loop" of le/3 
* CEs [32] --> Loop 22 
* CEs [29] --> Loop 23 
* CEs [30] --> Loop 24 
* CEs [31] --> Loop 25 

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

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


### Specialization of cost equations gcd/3 
* CE 15 is refined into CE [33] 
* CE 11 is refined into CE [34,35,36,37,38] 
* CE 14 is refined into CE [39] 
* CE 16 is refined into CE [40] 
* CE 13 is refined into CE [41,42,43,44] 
* CE 12 is refined into CE [45,46,47,48] 


### Cost equations --> "Loop" of gcd/3 
* CEs [48] --> Loop 26 
* CEs [44] --> Loop 27 
* CEs [47] --> Loop 28 
* CEs [43] --> Loop 29 
* CEs [41] --> Loop 30 
* CEs [42] --> Loop 31 
* CEs [45] --> Loop 32 
* CEs [46] --> Loop 33 
* CEs [34] --> Loop 34 
* CEs [33] --> Loop 35 
* CEs [35] --> Loop 36 
* CEs [36,37,38,39,40] --> Loop 37 

### Ranking functions of CR gcd(V1,V,Out) 
* RF of phase [26,27]: [V1+V-3]
* RF of phase [30]: [V1]

#### Partial ranking functions of CR gcd(V1,V,Out) 
* Partial RF of phase [26,27]:
  - RF of loop [26:1]:
    V-2
    V1/2+V/2-2
  - RF of loop [27:1]:
    V1-1 depends on loops [26:1] 
    V1-V+1 depends on loops [26:1] 
* Partial RF of phase [30]:
  - RF of loop [30:1]:
    V1


### Specialization of cost equations start/3 
* CE 3 is refined into CE [49,50,51,52] 
* CE 1 is refined into CE [53] 
* CE 2 is refined into CE [54,55,56,57] 
* CE 4 is refined into CE [58,59,60] 
* CE 5 is refined into CE [61,62] 
* CE 6 is refined into CE [63,64,65,66,67] 
* CE 7 is refined into CE [68,69,70] 


### Cost equations --> "Loop" of start/3 
* CEs [70] --> Loop 38 
* CEs [58,64,69] --> Loop 39 
* CEs [49,50,51,52] --> Loop 40 
* CEs [54,55,56,57] --> Loop 41 
* CEs [53,59,60,61,62,63,65,66,67,68] --> Loop 42 

### Ranking functions of CR start(V1,V,V2) 

#### Partial ranking functions of CR start(V1,V,V2) 


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

#### Cost of chains of pred(V1,Out):
* Chain [17]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [16]: 1
  with precondition: [V1=Out+1,V1>=1] 


#### Cost of chains of minus(V1,V,Out):
* Chain [[19],[18],21]: 3*it(18)+1
  Such that:aux(1) =< V
it(18) =< aux(1)

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

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

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

* Chain [[19],20]: 1*it(19)+0
  Such that:it(19) =< V

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

* Chain [[18],21]: 2*it(18)+1
  Such that:it(18) =< V

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

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

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


#### Cost of chains of le(V1,V,Out):
* Chain [[22],25]: 1*it(22)+1
  Such that:it(22) =< V1

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

* Chain [[22],24]: 1*it(22)+1
  Such that:it(22) =< V

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

* Chain [[22],23]: 1*it(22)+0
  Such that:it(22) =< V

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

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

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

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


#### Cost of chains of gcd(V1,V,Out):
* Chain [[30],37]: 6*it(30)+1*s(8)+2
  Such that:s(8) =< 1
aux(4) =< V1
it(30) =< aux(4)

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

* Chain [[30],34]: 4*it(30)+2
  Such that:it(30) =< V1

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

* Chain [[30],31,37]: 4*it(30)+1*s(8)+6
  Such that:s(8) =< 1
it(30) =< V1

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

* Chain [[26,27],37]: 4*it(26)+4*it(27)+6*s(6)+3*s(21)+2
  Such that:aux(10) =< V1-V+1
it(26) =< V1/2+V/2
aux(28) =< V1
aux(29) =< V1+V
aux(30) =< V
s(6) =< aux(29)
it(26) =< aux(29)
it(27) =< aux(29)
it(26) =< aux(30)
it(27) =< aux(30)+aux(10)
it(27) =< aux(30)+aux(28)
s(22) =< aux(30)+aux(28)
s(22) =< it(27)*aux(30)
s(21) =< s(22)

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

* Chain [[26,27],36]: 4*it(26)+4*it(27)+3*s(19)+3*s(21)+2
  Such that:aux(10) =< V1-V+1
aux(31) =< V1
aux(32) =< V1+V
aux(33) =< V1/2+V/2
aux(34) =< V
it(26) =< aux(33)
it(26) =< aux(32)
it(27) =< aux(32)
it(27) =< aux(33)
it(26) =< aux(34)
it(27) =< aux(34)+aux(10)
it(27) =< aux(34)+aux(31)
s(22) =< aux(34)+aux(31)
s(22) =< it(27)*aux(34)
s(21) =< s(22)
s(19) =< aux(32)

  with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 

* Chain [[26,27],33,37]: 4*it(26)+4*it(27)+1*s(8)+3*s(19)+3*s(21)+6
  Such that:s(8) =< 1
aux(10) =< V1-V+1
aux(35) =< V1
aux(36) =< V1+V
aux(37) =< V1/2+V/2
aux(38) =< V
it(26) =< aux(37)
it(26) =< aux(36)
it(27) =< aux(36)
it(27) =< aux(37)
it(26) =< aux(38)
it(27) =< aux(38)+aux(10)
it(27) =< aux(38)+aux(35)
s(22) =< aux(38)+aux(35)
s(22) =< it(27)*aux(38)
s(21) =< s(22)
s(19) =< aux(36)

  with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 

* Chain [[26,27],32,[30],37]: 4*it(26)+4*it(27)+9*it(30)+1*s(8)+3*s(21)+6
  Such that:s(8) =< 1
aux(10) =< V1-V+1
it(26) =< V1/2+V/2
aux(39) =< V1
aux(40) =< V1+V
aux(41) =< V
it(30) =< aux(40)
it(26) =< aux(40)
it(27) =< aux(40)
it(26) =< aux(41)
it(27) =< aux(41)+aux(10)
it(27) =< aux(41)+aux(39)
s(22) =< aux(41)+aux(39)
s(22) =< it(27)*aux(41)
s(21) =< s(22)

  with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 

* Chain [[26,27],32,[30],34]: 4*it(26)+4*it(27)+7*it(30)+3*s(21)+6
  Such that:aux(10) =< V1-V+1
it(26) =< V1/2+V/2
aux(42) =< V1
aux(43) =< V1+V
aux(44) =< V
it(30) =< aux(43)
it(26) =< aux(43)
it(27) =< aux(43)
it(26) =< aux(44)
it(27) =< aux(44)+aux(10)
it(27) =< aux(44)+aux(42)
s(22) =< aux(44)+aux(42)
s(22) =< it(27)*aux(44)
s(21) =< s(22)

  with precondition: [Out=0,V1>=3,V>=3,V+V1>=7] 

* Chain [[26,27],32,[30],31,37]: 4*it(26)+4*it(27)+7*it(30)+1*s(8)+3*s(21)+10
  Such that:s(8) =< 1
aux(10) =< V1-V+1
it(26) =< V1/2+V/2
aux(45) =< V1
aux(46) =< V1+V
aux(47) =< V
it(30) =< aux(46)
it(26) =< aux(46)
it(27) =< aux(46)
it(26) =< aux(47)
it(27) =< aux(47)+aux(10)
it(27) =< aux(47)+aux(45)
s(22) =< aux(47)+aux(45)
s(22) =< it(27)*aux(47)
s(21) =< s(22)

  with precondition: [Out=0,V1>=3,V>=3,V+V1>=7] 

* Chain [[26,27],32,37]: 4*it(26)+4*it(27)+5*s(6)+1*s(8)+3*s(21)+6
  Such that:s(8) =< 1
aux(10) =< V1-V+1
it(26) =< V1/2+V/2
aux(48) =< V1
aux(49) =< V1+V
aux(50) =< V
s(6) =< aux(49)
it(26) =< aux(49)
it(27) =< aux(49)
it(26) =< aux(50)
it(27) =< aux(50)+aux(10)
it(27) =< aux(50)+aux(48)
s(22) =< aux(50)+aux(48)
s(22) =< it(27)*aux(50)
s(21) =< s(22)

  with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 

* Chain [[26,27],32,34]: 4*it(26)+4*it(27)+3*s(19)+3*s(21)+6
  Such that:aux(10) =< V1-V+1
aux(51) =< V1
aux(52) =< V1+V
aux(53) =< V1/2+V/2
aux(54) =< V
it(26) =< aux(53)
it(26) =< aux(52)
it(27) =< aux(52)
it(27) =< aux(53)
it(26) =< aux(54)
it(27) =< aux(54)+aux(10)
it(27) =< aux(54)+aux(51)
s(22) =< aux(54)+aux(51)
s(22) =< it(27)*aux(54)
s(21) =< s(22)
s(19) =< aux(52)

  with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 

* Chain [[26,27],32,31,37]: 4*it(26)+4*it(27)+1*s(8)+3*s(19)+3*s(21)+10
  Such that:s(8) =< 1
aux(10) =< V1-V+1
aux(55) =< V1
aux(56) =< V1+V
aux(57) =< V1/2+V/2
aux(58) =< V
it(26) =< aux(57)
it(26) =< aux(56)
it(27) =< aux(56)
it(27) =< aux(57)
it(26) =< aux(58)
it(27) =< aux(58)+aux(10)
it(27) =< aux(58)+aux(55)
s(22) =< aux(58)+aux(55)
s(22) =< it(27)*aux(58)
s(21) =< s(22)
s(19) =< aux(56)

  with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 

* Chain [[26,27],29,37]: 4*it(26)+4*it(27)+7*s(8)+3*s(19)+3*s(21)+6
  Such that:aux(10) =< V1-V+1
aux(61) =< V1
aux(62) =< V1+V
aux(63) =< V1/2+V/2
aux(64) =< V
it(26) =< aux(63)
s(8) =< aux(63)
it(26) =< aux(62)
it(27) =< aux(62)
it(27) =< aux(63)
it(26) =< aux(64)
it(27) =< aux(64)+aux(10)
it(27) =< aux(64)+aux(61)
s(22) =< aux(64)+aux(61)
s(22) =< it(27)*aux(64)
s(21) =< s(22)
s(19) =< aux(62)

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

* Chain [[26,27],28,37]: 4*it(26)+4*it(27)+7*s(8)+3*s(19)+3*s(21)+6
  Such that:aux(10) =< V1-V+1
aux(67) =< V1
aux(68) =< V1+V
aux(69) =< V1/2+V/2
aux(70) =< V
it(26) =< aux(69)
s(8) =< aux(67)
it(26) =< aux(68)
it(27) =< aux(68)
it(27) =< aux(69)
it(26) =< aux(70)
it(27) =< aux(70)+aux(10)
it(27) =< aux(70)+aux(67)
s(22) =< aux(70)+aux(67)
s(22) =< it(27)*aux(70)
s(21) =< s(22)
s(19) =< aux(68)

  with precondition: [Out=0,V1>=3,V>=3,V+V1>=8] 

* Chain [37]: 2*s(6)+1*s(8)+2
  Such that:s(8) =< V
aux(3) =< V1
s(6) =< aux(3)

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

* Chain [36]: 2
  with precondition: [V1=1,Out=0,V>=2] 

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

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

* Chain [33,37]: 1*s(8)+6
  Such that:s(8) =< 1

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

* Chain [32,[30],37]: 6*it(30)+1*s(8)+6
  Such that:s(8) =< 1
aux(4) =< V
it(30) =< aux(4)

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

* Chain [32,[30],34]: 4*it(30)+6
  Such that:it(30) =< V

  with precondition: [V1=1,Out=0,V>=3] 

* Chain [32,[30],31,37]: 4*it(30)+1*s(8)+10
  Such that:s(8) =< 1
it(30) =< V

  with precondition: [V1=1,Out=0,V>=3] 

* Chain [32,37]: 2*s(6)+1*s(8)+6
  Such that:s(8) =< 1
aux(3) =< V
s(6) =< aux(3)

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

* Chain [32,34]: 6
  with precondition: [V1=1,Out=0,V>=2] 

* Chain [32,31,37]: 1*s(8)+10
  Such that:s(8) =< 1

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

* Chain [31,37]: 1*s(8)+6
  Such that:s(8) =< 1

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

* Chain [29,37]: 7*s(8)+6
  Such that:aux(60) =< V
s(8) =< aux(60)

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

* Chain [28,37]: 7*s(8)+6
  Such that:aux(66) =< V1
s(8) =< aux(66)

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


#### Cost of chains of start(V1,V,V2):
* Chain [42]: 33*s(169)+17*s(173)+10*s(180)+44*s(181)+24*s(183)+18*s(185)+52*s(186)+20*s(187)+15*s(189)+7*s(191)+10
  Such that:s(174) =< 1
s(176) =< V1-V+1
s(177) =< V1+V
s(178) =< V1/2+V/2
aux(79) =< V1
aux(80) =< V
s(173) =< aux(79)
s(169) =< aux(80)
s(180) =< s(174)
s(181) =< s(178)
s(181) =< s(177)
s(183) =< s(177)
s(183) =< s(178)
s(181) =< aux(80)
s(183) =< aux(80)+s(176)
s(183) =< aux(80)+aux(79)
s(184) =< aux(80)+aux(79)
s(184) =< s(183)*aux(80)
s(185) =< s(184)
s(186) =< s(177)
s(187) =< s(177)
s(187) =< aux(80)+s(176)
s(187) =< aux(80)+aux(79)
s(188) =< aux(80)+aux(79)
s(188) =< s(187)*aux(80)
s(189) =< s(188)
s(191) =< s(178)

  with precondition: [V1>=0] 

* Chain [41]: 57*s(198)+44*s(199)+24*s(201)+18*s(203)+134*s(204)+20*s(205)+15*s(207)+14*s(209)+107*s(215)+44*s(223)+24*s(225)+18*s(227)+20*s(229)+15*s(231)+7*s(233)+44*s(242)+24*s(244)+18*s(246)+20*s(248)+15*s(250)+16*s(251)+12
  Such that:s(237) =< -2*V+V2+1
s(218) =< -V+1
s(236) =< -V+V2
s(220) =< V/2
aux(85) =< 1
aux(86) =< V
aux(87) =< V2
aux(88) =< V2/2
s(198) =< aux(85)
s(204) =< aux(87)
s(223) =< s(220)
s(215) =< aux(86)
s(223) =< aux(86)
s(225) =< aux(86)
s(225) =< s(220)
s(225) =< aux(86)+s(218)
s(226) =< aux(86)
s(226) =< s(225)*aux(86)
s(227) =< s(226)
s(229) =< aux(86)
s(229) =< aux(86)+s(218)
s(230) =< aux(86)
s(230) =< s(229)*aux(86)
s(231) =< s(230)
s(233) =< s(220)
s(242) =< aux(88)
s(242) =< aux(87)
s(244) =< aux(87)
s(244) =< aux(88)
s(242) =< aux(86)
s(244) =< aux(86)+s(237)
s(244) =< aux(86)+s(236)
s(245) =< aux(86)+s(236)
s(245) =< s(244)*aux(86)
s(246) =< s(245)
s(248) =< aux(87)
s(248) =< aux(86)+s(237)
s(248) =< aux(86)+s(236)
s(249) =< aux(86)+s(236)
s(249) =< s(248)*aux(86)
s(250) =< s(249)
s(251) =< s(236)
s(209) =< aux(88)
s(199) =< aux(88)
s(199) =< aux(87)
s(201) =< aux(87)
s(201) =< aux(88)
s(199) =< aux(85)
s(201) =< aux(85)+aux(87)
s(202) =< aux(85)+aux(87)
s(202) =< s(201)*aux(85)
s(203) =< s(202)
s(205) =< aux(87)
s(205) =< aux(85)+aux(87)
s(206) =< aux(85)+aux(87)
s(206) =< s(205)*aux(85)
s(207) =< s(206)

  with precondition: [V1=1,V>=1,V2>=1] 

* Chain [40]: 57*s(259)+44*s(260)+24*s(262)+18*s(264)+134*s(265)+20*s(266)+15*s(268)+14*s(270)+107*s(276)+44*s(284)+24*s(286)+18*s(288)+20*s(290)+15*s(292)+7*s(294)+44*s(303)+24*s(305)+18*s(307)+20*s(309)+15*s(311)+16*s(312)+12
  Such that:s(298) =< V-2*V2+1
s(297) =< V-V2
s(279) =< -V2+1
s(281) =< V2/2
aux(93) =< 1
aux(94) =< V
aux(95) =< V/2
aux(96) =< V2
s(259) =< aux(93)
s(265) =< aux(94)
s(284) =< s(281)
s(276) =< aux(96)
s(284) =< aux(96)
s(286) =< aux(96)
s(286) =< s(281)
s(286) =< aux(96)+s(279)
s(287) =< aux(96)
s(287) =< s(286)*aux(96)
s(288) =< s(287)
s(290) =< aux(96)
s(290) =< aux(96)+s(279)
s(291) =< aux(96)
s(291) =< s(290)*aux(96)
s(292) =< s(291)
s(294) =< s(281)
s(303) =< aux(95)
s(303) =< aux(94)
s(305) =< aux(94)
s(305) =< aux(95)
s(303) =< aux(96)
s(305) =< aux(96)+s(298)
s(305) =< aux(96)+s(297)
s(306) =< aux(96)+s(297)
s(306) =< s(305)*aux(96)
s(307) =< s(306)
s(309) =< aux(94)
s(309) =< aux(96)+s(298)
s(309) =< aux(96)+s(297)
s(310) =< aux(96)+s(297)
s(310) =< s(309)*aux(96)
s(311) =< s(310)
s(312) =< s(297)
s(270) =< aux(95)
s(260) =< aux(95)
s(260) =< aux(94)
s(262) =< aux(94)
s(262) =< aux(95)
s(260) =< aux(93)
s(262) =< aux(93)+aux(94)
s(263) =< aux(93)+aux(94)
s(263) =< s(262)*aux(93)
s(264) =< s(263)
s(266) =< aux(94)
s(266) =< aux(93)+aux(94)
s(267) =< aux(93)+aux(94)
s(267) =< s(266)*aux(93)
s(268) =< s(267)

  with precondition: [V1=2,V>=1,V2>=1] 

* Chain [39]: 1
  with precondition: [V=0,V1>=0] 

* Chain [38]: 3*s(316)+14*s(317)+6
  Such that:s(314) =< 1
s(315) =< V1
s(316) =< s(314)
s(317) =< s(315)

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


Closed-form bounds of start(V1,V,V2): 
-------------------------------------
* Chain [42] with precondition: [V1>=0] 
    - Upper bound: 50*V1+20+nat(V)*66+nat(V1+V)*96+nat(V1/2+V/2)*51 
    - Complexity: n 
* Chain [41] with precondition: [V1=1,V>=1,V2>=1] 
    - Upper bound: 217*V+255*V2+146+nat(-V+V2)*49+51/2*V+29*V2 
    - Complexity: n 
* Chain [40] with precondition: [V1=2,V>=1,V2>=1] 
    - Upper bound: 255*V+217*V2+146+nat(V-V2)*49+29*V+51/2*V2 
    - Complexity: n 
* Chain [39] with precondition: [V=0,V1>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [38] with precondition: [V=1,V1>=1] 
    - Upper bound: 14*V1+9 
    - Complexity: n 

### Maximum cost of start(V1,V,V2): max([14*V1+8,nat(V)*66+19+max([nat(V1+V)*96+50*V1+nat(V1/2+V/2)*51,nat(V)*151+126+nat(V2)*217+nat(V/2)*51+nat(V2/2)*51+max([nat(-V+V2)*49+nat(V2)*38+nat(V2/2)*7,nat(V-V2)*49+nat(V)*38+nat(V/2)*7])])])+1 
Asymptotic class: n 
* Total analysis performed in 818 ms.


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

(10)
BOUNDS(1, n^1)

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

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

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   minus(X, s(Y)) -> pred(minus(X, Y))
   minus(X, 0') -> X
   pred(s(X)) -> X
   le(s(X), s(Y)) -> le(X, Y)
   le(s(X), 0') -> false
   le(0', Y) -> true
   gcd(0', Y) -> 0'
   gcd(s(X), 0') -> s(X)
   gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
   if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
   if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

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

(14)
Obligation:
Innermost TRS:
Rules:
minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0') -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0') -> false
le(0', Y) -> true
gcd(0', Y) -> 0'
gcd(s(X), 0') -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))

Types:
minus :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
pred :: s:0' -> s:0'
0' :: s:0'
le :: s:0' -> s:0' -> false:true
false :: false:true
true :: false:true
gcd :: s:0' -> s:0' -> s:0'
if :: false:true -> s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
gen_s:0'3_0 :: Nat -> s:0'

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

(15) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
minus, le, gcd

They will be analysed ascendingly in the following order:
minus < gcd
le < gcd

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

(16)
Obligation:
Innermost TRS:
Rules:
minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0') -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0') -> false
le(0', Y) -> true
gcd(0', Y) -> 0'
gcd(s(X), 0') -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))

Types:
minus :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
pred :: s:0' -> s:0'
0' :: s:0'
le :: s:0' -> s:0' -> false:true
false :: false:true
true :: false:true
gcd :: s:0' -> s:0' -> s:0'
if :: false:true -> s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
gen_s:0'3_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
minus, le, gcd

They will be analysed ascendingly in the following order:
minus < gcd
le < gcd

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)

Induction Base:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, 0)))

Induction Step:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
pred(minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0)))) ->_IH
pred(*4_0)

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

(18)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0') -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0') -> false
le(0', Y) -> true
gcd(0', Y) -> 0'
gcd(s(X), 0') -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))

Types:
minus :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
pred :: s:0' -> s:0'
0' :: s:0'
le :: s:0' -> s:0' -> false:true
false :: false:true
true :: false:true
gcd :: s:0' -> s:0' -> s:0'
if :: false:true -> s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
gen_s:0'3_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
minus, le, gcd

They will be analysed ascendingly in the following order:
minus < gcd
le < gcd

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0') -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0') -> false
le(0', Y) -> true
gcd(0', Y) -> 0'
gcd(s(X), 0') -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))

Types:
minus :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
pred :: s:0' -> s:0'
0' :: s:0'
le :: s:0' -> s:0' -> false:true
false :: false:true
true :: false:true
gcd :: s:0' -> s:0' -> s:0'
if :: false:true -> s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
le, gcd

They will be analysed ascendingly in the following order:
le < gcd

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_s:0'3_0(+(1, n3087_0)), gen_s:0'3_0(n3087_0)) -> false, rt in Omega(1 + n3087_0)

Induction Base:
le(gen_s:0'3_0(+(1, 0)), gen_s:0'3_0(0)) ->_R^Omega(1)
false

Induction Step:
le(gen_s:0'3_0(+(1, +(n3087_0, 1))), gen_s:0'3_0(+(n3087_0, 1))) ->_R^Omega(1)
le(gen_s:0'3_0(+(1, n3087_0)), gen_s:0'3_0(n3087_0)) ->_IH
false

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(24)
Obligation:
Innermost TRS:
Rules:
minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0') -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0') -> false
le(0', Y) -> true
gcd(0', Y) -> 0'
gcd(s(X), 0') -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))

Types:
minus :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
pred :: s:0' -> s:0'
0' :: s:0'
le :: s:0' -> s:0' -> false:true
false :: false:true
true :: false:true
gcd :: s:0' -> s:0' -> s:0'
if :: false:true -> s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
le(gen_s:0'3_0(+(1, n3087_0)), gen_s:0'3_0(n3087_0)) -> false, rt in Omega(1 + n3087_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
gcd