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


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WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 542 ms]
    (12) BOUNDS(1, n^2)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) typed CpxTrs
    (17) OrderProof [LOWER BOUND(ID), 0 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 290 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 27 ms]
            (26) typed CpxTrs


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

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


The TRS R consists of the following rules:

   diff(x, y) -> cond1(equal(x, y), x, y)
   cond1(true, x, y) -> 0
   cond1(false, x, y) -> cond2(gt(x, y), x, y)
   cond2(true, x, y) -> s(diff(x, s(y)))
   cond2(false, x, y) -> s(diff(s(x), y))
   gt(0, v) -> false
   gt(s(u), 0) -> true
   gt(s(u), s(v)) -> gt(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: FULL
----------------------------------------

(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

The duplicating contexts are:
diff([], y)
diff(x, [])
cond1(false, [], y)
cond1(false, x, [])


The defined contexts are:
cond2([], x1, x2)
cond1([], x1, x2)


[] just represents basic- or constructor-terms in the following defined contexts:
cond2([], x1, x2)
cond1([], x1, x2)


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
----------------------------------------

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


The TRS R consists of the following rules:

   diff(x, y) -> cond1(equal(x, y), x, y)
   cond1(true, x, y) -> 0
   cond1(false, x, y) -> cond2(gt(x, y), x, y)
   cond2(true, x, y) -> s(diff(x, s(y)))
   cond2(false, x, y) -> s(diff(s(x), y))
   gt(0, v) -> false
   gt(s(u), 0) -> true
   gt(s(u), s(v)) -> gt(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: INNERMOST
----------------------------------------

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   diff(x, y) -> cond1(equal(x, y), x, y) [1]
   cond1(true, x, y) -> 0 [1]
   cond1(false, x, y) -> cond2(gt(x, y), x, y) [1]
   cond2(true, x, y) -> s(diff(x, s(y))) [1]
   cond2(false, x, y) -> s(diff(s(x), y)) [1]
   gt(0, v) -> false [1]
   gt(s(u), 0) -> true [1]
   gt(s(u), s(v)) -> gt(u, v) [1]
   equal(0, 0) -> true [1]
   equal(s(x), 0) -> false [1]
   equal(0, s(y)) -> false [1]
   equal(s(x), s(y)) -> equal(x, y) [1]

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

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   diff(x, y) -> cond1(equal(x, y), x, y) [1]
   cond1(true, x, y) -> 0 [1]
   cond1(false, x, y) -> cond2(gt(x, y), x, y) [1]
   cond2(true, x, y) -> s(diff(x, s(y))) [1]
   cond2(false, x, y) -> s(diff(s(x), y)) [1]
   gt(0, v) -> false [1]
   gt(s(u), 0) -> true [1]
   gt(s(u), s(v)) -> gt(u, v) [1]
   equal(0, 0) -> true [1]
   equal(s(x), 0) -> false [1]
   equal(0, s(y)) -> false [1]
   equal(s(x), s(y)) -> equal(x, y) [1]

The TRS has the following type information:
diff :: 0:s -> 0:s -> 0:s
cond1 :: true:false -> 0:s -> 0:s -> 0:s
equal :: 0:s -> 0:s -> true:false
true :: true:false
0 :: 0:s
false :: true:false
cond2 :: true:false -> 0:s -> 0:s -> 0:s
gt :: 0:s -> 0:s -> true:false
s :: 0:s -> 0:s

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

(7) 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:
none

And the following fresh constants: none

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

(8)
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:

   diff(x, y) -> cond1(equal(x, y), x, y) [1]
   cond1(true, x, y) -> 0 [1]
   cond1(false, x, y) -> cond2(gt(x, y), x, y) [1]
   cond2(true, x, y) -> s(diff(x, s(y))) [1]
   cond2(false, x, y) -> s(diff(s(x), y)) [1]
   gt(0, v) -> false [1]
   gt(s(u), 0) -> true [1]
   gt(s(u), s(v)) -> gt(u, v) [1]
   equal(0, 0) -> true [1]
   equal(s(x), 0) -> false [1]
   equal(0, s(y)) -> false [1]
   equal(s(x), s(y)) -> equal(x, y) [1]

The TRS has the following type information:
diff :: 0:s -> 0:s -> 0:s
cond1 :: true:false -> 0:s -> 0:s -> 0:s
equal :: 0:s -> 0:s -> true:false
true :: true:false
0 :: 0:s
false :: true:false
cond2 :: true:false -> 0:s -> 0:s -> 0:s
gt :: 0:s -> 0:s -> true:false
s :: 0:s -> 0:s

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

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

   true => 1
   0 => 0
   false => 0

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   cond1(z, z', z'') -{ 1 }-> cond2(gt(x, y), x, y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   cond1(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   cond2(z, z', z'') -{ 1 }-> 1 + diff(x, 1 + y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   cond2(z, z', z'') -{ 1 }-> 1 + diff(1 + x, y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   diff(z, z') -{ 1 }-> cond1(equal(x, y), x, y) :|: x >= 0, y >= 0, z = x, z' = y
   equal(z, z') -{ 1 }-> equal(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   equal(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   equal(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   equal(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0
   gt(z, z') -{ 1 }-> gt(u, v) :|: v >= 0, z' = 1 + v, z = 1 + u, u >= 0
   gt(z, z') -{ 1 }-> 1 :|: z = 1 + u, z' = 0, u >= 0
   gt(z, z') -{ 1 }-> 0 :|: v >= 0, z' = v, z = 0

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V, V5),0,[diff(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V5),0,[cond1(V1, V, V5, Out)],[V1 >= 0,V >= 0,V5 >= 0]).
eq(start(V1, V, V5),0,[cond2(V1, V, V5, Out)],[V1 >= 0,V >= 0,V5 >= 0]).
eq(start(V1, V, V5),0,[gt(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V5),0,[equal(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(diff(V1, V, Out),1,[equal(V3, V2, Ret0),cond1(Ret0, V3, V2, Ret)],[Out = Ret,V3 >= 0,V2 >= 0,V1 = V3,V = V2]).
eq(cond1(V1, V, V5, Out),1,[],[Out = 0,V = V4,V5 = V6,V1 = 1,V4 >= 0,V6 >= 0]).
eq(cond1(V1, V, V5, Out),1,[gt(V8, V7, Ret01),cond2(Ret01, V8, V7, Ret1)],[Out = Ret1,V = V8,V5 = V7,V8 >= 0,V7 >= 0,V1 = 0]).
eq(cond2(V1, V, V5, Out),1,[diff(V9, 1 + V10, Ret11)],[Out = 1 + Ret11,V = V9,V5 = V10,V1 = 1,V9 >= 0,V10 >= 0]).
eq(cond2(V1, V, V5, Out),1,[diff(1 + V12, V11, Ret12)],[Out = 1 + Ret12,V = V12,V5 = V11,V12 >= 0,V11 >= 0,V1 = 0]).
eq(gt(V1, V, Out),1,[],[Out = 0,V13 >= 0,V = V13,V1 = 0]).
eq(gt(V1, V, Out),1,[],[Out = 1,V1 = 1 + V14,V = 0,V14 >= 0]).
eq(gt(V1, V, Out),1,[gt(V16, V15, Ret2)],[Out = Ret2,V15 >= 0,V = 1 + V15,V1 = 1 + V16,V16 >= 0]).
eq(equal(V1, V, Out),1,[],[Out = 1,V1 = 0,V = 0]).
eq(equal(V1, V, Out),1,[],[Out = 0,V17 >= 0,V1 = 1 + V17,V = 0]).
eq(equal(V1, V, Out),1,[],[Out = 0,V = 1 + V18,V18 >= 0,V1 = 0]).
eq(equal(V1, V, Out),1,[equal(V20, V19, Ret3)],[Out = Ret3,V = 1 + V19,V20 >= 0,V19 >= 0,V1 = 1 + V20]).
input_output_vars(diff(V1,V,Out),[V1,V],[Out]).
input_output_vars(cond1(V1,V,V5,Out),[V1,V,V5],[Out]).
input_output_vars(cond2(V1,V,V5,Out),[V1,V,V5],[Out]).
input_output_vars(gt(V1,V,Out),[V1,V],[Out]).
input_output_vars(equal(V1,V,Out),[V1,V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [equal/3]
1. recursive  : [gt/3]
2. recursive  : [cond1/4,cond2/4,diff/3]
3. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into equal/3
1. SCC is partially evaluated into gt/3
2. SCC is partially evaluated into diff/3
3. 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 [19] 
* CE 16 is refined into CE [20] 
* CE 17 is refined into CE [21] 
* CE 15 is refined into CE [22] 


### Cost equations --> "Loop" of equal/3 
* CEs [20] --> Loop 13 
* CEs [21] --> Loop 14 
* CEs [22] --> Loop 15 
* CEs [19] --> Loop 16 

### Ranking functions of CR equal(V1,V,Out) 
* RF of phase [16]: [V,V1]

#### Partial ranking functions of CR equal(V1,V,Out) 
* Partial RF of phase [16]:
  - RF of loop [16:1]:
    V
    V1


### Specialization of cost equations gt/3 
* CE 11 is refined into CE [23] 
* CE 10 is refined into CE [24] 
* CE 9 is refined into CE [25] 


### Cost equations --> "Loop" of gt/3 
* CEs [24] --> Loop 17 
* CEs [25] --> Loop 18 
* CEs [23] --> Loop 19 

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

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


### Specialization of cost equations diff/3 
* CE 13 is refined into CE [26,27] 
* CE 14 is refined into CE [28,29] 
* CE 12 is refined into CE [30,31] 


### Cost equations --> "Loop" of diff/3 
* CEs [31] --> Loop 20 
* CEs [30] --> Loop 21 
* CEs [27] --> Loop 22 
* CEs [29] --> Loop 23 
* CEs [26] --> Loop 24 
* CEs [28] --> Loop 25 

### Ranking functions of CR diff(V1,V,Out) 
* RF of phase [22]: [V1-V]
* RF of phase [23]: [-V1+V]

#### Partial ranking functions of CR diff(V1,V,Out) 
* Partial RF of phase [22]:
  - RF of loop [22:1]:
    V1-V
* Partial RF of phase [23]:
  - RF of loop [23:1]:
    -V1+V


### Specialization of cost equations start/3 
* CE 1 is refined into CE [32] 
* CE 5 is refined into CE [33,34,35,36,37] 
* CE 2 is refined into CE [38,39,40,41] 
* CE 3 is refined into CE [42,43,44,45,46,47] 
* CE 4 is refined into CE [48,49,50,51,52] 
* CE 6 is refined into CE [53,54,55,56,57,58,59,60] 
* CE 7 is refined into CE [61,62,63,64] 
* CE 8 is refined into CE [65,66,67,68,69,70] 


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

### Ranking functions of CR start(V1,V,V5) 

#### Partial ranking functions of CR start(V1,V,V5) 


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

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

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

* Chain [[16],14]: 1*it(16)+1
  Such that:it(16) =< V1

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

* Chain [[16],13]: 1*it(16)+1
  Such that:it(16) =< V

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

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

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

* Chain [13]: 1
  with precondition: [V=0,Out=0,V1>=1] 


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

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

* Chain [[19],17]: 1*it(19)+1
  Such that:it(19) =< V

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

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

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


#### Cost of chains of diff(V1,V,Out):
* Chain [[23],20]: 5*it(23)+1*s(1)+2*s(6)+3
  Such that:it(23) =< Out
aux(3) =< V
s(1) =< aux(3)
s(7) =< it(23)*aux(3)
s(6) =< s(7)

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

* Chain [[22],20]: 5*it(22)+1*s(1)+2*s(12)+3
  Such that:it(22) =< Out
aux(6) =< V+Out
s(1) =< aux(6)
s(13) =< it(22)*aux(6)
s(12) =< s(13)

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

* Chain [25,[23],20]: 6*it(23)+2*s(6)+8
  Such that:aux(7) =< Out
it(23) =< aux(7)
s(7) =< it(23)*aux(7)
s(6) =< s(7)

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

* Chain [25,20]: 1*s(1)+8
  Such that:s(1) =< 1

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

* Chain [24,[22],20]: 6*it(22)+2*s(12)+8
  Such that:aux(8) =< Out
it(22) =< aux(8)
s(13) =< it(22)*aux(8)
s(12) =< s(13)

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

* Chain [24,20]: 1*s(1)+8
  Such that:s(1) =< 1

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

* Chain [21]: 3
  with precondition: [V1=0,V=0,Out=0] 

* Chain [20]: 1*s(1)+3
  Such that:s(1) =< V

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


#### Cost of chains of start(V1,V,V5):
* Chain [35]: 9*s(14)+5*s(15)+8*s(17)+2*s(19)+3*s(20)+2*s(26)+10*s(28)+4*s(32)+5*s(34)+1*s(36)+2*s(38)+2*s(47)+11
  Such that:s(35) =< V+1
s(15) =< V-V5
s(34) =< V-V5+1
aux(10) =< 1
aux(11) =< -V+V5
aux(12) =< V
aux(13) =< V5
s(20) =< aux(10)
s(28) =< aux(11)
s(17) =< aux(12)
s(14) =< aux(13)
s(31) =< s(28)*aux(13)
s(32) =< s(31)
s(18) =< s(15)*aux(12)
s(19) =< s(18)
s(46) =< s(17)*aux(12)
s(47) =< s(46)
s(36) =< s(35)
s(37) =< s(34)*s(35)
s(38) =< s(37)
s(25) =< s(14)*aux(13)
s(26) =< s(25)

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

* Chain [34]: 10
  with precondition: [V1=0,V=1] 

* Chain [33]: 6*s(50)+2*s(54)+6*s(56)+2*s(58)+9
  Such that:aux(15) =< V
s(55) =< V+1
s(50) =< aux(15)
s(53) =< s(50)*aux(15)
s(54) =< s(53)
s(56) =< s(55)
s(57) =< s(56)*s(55)
s(58) =< s(57)

  with precondition: [V1=0,V5=0,V>=1] 

* Chain [32]: 2*s(59)+1*s(61)+6
  Such that:s(61) =< V+1
aux(16) =< V5
s(59) =< aux(16)

  with precondition: [V1=0,V+1=V5,V>=0] 

* Chain [31]: 1*s(62)+5*s(63)+1*s(65)+2*s(67)+6
  Such that:s(63) =< 1
s(62) =< V5
s(64) =< V5+1
s(65) =< s(64)
s(66) =< s(63)*s(64)
s(67) =< s(66)

  with precondition: [V1=0,V=V5,V>=1] 

* Chain [30]: 1*s(68)+1*s(69)+6
  Such that:s(68) =< V5
s(69) =< V5+1

  with precondition: [V1=0,V=V5+1,V>=2] 

* Chain [29]: 2*s(70)+6*s(72)+2*s(74)+6*s(77)+2*s(79)+9
  Such that:s(76) =< V1
s(71) =< V5+1
aux(17) =< 1
s(70) =< aux(17)
s(77) =< s(76)
s(78) =< s(77)*s(76)
s(79) =< s(78)
s(72) =< s(71)
s(73) =< s(72)*s(71)
s(74) =< s(73)

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

* Chain [28]: 2*s(80)+5*s(81)+1*s(83)+2*s(85)+5*s(86)+2*s(90)+4
  Such that:s(86) =< -V+V5+1
s(82) =< V
s(81) =< V-V5
aux(18) =< V5+1
s(80) =< aux(18)
s(89) =< s(86)*aux(18)
s(90) =< s(89)
s(83) =< s(82)
s(84) =< s(81)*s(82)
s(85) =< s(84)

  with precondition: [V1=1,V>=0,V5>=0] 

* Chain [27]: 3*s(91)+5*s(92)+2*s(96)+2*s(97)+3
  Such that:s(92) =< -V1+V
aux(19) =< V1
aux(20) =< V
s(97) =< aux(19)
s(91) =< aux(20)
s(95) =< s(92)*aux(20)
s(96) =< s(95)

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

* Chain [26]: 5*s(100)+1*s(102)+2*s(104)+2*s(105)+3
  Such that:s(101) =< V1
s(100) =< V1-V
aux(21) =< V
s(105) =< aux(21)
s(102) =< s(101)
s(103) =< s(100)*s(101)
s(104) =< s(103)

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


Closed-form bounds of start(V1,V,V5): 
-------------------------------------
* Chain [35] with precondition: [V1=0,V>=0] 
    - Upper bound: 8*V+14+2*V*V+2*V*nat(V-V5)+nat(V5)*9+nat(V5)*2*nat(V5)+nat(V5)*4*nat(-V+V5)+(V+1)+(2*V+2)*nat(V-V5+1)+nat(-V+V5)*10+nat(V-V5+1)*5+nat(V-V5)*5 
    - Complexity: n^2 
* Chain [34] with precondition: [V1=0,V=1] 
    - Upper bound: 10 
    - Complexity: constant 
* Chain [33] with precondition: [V1=0,V5=0,V>=1] 
    - Upper bound: 6*V+9+2*V*V+(6*V+6)+(2*V+2)*(V+1) 
    - Complexity: n^2 
* Chain [32] with precondition: [V1=0,V+1=V5,V>=0] 
    - Upper bound: V+2*V5+7 
    - Complexity: n 
* Chain [31] with precondition: [V1=0,V=V5,V>=1] 
    - Upper bound: 4*V5+14 
    - Complexity: n 
* Chain [30] with precondition: [V1=0,V=V5+1,V>=2] 
    - Upper bound: 2*V5+7 
    - Complexity: n 
* Chain [29] with precondition: [V=0,V1>=1] 
    - Upper bound: 6*V1+11+2*V1*V1+nat(V5+1)*6+nat(V5+1)*2*nat(V5+1) 
    - Complexity: n^2 
* Chain [28] with precondition: [V1=1,V>=0,V5>=0] 
    - Upper bound: V+4+2*V*nat(V-V5)+(2*V5+2)+(2*V5+2)*nat(-V+V5+1)+nat(-V+V5+1)*5+nat(V-V5)*5 
    - Complexity: n^2 
* Chain [27] with precondition: [V1>=1,V>=V1] 
    - Upper bound: 2*V1+3*V+3+(-V1+V)*(2*V)+(-5*V1+5*V) 
    - Complexity: n^2 
* Chain [26] with precondition: [V>=1,V1>=V+1] 
    - Upper bound: 5*V1-5*V+(V1+3+(V1-V)*(2*V1)+2*V) 
    - Complexity: n^2 

### Maximum cost of start(V1,V,V5): max([max([max([max([2*V*nat(-V1+V)+2*V1+nat(-V1+V)*5,3*V+6+2*V*V+(V+1)+max([5*V+5+(2*V+2)*(V+1),2*V+5+2*V*nat(V-V5)+nat(V5)*9+nat(V5)*2*nat(V5)+nat(V5)*4*nat(-V+V5)+(2*V+2)*nat(V-V5+1)+nat(-V+V5)*10+nat(V-V5+1)*5+nat(V-V5)*5])])+V,2*V1*nat(V1-V)+V1+nat(V1-V)*5])+V,2*V*nat(V-V5)+1+nat(V5+1)*2+nat(V5+1)*2*nat(-V+V5+1)+nat(-V+V5+1)*5+nat(V-V5)*5])+V,max([nat(V5)+max([V+1+nat(V5),nat(V5+1)*2+5+nat(V5+1)]),6*V1+1+2*V1*V1+nat(V5+1)*6+nat(V5+1)*2*nat(V5+1)+4])+3])+3 
Asymptotic class: n^2 
* Total analysis performed in 451 ms.


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

(12)
BOUNDS(1, n^2)

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

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

(14)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   diff(x, y) -> cond1(equal(x, y), x, y)
   cond1(true, x, y) -> 0'
   cond1(false, x, y) -> cond2(gt(x, y), x, y)
   cond2(true, x, y) -> s(diff(x, s(y)))
   cond2(false, x, y) -> s(diff(s(x), y))
   gt(0', v) -> false
   gt(s(u), 0') -> true
   gt(s(u), s(v)) -> gt(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: FULL
----------------------------------------

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

(16)
Obligation:
TRS:
Rules:
diff(x, y) -> cond1(equal(x, y), x, y)
cond1(true, x, y) -> 0'
cond1(false, x, y) -> cond2(gt(x, y), x, y)
cond2(true, x, y) -> s(diff(x, s(y)))
cond2(false, x, y) -> s(diff(s(x), y))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)
equal(0', 0') -> true
equal(s(x), 0') -> false
equal(0', s(y)) -> false
equal(s(x), s(y)) -> equal(x, y)

Types:
diff :: 0':s -> 0':s -> 0':s
cond1 :: true:false -> 0':s -> 0':s -> 0':s
equal :: 0':s -> 0':s -> true:false
true :: true:false
0' :: 0':s
false :: true:false
cond2 :: true:false -> 0':s -> 0':s -> 0':s
gt :: 0':s -> 0':s -> true:false
s :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s

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

(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
diff, equal, gt

They will be analysed ascendingly in the following order:
equal < diff
gt < diff

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

(18)
Obligation:
TRS:
Rules:
diff(x, y) -> cond1(equal(x, y), x, y)
cond1(true, x, y) -> 0'
cond1(false, x, y) -> cond2(gt(x, y), x, y)
cond2(true, x, y) -> s(diff(x, s(y)))
cond2(false, x, y) -> s(diff(s(x), y))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)
equal(0', 0') -> true
equal(s(x), 0') -> false
equal(0', s(y)) -> false
equal(s(x), s(y)) -> equal(x, y)

Types:
diff :: 0':s -> 0':s -> 0':s
cond1 :: true:false -> 0':s -> 0':s -> 0':s
equal :: 0':s -> 0':s -> true:false
true :: true:false
0' :: 0':s
false :: true:false
cond2 :: true:false -> 0':s -> 0':s -> 0':s
gt :: 0':s -> 0':s -> true:false
s :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
equal, diff, gt

They will be analysed ascendingly in the following order:
equal < diff
gt < diff

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
equal(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)

Induction Base:
equal(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
true

Induction Step:
equal(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
equal(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
true

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

(20)
Complex Obligation (BEST)

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

(21)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
diff(x, y) -> cond1(equal(x, y), x, y)
cond1(true, x, y) -> 0'
cond1(false, x, y) -> cond2(gt(x, y), x, y)
cond2(true, x, y) -> s(diff(x, s(y)))
cond2(false, x, y) -> s(diff(s(x), y))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)
equal(0', 0') -> true
equal(s(x), 0') -> false
equal(0', s(y)) -> false
equal(s(x), s(y)) -> equal(x, y)

Types:
diff :: 0':s -> 0':s -> 0':s
cond1 :: true:false -> 0':s -> 0':s -> 0':s
equal :: 0':s -> 0':s -> true:false
true :: true:false
0' :: 0':s
false :: true:false
cond2 :: true:false -> 0':s -> 0':s -> 0':s
gt :: 0':s -> 0':s -> true:false
s :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
equal, diff, gt

They will be analysed ascendingly in the following order:
equal < diff
gt < diff

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
TRS:
Rules:
diff(x, y) -> cond1(equal(x, y), x, y)
cond1(true, x, y) -> 0'
cond1(false, x, y) -> cond2(gt(x, y), x, y)
cond2(true, x, y) -> s(diff(x, s(y)))
cond2(false, x, y) -> s(diff(s(x), y))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)
equal(0', 0') -> true
equal(s(x), 0') -> false
equal(0', s(y)) -> false
equal(s(x), s(y)) -> equal(x, y)

Types:
diff :: 0':s -> 0':s -> 0':s
cond1 :: true:false -> 0':s -> 0':s -> 0':s
equal :: 0':s -> 0':s -> true:false
true :: true:false
0' :: 0':s
false :: true:false
cond2 :: true:false -> 0':s -> 0':s -> 0':s
gt :: 0':s -> 0':s -> true:false
s :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
equal(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
gt, diff

They will be analysed ascendingly in the following order:
gt < diff

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gt(gen_0':s3_0(n476_0), gen_0':s3_0(n476_0)) -> false, rt in Omega(1 + n476_0)

Induction Base:
gt(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
false

Induction Step:
gt(gen_0':s3_0(+(n476_0, 1)), gen_0':s3_0(+(n476_0, 1))) ->_R^Omega(1)
gt(gen_0':s3_0(n476_0), gen_0':s3_0(n476_0)) ->_IH
false

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

(26)
Obligation:
TRS:
Rules:
diff(x, y) -> cond1(equal(x, y), x, y)
cond1(true, x, y) -> 0'
cond1(false, x, y) -> cond2(gt(x, y), x, y)
cond2(true, x, y) -> s(diff(x, s(y)))
cond2(false, x, y) -> s(diff(s(x), y))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)
equal(0', 0') -> true
equal(s(x), 0') -> false
equal(0', s(y)) -> false
equal(s(x), s(y)) -> equal(x, y)

Types:
diff :: 0':s -> 0':s -> 0':s
cond1 :: true:false -> 0':s -> 0':s -> 0':s
equal :: 0':s -> 0':s -> true:false
true :: true:false
0' :: 0':s
false :: true:false
cond2 :: true:false -> 0':s -> 0':s -> 0':s
gt :: 0':s -> 0':s -> true:false
s :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
equal(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
gt(gen_0':s3_0(n476_0), gen_0':s3_0(n476_0)) -> false, rt in Omega(1 + n476_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
diff