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


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


WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 2 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 602 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), 291 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), 46 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:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   minus(x, 0) -> x
   minus(0, x) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, x, y) -> gcd(minus(x, y), y)
   if_gcd(false, x, y) -> gcd(minus(y, x), 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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(x, 0) -> x [1]
   minus(0, x) -> 0 [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   gcd(0, y) -> y [1]
   gcd(s(x), 0) -> s(x) [1]
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1]
   if_gcd(true, x, y) -> gcd(minus(x, y), y) [1]
   if_gcd(false, x, y) -> gcd(minus(y, x), 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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(x, 0) -> x [1]
   minus(0, x) -> 0 [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   gcd(0, y) -> y [1]
   gcd(s(x), 0) -> s(x) [1]
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1]
   if_gcd(true, x, y) -> gcd(minus(x, y), y) [1]
   if_gcd(false, x, y) -> gcd(minus(y, x), x) [1]

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

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:
none

And the following fresh constants: none

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

(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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(x, 0) -> x [1]
   minus(0, x) -> 0 [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   gcd(0, y) -> y [1]
   gcd(s(x), 0) -> s(x) [1]
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1]
   if_gcd(true, x, y) -> gcd(minus(x, y), y) [1]
   if_gcd(false, x, y) -> gcd(minus(y, x), x) [1]

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

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
   true => 1
   false => 0

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

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

   gcd(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   gcd(z, z') -{ 1 }-> if_gcd(le(y, x), 1 + x, 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   gcd(z, z') -{ 1 }-> 1 + x :|: x >= 0, z = 1 + x, z' = 0
   if_gcd(z, z', z'') -{ 1 }-> gcd(minus(x, y), y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if_gcd(z, z', z'') -{ 1 }-> gcd(minus(y, x), x) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
   le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0

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, V15),0,[le(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V15),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V15),0,[gcd(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V15),0,[fun(V1, V, V15, Out)],[V1 >= 0,V >= 0,V15 >= 0]).
eq(le(V1, V, Out),1,[],[Out = 1,V2 >= 0,V1 = 0,V = V2]).
eq(le(V1, V, Out),1,[],[Out = 0,V3 >= 0,V1 = 1 + V3,V = 0]).
eq(le(V1, V, Out),1,[le(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]).
eq(minus(V1, V, Out),1,[],[Out = V6,V6 >= 0,V1 = V6,V = 0]).
eq(minus(V1, V, Out),1,[],[Out = 0,V = V7,V7 >= 0,V1 = 0]).
eq(minus(V1, V, Out),1,[minus(V8, V9, Ret1)],[Out = Ret1,V = 1 + V9,V8 >= 0,V9 >= 0,V1 = 1 + V8]).
eq(gcd(V1, V, Out),1,[],[Out = V10,V10 >= 0,V1 = 0,V = V10]).
eq(gcd(V1, V, Out),1,[],[Out = 1 + V11,V11 >= 0,V1 = 1 + V11,V = 0]).
eq(gcd(V1, V, Out),1,[le(V12, V13, Ret0),fun(Ret0, 1 + V13, 1 + V12, Ret2)],[Out = Ret2,V = 1 + V12,V13 >= 0,V12 >= 0,V1 = 1 + V13]).
eq(fun(V1, V, V15, Out),1,[minus(V16, V14, Ret01),gcd(Ret01, V14, Ret3)],[Out = Ret3,V = V16,V15 = V14,V1 = 1,V16 >= 0,V14 >= 0]).
eq(fun(V1, V, V15, Out),1,[minus(V18, V17, Ret02),gcd(Ret02, V17, Ret4)],[Out = Ret4,V = V17,V15 = V18,V17 >= 0,V18 >= 0,V1 = 0]).
input_output_vars(le(V1,V,Out),[V1,V],[Out]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(gcd(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun(V1,V,V15,Out),[V1,V,V15],[Out]).


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

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

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

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

### Specialization of cost equations le/3 
* CE 15 is refined into CE [16] 
* CE 14 is refined into CE [17] 
* CE 13 is refined into CE [18] 


### Cost equations --> "Loop" of le/3 
* CEs [17] --> Loop 13 
* CEs [18] --> Loop 14 
* CEs [16] --> Loop 15 

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

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


### Specialization of cost equations minus/3 
* CE 8 is refined into CE [19] 
* CE 6 is refined into CE [20] 
* CE 7 is refined into CE [21] 


### Cost equations --> "Loop" of minus/3 
* CEs [20] --> Loop 16 
* CEs [21] --> Loop 17 
* CEs [19] --> Loop 18 

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

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


### Specialization of cost equations gcd/3 
* CE 12 is refined into CE [22] 
* CE 11 is refined into CE [23] 
* CE 10 is refined into CE [24,25,26,27] 
* CE 9 is refined into CE [28,29] 


### Cost equations --> "Loop" of gcd/3 
* CEs [29] --> Loop 19 
* CEs [26,27] --> Loop 20 
* CEs [28] --> Loop 21 
* CEs [24,25] --> Loop 22 
* CEs [22] --> Loop 23 
* CEs [23] --> Loop 24 

### Ranking functions of CR gcd(V1,V,Out) 
* RF of phase [19,20]: [V1+V-3]
* RF of phase [22]: [V1]

#### Partial ranking functions of CR gcd(V1,V,Out) 
* Partial RF of phase [19,20]:
  - RF of loop [19:1]:
    V-2
    V1/2+V/2-2
  - RF of loop [20:1]:
    V1-1 depends on loops [19:1] 
    V1-V+1 depends on loops [19:1] 
* Partial RF of phase [22]:
  - RF of loop [22:1]:
    V1


### Specialization of cost equations start/3 
* CE 2 is refined into CE [30,31,32,33,34,35,36,37,38] 
* CE 1 is refined into CE [39,40,41,42,43,44,45,46,47] 
* CE 3 is refined into CE [48,49,50,51] 
* CE 4 is refined into CE [52,53,54,55] 
* CE 5 is refined into CE [56,57,58,59,60,61] 


### Cost equations --> "Loop" of start/3 
* CEs [60,61] --> Loop 25 
* CEs [50,55,59] --> Loop 26 
* CEs [35] --> Loop 27 
* CEs [33,34] --> Loop 28 
* CEs [36,37,38,57] --> Loop 29 
* CEs [32,51,54] --> Loop 30 
* CEs [30,31] --> Loop 31 
* CEs [43] --> Loop 32 
* CEs [44] --> Loop 33 
* CEs [45] --> Loop 34 
* CEs [41,49,53,58] --> Loop 35 
* CEs [39,40,42,46,47,48,52,56] --> Loop 36 

### Ranking functions of CR start(V1,V,V15) 

#### Partial ranking functions of CR start(V1,V,V15) 


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

#### Cost of chains of le(V1,V,Out):
* Chain [[15],14]: 1*it(15)+1
  Such that:it(15) =< V1

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

* Chain [[15],13]: 1*it(15)+1
  Such that:it(15) =< V

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

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

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


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

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

* Chain [[18],16]: 1*it(18)+1
  Such that:it(18) =< V

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

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

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


#### Cost of chains of gcd(V1,V,Out):
* Chain [[22],24]: 6*it(22)+1
  Such that:it(22) =< V1

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

* Chain [[19,20],24]: 4*it(19)+4*it(20)+2*s(13)+4*s(15)+1
  Such that:aux(9) =< V1-V+1
aux(21) =< V1+V
aux(22) =< V1+V-Out
it(19) =< V1/2+V/2
aux(24) =< V1/2+V/2-Out/2
aux(25) =< V
aux(26) =< V-Out
aux(8) =< 2*V-2*Out
aux(27) =< V1
it(19) =< aux(21)
it(20) =< aux(21)
s(14) =< aux(21)
it(19) =< aux(22)
it(20) =< aux(22)
s(14) =< aux(22)
it(19) =< aux(24)
it(20) =< aux(24)
aux(6) =< aux(25)
it(19) =< aux(25)
aux(6) =< aux(26)
it(19) =< aux(26)
it(20) =< aux(8)+aux(9)
it(20) =< aux(6)+aux(27)
s(16) =< aux(6)+aux(27)
s(16) =< it(20)*aux(25)
s(15) =< s(16)
s(13) =< s(14)

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

* Chain [[19,20],21,[22],24]: 4*it(19)+4*it(20)+6*it(22)+2*s(13)+4*s(15)+1*s(17)+5
  Such that:s(17) =< 1
aux(9) =< V1-V+1
it(19) =< V1/2+V/2
aux(8) =< 2*V
aux(28) =< V1
aux(29) =< V1+V
aux(30) =< V
it(22) =< aux(30)
it(19) =< aux(29)
it(20) =< aux(29)
it(19) =< aux(30)
it(20) =< aux(8)+aux(9)
it(20) =< aux(30)+aux(28)
s(16) =< aux(30)+aux(28)
s(16) =< it(20)*aux(30)
s(15) =< s(16)
s(13) =< aux(29)

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

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

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

* Chain [21,[22],24]: 6*it(22)+1*s(17)+5
  Such that:s(17) =< 1
it(22) =< V

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


#### Cost of chains of start(V1,V,V15):
* Chain [36]: 5*s(18)+8*s(19)+1*s(20)+8*s(22)+4*s(28)+4*s(30)+4*s(42)+4*s(46)+7
  Such that:s(20) =< 1
aux(35) =< -2*V+V15+1
aux(36) =< -V+V15
aux(37) =< V
aux(38) =< 2*V
aux(39) =< V15
aux(40) =< V15/2
s(18) =< aux(39)
s(22) =< aux(40)
s(19) =< aux(37)
s(22) =< aux(39)
s(28) =< aux(39)
s(22) =< aux(37)
s(28) =< aux(38)+aux(35)
s(28) =< aux(37)+aux(36)
s(29) =< aux(37)+aux(36)
s(29) =< s(28)*aux(37)
s(30) =< s(29)
s(42) =< aux(39)
s(42) =< aux(40)
s(42) =< aux(38)+aux(35)
s(42) =< aux(37)+aux(36)
s(45) =< aux(37)+aux(36)
s(45) =< s(42)*aux(37)
s(46) =< s(45)

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

* Chain [35]: 3
  with precondition: [V=0,V1>=0] 

* Chain [34]: 1*s(48)+6*s(49)+3
  Such that:s(48) =< 1
s(49) =< V15

  with precondition: [V1=0,V=1,V15>=2] 

* Chain [33]: 7*s(50)+1*s(51)+7
  Such that:s(51) =< 1
aux(41) =< V
s(50) =< aux(41)

  with precondition: [V1=0,V+1=V15,V>=2] 

* Chain [32]: 1*s(53)+3
  Such that:s(53) =< V15

  with precondition: [V1=0,V=V15,V>=1] 

* Chain [31]: 3
  with precondition: [V1=1,V=0,V15>=0] 

* Chain [30]: 2*s(54)+3
  Such that:aux(42) =< V1
s(54) =< aux(42)

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

* Chain [29]: 3*s(56)+16*s(57)+8*s(58)+8*s(61)+4*s(67)+4*s(69)+4*s(81)+4*s(85)+7
  Such that:aux(47) =< 1
aux(48) =< V
aux(49) =< V-2*V15+1
aux(50) =< V-V15
aux(51) =< V/2
aux(52) =< V15
aux(53) =< 2*V15
s(56) =< aux(47)
s(57) =< aux(48)
s(61) =< aux(51)
s(58) =< aux(52)
s(61) =< aux(48)
s(67) =< aux(48)
s(61) =< aux(52)
s(67) =< aux(53)+aux(49)
s(67) =< aux(52)+aux(50)
s(68) =< aux(52)+aux(50)
s(68) =< s(67)*aux(52)
s(69) =< s(68)
s(81) =< aux(48)
s(81) =< aux(51)
s(81) =< aux(53)+aux(49)
s(81) =< aux(52)+aux(50)
s(84) =< aux(52)+aux(50)
s(84) =< s(81)*aux(52)
s(85) =< s(84)

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

* Chain [28]: 1*s(89)+1*s(90)+3
  Such that:s(89) =< V
s(90) =< V15

  with precondition: [V1=1,V>=1,V15>=V] 

* Chain [27]: 7*s(91)+1*s(92)+7
  Such that:s(92) =< 1
aux(54) =< V
s(91) =< aux(54)

  with precondition: [V1=1,V=V15+1,V>=3] 

* Chain [26]: 2*s(94)+6*s(96)+1
  Such that:s(96) =< V1
aux(55) =< V
s(94) =< aux(55)

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

* Chain [25]: 1*s(97)+8*s(99)+6*s(104)+4*s(105)+4*s(107)+4*s(108)+4*s(118)+4*s(122)+5
  Such that:s(97) =< 1
aux(59) =< V1
aux(60) =< V1-V+1
aux(61) =< V1+V
aux(62) =< V1/2+V/2
aux(63) =< V
aux(64) =< 2*V
s(99) =< aux(62)
s(104) =< aux(63)
s(99) =< aux(61)
s(105) =< aux(61)
s(99) =< aux(63)
s(105) =< aux(64)+aux(60)
s(105) =< aux(63)+aux(59)
s(106) =< aux(63)+aux(59)
s(106) =< s(105)*aux(63)
s(107) =< s(106)
s(108) =< aux(61)
s(118) =< aux(61)
s(118) =< aux(62)
s(118) =< aux(64)+aux(60)
s(118) =< aux(63)+aux(59)
s(121) =< aux(63)+aux(59)
s(121) =< s(118)*aux(63)
s(122) =< s(121)

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


Closed-form bounds of start(V1,V,V15): 
-------------------------------------
* Chain [36] with precondition: [V1=0,V>=0] 
    - Upper bound: 16*V+8+nat(V15)*13+nat(-V+V15)*8+nat(V15/2)*8 
    - Complexity: n 
* Chain [35] with precondition: [V=0,V1>=0] 
    - Upper bound: 3 
    - Complexity: constant 
* Chain [34] with precondition: [V1=0,V=1,V15>=2] 
    - Upper bound: 6*V15+4 
    - Complexity: n 
* Chain [33] with precondition: [V1=0,V+1=V15,V>=2] 
    - Upper bound: 7*V+8 
    - Complexity: n 
* Chain [32] with precondition: [V1=0,V=V15,V>=1] 
    - Upper bound: V15+3 
    - Complexity: n 
* Chain [31] with precondition: [V1=1,V=0,V15>=0] 
    - Upper bound: 3 
    - Complexity: constant 
* Chain [30] with precondition: [V1>=1,V>=V1] 
    - Upper bound: 2*V1+3 
    - Complexity: n 
* Chain [29] with precondition: [V1=1,V>=2] 
    - Upper bound: 24*V+10+nat(V15)*16+nat(V-V15)*8+4*V 
    - Complexity: n 
* Chain [28] with precondition: [V1=1,V>=1,V15>=V] 
    - Upper bound: V+V15+3 
    - Complexity: n 
* Chain [27] with precondition: [V1=1,V=V15+1,V>=3] 
    - Upper bound: 7*V+8 
    - Complexity: n 
* Chain [26] with precondition: [V>=1,V1>=V] 
    - Upper bound: 6*V1+2*V+1 
    - Complexity: n 
* Chain [25] with precondition: [V1>=2,V>=2] 
    - Upper bound: 24*V1+30*V+6 
    - Complexity: n 

### Maximum cost of start(V1,V,V15): max([max([2*V1,nat(V15)*5+1+nat(V15)])+2,max([max([6*V1,5*V+5+max([2,7*V+max([24*V1+16*V,2*V+2+nat(V15)*13+max([nat(V15/2)*8+nat(-V+V15)*8,8*V+2+nat(V15)*3+nat(V-V15)*8+4*V])])])])+V,nat(V15)+2])+V])+1 
Asymptotic class: n 
* Total analysis performed in 499 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:

   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   minus(x, 0') -> x
   minus(0', x) -> 0'
   minus(s(x), s(y)) -> minus(x, y)
   gcd(0', y) -> y
   gcd(s(x), 0') -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, x, y) -> gcd(minus(x, y), y)
   if_gcd(false, x, y) -> gcd(minus(y, x), x)

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

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

(14)
Obligation:
Innermost TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(0', x) -> 0'
minus(s(x), s(y)) -> minus(x, y)
gcd(0', y) -> y
gcd(s(x), 0') -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
minus :: 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s
if_gcd :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s

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

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

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

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

(16)
Obligation:
Innermost TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(0', x) -> 0'
minus(s(x), s(y)) -> minus(x, y)
gcd(0', y) -> y
gcd(s(x), 0') -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
minus :: 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s
if_gcd :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
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:
le, minus, gcd

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

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

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

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

Induction Step:
le(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
le(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).
----------------------------------------

(18)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(0', x) -> 0'
minus(s(x), s(y)) -> minus(x, y)
gcd(0', y) -> y
gcd(s(x), 0') -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
minus :: 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s
if_gcd :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
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:
le, minus, gcd

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

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(0', x) -> 0'
minus(s(x), s(y)) -> minus(x, y)
gcd(0', y) -> y
gcd(s(x), 0') -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
minus :: 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s
if_gcd :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
le(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:
minus, gcd

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

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s3_0(n306_0), gen_0':s3_0(n306_0)) -> gen_0':s3_0(0), rt in Omega(1 + n306_0)

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

Induction Step:
minus(gen_0':s3_0(+(n306_0, 1)), gen_0':s3_0(+(n306_0, 1))) ->_R^Omega(1)
minus(gen_0':s3_0(n306_0), gen_0':s3_0(n306_0)) ->_IH
gen_0':s3_0(0)

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

(24)
Obligation:
Innermost TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(0', x) -> 0'
minus(s(x), s(y)) -> minus(x, y)
gcd(0', y) -> y
gcd(s(x), 0') -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
minus :: 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s
if_gcd :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
minus(gen_0':s3_0(n306_0), gen_0':s3_0(n306_0)) -> gen_0':s3_0(0), rt in Omega(1 + n306_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:
gcd