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


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 316 ms]
    (12) BOUNDS(1, n^2)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) SlicingProof [LOWER BOUND(ID), 0 ms]
    (16) CpxTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 0 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 250 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (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:

   lt(0, s(x)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   fac(x) -> help(x, 0)
   help(x, c) -> if(lt(c, x), x, c)
   if(true, x, c) -> times(s(c), help(x, s(c)))
   if(false, x, c) -> s(0)

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

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

The duplicating contexts are:
help(x, [])
help([], c)
if(true, x, [])


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


[] just represents basic- or constructor-terms in the following defined contexts:
if([], 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:

   lt(0, s(x)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   fac(x) -> help(x, 0)
   help(x, c) -> if(lt(c, x), x, c)
   if(true, x, c) -> times(s(c), help(x, s(c)))
   if(false, x, c) -> s(0)

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:

   lt(0, s(x)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   fac(x) -> help(x, 0) [1]
   help(x, c) -> if(lt(c, x), x, c) [1]
   if(true, x, c) -> times(s(c), help(x, s(c))) [1]
   if(false, x, c) -> s(0) [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:

   lt(0, s(x)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   fac(x) -> help(x, 0) [1]
   help(x, c) -> if(lt(c, x), x, c) [1]
   if(true, x, c) -> times(s(c), help(x, s(c))) [1]
   if(false, x, c) -> s(0) [1]

The TRS has the following type information:
lt :: 0:s:times -> 0:s:times -> true:false
0 :: 0:s:times
s :: 0:s:times -> 0:s:times
true :: true:false
false :: true:false
fac :: 0:s:times -> 0:s:times
help :: 0:s:times -> 0:s:times -> 0:s:times
if :: true:false -> 0:s:times -> 0:s:times -> 0:s:times
times :: 0:s:times -> 0:s:times -> 0:s:times

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:

   lt(v0, v1) -> null_lt [0]
   if(v0, v1, v2) -> null_if [0]

And the following fresh constants:    null_lt, null_if

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

(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:

   lt(0, s(x)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   fac(x) -> help(x, 0) [1]
   help(x, c) -> if(lt(c, x), x, c) [1]
   if(true, x, c) -> times(s(c), help(x, s(c))) [1]
   if(false, x, c) -> s(0) [1]
   lt(v0, v1) -> null_lt [0]
   if(v0, v1, v2) -> null_if [0]

The TRS has the following type information:
lt :: 0:s:times:null_if -> 0:s:times:null_if -> true:false:null_lt
0 :: 0:s:times:null_if
s :: 0:s:times:null_if -> 0:s:times:null_if
true :: true:false:null_lt
false :: true:false:null_lt
fac :: 0:s:times:null_if -> 0:s:times:null_if
help :: 0:s:times:null_if -> 0:s:times:null_if -> 0:s:times:null_if
if :: true:false:null_lt -> 0:s:times:null_if -> 0:s:times:null_if -> 0:s:times:null_if
times :: 0:s:times:null_if -> 0:s:times:null_if -> 0:s:times:null_if
null_lt :: true:false:null_lt
null_if :: 0:s:times:null_if

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:

   0 => 0
   true => 2
   false => 1
   null_lt => 0
   null_if => 0

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

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

   fac(z) -{ 1 }-> help(x, 0) :|: x >= 0, z = x
   help(z, z') -{ 1 }-> if(lt(c, x), x, c) :|: c >= 0, x >= 0, z' = c, z = x
   if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' = x, c >= 0, z = 1, x >= 0, z'' = c
   if(z, z', z'') -{ 1 }-> 1 + (1 + c) + help(x, 1 + c) :|: z = 2, z' = x, c >= 0, x >= 0, z'' = c
   lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lt(z, z') -{ 1 }-> 2 :|: z' = 1 + x, x >= 0, z = 0
   lt(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

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, V9),0,[lt(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V9),0,[fac(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V9),0,[help(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V9),0,[if(V1, V, V9, Out)],[V1 >= 0,V >= 0,V9 >= 0]).
eq(lt(V1, V, Out),1,[],[Out = 2,V = 1 + V2,V2 >= 0,V1 = 0]).
eq(lt(V1, V, Out),1,[],[Out = 1,V3 >= 0,V1 = V3,V = 0]).
eq(lt(V1, V, Out),1,[lt(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]).
eq(fac(V1, Out),1,[help(V6, 0, Ret1)],[Out = Ret1,V6 >= 0,V1 = V6]).
eq(help(V1, V, Out),1,[lt(V8, V7, Ret0),if(Ret0, V7, V8, Ret2)],[Out = Ret2,V8 >= 0,V7 >= 0,V = V8,V1 = V7]).
eq(if(V1, V, V9, Out),1,[help(V10, 1 + V11, Ret11)],[Out = 2 + Ret11 + V11,V1 = 2,V = V10,V11 >= 0,V10 >= 0,V9 = V11]).
eq(if(V1, V, V9, Out),1,[],[Out = 1,V = V13,V12 >= 0,V1 = 1,V13 >= 0,V9 = V12]).
eq(lt(V1, V, Out),0,[],[Out = 0,V15 >= 0,V14 >= 0,V1 = V15,V = V14]).
eq(if(V1, V, V9, Out),0,[],[Out = 0,V17 >= 0,V9 = V18,V16 >= 0,V1 = V17,V = V16,V18 >= 0]).
input_output_vars(lt(V1,V,Out),[V1,V],[Out]).
input_output_vars(fac(V1,Out),[V1],[Out]).
input_output_vars(help(V1,V,Out),[V1,V],[Out]).
input_output_vars(if(V1,V,V9,Out),[V1,V,V9],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [lt/3]
1. recursive  : [help/3,if/4]
2. non_recursive  : [fac/2]
3. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into lt/3
1. SCC is partially evaluated into help/3
2. SCC is completely evaluated into other SCCs
3. SCC is partially evaluated into start/3

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

### Specialization of cost equations lt/3 
* CE 13 is refined into CE [14] 
* CE 11 is refined into CE [15] 
* CE 10 is refined into CE [16] 
* CE 12 is refined into CE [17] 


### Cost equations --> "Loop" of lt/3 
* CEs [17] --> Loop 10 
* CEs [14] --> Loop 11 
* CEs [15] --> Loop 12 
* CEs [16] --> Loop 13 

### Ranking functions of CR lt(V1,V,Out) 
* RF of phase [10]: [V,V1]

#### Partial ranking functions of CR lt(V1,V,Out) 
* Partial RF of phase [10]:
  - RF of loop [10:1]:
    V
    V1


### Specialization of cost equations help/3 
* CE 9 is refined into CE [18,19] 
* CE 8 is refined into CE [20,21] 
* CE 7 is refined into CE [22,23,24,25,26] 


### Cost equations --> "Loop" of help/3 
* CEs [21] --> Loop 14 
* CEs [22] --> Loop 15 
* CEs [20] --> Loop 16 
* CEs [23,24,25,26] --> Loop 17 
* CEs [19] --> Loop 18 
* CEs [18] --> Loop 19 

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

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


### Specialization of cost equations start/3 
* CE 3 is refined into CE [27,28,29,30] 
* CE 1 is refined into CE [31] 
* CE 2 is refined into CE [32] 
* CE 4 is refined into CE [33,34,35,36,37] 
* CE 5 is refined into CE [38,39,40,41,42] 
* CE 6 is refined into CE [43,44,45,46,47,48,49] 


### Cost equations --> "Loop" of start/3 
* CEs [34,46,47] --> Loop 20 
* CEs [27,28,29,30] --> Loop 21 
* CEs [32,39,44] --> Loop 22 
* CEs [31,33,35,36,37,38,40,41,42,43,45,48,49] --> Loop 23 

### Ranking functions of CR start(V1,V,V9) 

#### Partial ranking functions of CR start(V1,V,V9) 


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

#### Cost of chains of lt(V1,V,Out):
* Chain [[10],13]: 1*it(10)+1
  Such that:it(10) =< V1

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

* Chain [[10],12]: 1*it(10)+1
  Such that:it(10) =< V

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

* Chain [[10],11]: 1*it(10)+0
  Such that:it(10) =< V

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

* Chain [13]: 1
  with precondition: [V1=0,Out=2,V>=1] 

* Chain [12]: 1
  with precondition: [V=0,Out=1,V1>=0] 

* Chain [11]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of help(V1,V,Out):
* Chain [[18],17]: 3*it(18)+3*s(2)+1*s(7)+2
  Such that:it(18) =< V1-V
aux(3) =< V1
s(2) =< aux(3)
s(7) =< it(18)*aux(3)

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

* Chain [[18],14]: 3*it(18)+1*s(7)+1*s(8)+3
  Such that:it(18) =< V1-V
aux(4) =< V1
s(8) =< aux(4)
s(7) =< it(18)*aux(4)

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

* Chain [19,[18],17]: 6*it(18)+1*s(7)+5
  Such that:aux(5) =< V1
it(18) =< aux(5)
s(7) =< it(18)*aux(5)

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

* Chain [19,[18],14]: 4*it(18)+1*s(7)+6
  Such that:aux(6) =< V1
it(18) =< aux(6)
s(7) =< it(18)*aux(6)

  with precondition: [V=0,V1>=2,Out+2>=4*V1] 

* Chain [19,17]: 2*s(2)+1*s(4)+5
  Such that:s(4) =< 1
aux(1) =< V1
s(2) =< aux(1)

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

* Chain [19,14]: 1*s(8)+6
  Such that:s(8) =< 1

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

* Chain [17]: 2*s(2)+1*s(4)+2
  Such that:s(4) =< V
aux(1) =< V1
s(2) =< aux(1)

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

* Chain [16]: 3
  with precondition: [V1=0,Out=1,V>=0] 

* Chain [15]: 2
  with precondition: [V=0,Out=0,V1>=1] 

* Chain [14]: 1*s(8)+3
  Such that:s(8) =< V1

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


#### Cost of chains of start(V1,V,V9):
* Chain [23]: 3*s(26)+22*s(28)+1*s(32)+2*s(37)+6*s(44)+2*s(46)+7
  Such that:s(32) =< 1
s(43) =< V1-V
aux(10) =< V1
aux(11) =< V
s(28) =< aux(10)
s(26) =< aux(11)
s(37) =< s(28)*aux(10)
s(44) =< s(43)
s(46) =< s(44)*aux(10)

  with precondition: [V1>=0] 

* Chain [22]: 9
  with precondition: [V1=1] 

* Chain [21]: 1*s(50)+7*s(51)+6*s(55)+2*s(57)+4
  Such that:s(54) =< V-V9
s(50) =< V9+1
aux(13) =< V
s(51) =< aux(13)
s(55) =< s(54)
s(57) =< s(55)*aux(13)

  with precondition: [V1=2,V>=0,V9>=0] 

* Chain [20]: 1*s(58)+12*s(60)+2*s(63)+6
  Such that:s(58) =< 1
aux(14) =< V1
s(60) =< aux(14)
s(63) =< s(60)*aux(14)

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


Closed-form bounds of start(V1,V,V9): 
-------------------------------------
* Chain [23] with precondition: [V1>=0] 
    - Upper bound: 22*V1+8+2*V1*V1+2*V1*nat(V1-V)+nat(V)*3+nat(V1-V)*6 
    - Complexity: n^2 
* Chain [22] with precondition: [V1=1] 
    - Upper bound: 9 
    - Complexity: constant 
* Chain [21] with precondition: [V1=2,V>=0,V9>=0] 
    - Upper bound: 7*V+4+2*V*nat(V-V9)+(V9+1)+nat(V-V9)*6 
    - Complexity: n^2 
* Chain [20] with precondition: [V=0,V1>=0] 
    - Upper bound: 12*V1+7+2*V1*V1 
    - Complexity: n^2 

### Maximum cost of start(V1,V,V9): max([max([5,nat(V)*2*nat(V-V9)+nat(V)*7+nat(V9+1)+nat(V-V9)*6]),10*V1+1+2*V1*nat(V1-V)+nat(V)*3+nat(V1-V)*6+(12*V1+3+2*V1*V1)])+4 
Asymptotic class: n^2 
* Total analysis performed in 248 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:

   lt(0', s(x)) -> true
   lt(x, 0') -> false
   lt(s(x), s(y)) -> lt(x, y)
   fac(x) -> help(x, 0')
   help(x, c) -> if(lt(c, x), x, c)
   if(true, x, c) -> times(s(c), help(x, s(c)))
   if(false, x, c) -> s(0')

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

(15) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
times/0

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

(16)
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:

   lt(0', s(x)) -> true
   lt(x, 0') -> false
   lt(s(x), s(y)) -> lt(x, y)
   fac(x) -> help(x, 0')
   help(x, c) -> if(lt(c, x), x, c)
   if(true, x, c) -> times(help(x, s(c)))
   if(false, x, c) -> s(0')

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

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

(18)
Obligation:
TRS:
Rules:
lt(0', s(x)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
fac(x) -> help(x, 0')
help(x, c) -> if(lt(c, x), x, c)
if(true, x, c) -> times(help(x, s(c)))
if(false, x, c) -> s(0')

Types:
lt :: 0':s:times -> 0':s:times -> true:false
0' :: 0':s:times
s :: 0':s:times -> 0':s:times
true :: true:false
false :: true:false
fac :: 0':s:times -> 0':s:times
help :: 0':s:times -> 0':s:times -> 0':s:times
if :: true:false -> 0':s:times -> 0':s:times -> 0':s:times
times :: 0':s:times -> 0':s:times
hole_true:false1_0 :: true:false
hole_0':s:times2_0 :: 0':s:times
gen_0':s:times3_0 :: Nat -> 0':s:times

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
lt, help

They will be analysed ascendingly in the following order:
lt < help

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

(20)
Obligation:
TRS:
Rules:
lt(0', s(x)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
fac(x) -> help(x, 0')
help(x, c) -> if(lt(c, x), x, c)
if(true, x, c) -> times(help(x, s(c)))
if(false, x, c) -> s(0')

Types:
lt :: 0':s:times -> 0':s:times -> true:false
0' :: 0':s:times
s :: 0':s:times -> 0':s:times
true :: true:false
false :: true:false
fac :: 0':s:times -> 0':s:times
help :: 0':s:times -> 0':s:times -> 0':s:times
if :: true:false -> 0':s:times -> 0':s:times -> 0':s:times
times :: 0':s:times -> 0':s:times
hole_true:false1_0 :: true:false
hole_0':s:times2_0 :: 0':s:times
gen_0':s:times3_0 :: Nat -> 0':s:times


Generator Equations:
gen_0':s:times3_0(0) <=> 0'
gen_0':s:times3_0(+(x, 1)) <=> s(gen_0':s:times3_0(x))


The following defined symbols remain to be analysed:
lt, help

They will be analysed ascendingly in the following order:
lt < help

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lt(gen_0':s:times3_0(n5_0), gen_0':s:times3_0(+(1, n5_0))) -> true, rt in Omega(1 + n5_0)

Induction Base:
lt(gen_0':s:times3_0(0), gen_0':s:times3_0(+(1, 0))) ->_R^Omega(1)
true

Induction Step:
lt(gen_0':s:times3_0(+(n5_0, 1)), gen_0':s:times3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
lt(gen_0':s:times3_0(n5_0), gen_0':s:times3_0(+(1, n5_0))) ->_IH
true

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

(22)
Complex Obligation (BEST)

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

(23)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
lt(0', s(x)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
fac(x) -> help(x, 0')
help(x, c) -> if(lt(c, x), x, c)
if(true, x, c) -> times(help(x, s(c)))
if(false, x, c) -> s(0')

Types:
lt :: 0':s:times -> 0':s:times -> true:false
0' :: 0':s:times
s :: 0':s:times -> 0':s:times
true :: true:false
false :: true:false
fac :: 0':s:times -> 0':s:times
help :: 0':s:times -> 0':s:times -> 0':s:times
if :: true:false -> 0':s:times -> 0':s:times -> 0':s:times
times :: 0':s:times -> 0':s:times
hole_true:false1_0 :: true:false
hole_0':s:times2_0 :: 0':s:times
gen_0':s:times3_0 :: Nat -> 0':s:times


Generator Equations:
gen_0':s:times3_0(0) <=> 0'
gen_0':s:times3_0(+(x, 1)) <=> s(gen_0':s:times3_0(x))


The following defined symbols remain to be analysed:
lt, help

They will be analysed ascendingly in the following order:
lt < help

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

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

(25)
BOUNDS(n^1, INF)

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

(26)
Obligation:
TRS:
Rules:
lt(0', s(x)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
fac(x) -> help(x, 0')
help(x, c) -> if(lt(c, x), x, c)
if(true, x, c) -> times(help(x, s(c)))
if(false, x, c) -> s(0')

Types:
lt :: 0':s:times -> 0':s:times -> true:false
0' :: 0':s:times
s :: 0':s:times -> 0':s:times
true :: true:false
false :: true:false
fac :: 0':s:times -> 0':s:times
help :: 0':s:times -> 0':s:times -> 0':s:times
if :: true:false -> 0':s:times -> 0':s:times -> 0':s:times
times :: 0':s:times -> 0':s:times
hole_true:false1_0 :: true:false
hole_0':s:times2_0 :: 0':s:times
gen_0':s:times3_0 :: Nat -> 0':s:times


Lemmas:
lt(gen_0':s:times3_0(n5_0), gen_0':s:times3_0(+(1, n5_0))) -> true, rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s:times3_0(0) <=> 0'
gen_0':s:times3_0(+(x, 1)) <=> s(gen_0':s:times3_0(x))


The following defined symbols remain to be analysed:
help