WORST_CASE(Omega(n^1),O(n^1))
proof of input_yQldeqosoz.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(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, 283 ms]
    (10) BOUNDS(1, n^1)
(11) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CdtProblem
    (13) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRelTRS
    (15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRelTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 11 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 253 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 190 ms]
    (24) BEST
        (25) proven lower bound
            (26) LowerBoundPropagationProof [FINISHED, 0 ms]
            (27) BOUNDS(n^1, INF)
        (28) typed CpxTrs


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

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


The TRS R consists of the following rules:

   half(x) -> if(ge(x, s(s(0))), x)
   if(false, x) -> 0
   if(true, x) -> s(half(p(p(x))))
   p(0) -> 0
   p(s(x)) -> x
   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   log(0) -> 0
   log(s(x)) -> s(log(half(s(x))))

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

(1) RelTrsToWeightedTrsProof (UPPER BOUND(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:

   half(x) -> if(ge(x, s(s(0))), x) [1]
   if(false, x) -> 0 [1]
   if(true, x) -> s(half(p(p(x)))) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   log(0) -> 0 [1]
   log(s(x)) -> s(log(half(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:

   half(x) -> if(ge(x, s(s(0))), x) [1]
   if(false, x) -> 0 [1]
   if(true, x) -> s(half(p(p(x)))) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   log(0) -> 0 [1]
   log(s(x)) -> s(log(half(s(x)))) [1]

The TRS has the following type information:
half :: 0:s -> 0:s
if :: false:true -> 0:s -> 0:s
ge :: 0:s -> 0:s -> false:true
s :: 0:s -> 0:s
0 :: 0:s
false :: false:true
true :: false:true
p :: 0:s -> 0:s
log :: 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:

   half(x) -> if(ge(x, s(s(0))), x) [1]
   if(false, x) -> 0 [1]
   if(true, x) -> s(half(p(p(x)))) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   log(0) -> 0 [1]
   log(s(x)) -> s(log(half(s(x)))) [1]

The TRS has the following type information:
half :: 0:s -> 0:s
if :: false:true -> 0:s -> 0:s
ge :: 0:s -> 0:s -> false:true
s :: 0:s -> 0:s
0 :: 0:s
false :: false:true
true :: false:true
p :: 0:s -> 0:s
log :: 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
   false => 0
   true => 1

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

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

   ge(z, z') -{ 1 }-> ge(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   ge(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0
   half(z) -{ 1 }-> if(ge(x, 1 + (1 + 0)), x) :|: x >= 0, z = x
   if(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
   if(z, z') -{ 1 }-> 1 + half(p(p(x))) :|: z' = x, z = 1, x >= 0
   log(z) -{ 1 }-> 0 :|: z = 0
   log(z) -{ 1 }-> 1 + log(half(1 + x)) :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 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(V, V2),0,[half(V, Out)],[V >= 0]).
eq(start(V, V2),0,[if(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(start(V, V2),0,[p(V, Out)],[V >= 0]).
eq(start(V, V2),0,[ge(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(start(V, V2),0,[log(V, Out)],[V >= 0]).
eq(half(V, Out),1,[ge(V1, 1 + (1 + 0), Ret0),if(Ret0, V1, Ret)],[Out = Ret,V1 >= 0,V = V1]).
eq(if(V, V2, Out),1,[],[Out = 0,V2 = V3,V3 >= 0,V = 0]).
eq(if(V, V2, Out),1,[p(V4, Ret100),p(Ret100, Ret10),half(Ret10, Ret1)],[Out = 1 + Ret1,V2 = V4,V = 1,V4 >= 0]).
eq(p(V, Out),1,[],[Out = 0,V = 0]).
eq(p(V, Out),1,[],[Out = V5,V5 >= 0,V = 1 + V5]).
eq(ge(V, V2, Out),1,[],[Out = 1,V6 >= 0,V = V6,V2 = 0]).
eq(ge(V, V2, Out),1,[],[Out = 0,V2 = 1 + V7,V7 >= 0,V = 0]).
eq(ge(V, V2, Out),1,[ge(V8, V9, Ret2)],[Out = Ret2,V2 = 1 + V9,V8 >= 0,V9 >= 0,V = 1 + V8]).
eq(log(V, Out),1,[],[Out = 0,V = 0]).
eq(log(V, Out),1,[half(1 + V10, Ret101),log(Ret101, Ret11)],[Out = 1 + Ret11,V10 >= 0,V = 1 + V10]).
input_output_vars(half(V,Out),[V],[Out]).
input_output_vars(if(V,V2,Out),[V,V2],[Out]).
input_output_vars(p(V,Out),[V],[Out]).
input_output_vars(ge(V,V2,Out),[V,V2],[Out]).
input_output_vars(log(V,Out),[V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [ge/3]
1. non_recursive  : [p/2]
2. recursive  : [half/2,if/3]
3. recursive  : [log/2]
4. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into ge/3
1. SCC is partially evaluated into p/2
2. SCC is partially evaluated into half/2
3. SCC is partially evaluated into log/2
4. SCC is partially evaluated into start/2

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

### Specialization of cost equations ge/3 
* CE 13 is refined into CE [16] 
* CE 11 is refined into CE [17] 
* CE 12 is refined into CE [18] 


### Cost equations --> "Loop" of ge/3 
* CEs [17] --> Loop 12 
* CEs [18] --> Loop 13 
* CEs [16] --> Loop 14 

### Ranking functions of CR ge(V,V2,Out) 
* RF of phase [14]: [V,V2]

#### Partial ranking functions of CR ge(V,V2,Out) 
* Partial RF of phase [14]:
  - RF of loop [14:1]:
    V
    V2


### Specialization of cost equations p/2 
* CE 8 is refined into CE [19] 
* CE 7 is refined into CE [20] 


### Cost equations --> "Loop" of p/2 
* CEs [19] --> Loop 15 
* CEs [20] --> Loop 16 

### Ranking functions of CR p(V,Out) 

#### Partial ranking functions of CR p(V,Out) 


### Specialization of cost equations half/2 
* CE 10 is refined into CE [21,22] 
* CE 9 is refined into CE [23] 


### Cost equations --> "Loop" of half/2 
* CEs [23] --> Loop 17 
* CEs [22] --> Loop 18 
* CEs [21] --> Loop 19 

### Ranking functions of CR half(V,Out) 
* RF of phase [17]: [V-1]

#### Partial ranking functions of CR half(V,Out) 
* Partial RF of phase [17]:
  - RF of loop [17:1]:
    V-1


### Specialization of cost equations log/2 
* CE 15 is refined into CE [24,25,26] 
* CE 14 is refined into CE [27] 


### Cost equations --> "Loop" of log/2 
* CEs [27] --> Loop 20 
* CEs [26] --> Loop 21 
* CEs [25] --> Loop 22 
* CEs [24] --> Loop 23 

### Ranking functions of CR log(V,Out) 
* RF of phase [21,22]: [V-1]

#### Partial ranking functions of CR log(V,Out) 
* Partial RF of phase [21,22]:
  - RF of loop [21:1]:
    V/2-1
  - RF of loop [22:1]:
    V-1


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


### Cost equations --> "Loop" of start/2 
* CEs [42] --> Loop 24 
* CEs [28,29,30,31,32,33,36,37,38,40,43,44,46,47] --> Loop 25 
* CEs [34,35,39,41,45] --> Loop 26 

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

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


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

#### Cost of chains of ge(V,V2,Out):
* Chain [[14],13]: 1*it(14)+1
  Such that:it(14) =< V

  with precondition: [Out=0,V>=1,V2>=V+1] 

* Chain [[14],12]: 1*it(14)+1
  Such that:it(14) =< V2

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

* Chain [13]: 1
  with precondition: [V=0,Out=0,V2>=1] 

* Chain [12]: 1
  with precondition: [V2=0,Out=1,V>=0] 


#### Cost of chains of p(V,Out):
* Chain [16]: 1
  with precondition: [V=0,Out=0] 

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


#### Cost of chains of half(V,Out):
* Chain [[17],19]: 5*it(17)+1*s(3)+3
  Such that:aux(4) =< 2*Out
it(17) =< aux(4)
s(3) =< aux(4)*2

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

* Chain [[17],18]: 5*it(17)+1*s(3)+1*s(4)+3
  Such that:s(4) =< 1
aux(5) =< V
it(17) =< aux(5)
s(3) =< aux(5)*2

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

* Chain [19]: 3
  with precondition: [V=0,Out=0] 

* Chain [18]: 1*s(4)+3
  Such that:s(4) =< 1

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


#### Cost of chains of log(V,Out):
* Chain [[21,22],23,20]: 4*it(21)+4*it(22)+1*s(5)+1*s(20)+10*s(21)+2*s(22)+5
  Such that:s(5) =< 1
aux(13) =< V
aux(14) =< 2*V
aux(15) =< V/2
aux(6) =< aux(13)
it(21) =< aux(13)
it(22) =< aux(13)
it(22) =< aux(14)
aux(6) =< aux(15)
it(21) =< aux(15)
s(20) =< aux(6)
s(21) =< aux(14)
s(22) =< aux(14)*2

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

* Chain [23,20]: 1*s(5)+5
  Such that:s(5) =< 1

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

* Chain [20]: 1
  with precondition: [V=0,Out=0] 


#### Cost of chains of start(V,V2):
* Chain [26]: 3
  with precondition: [V=0] 

* Chain [25]: 6*s(27)+11*s(29)+2*s(30)+11*s(37)+2*s(38)+4*s(51)+4*s(52)+1*s(53)+10*s(54)+2*s(55)+6
  Such that:s(48) =< 2*V
s(49) =< V/2
aux(16) =< 1
aux(17) =< V
aux(18) =< V2
s(27) =< aux(16)
s(37) =< aux(17)
s(29) =< aux(18)
s(30) =< aux(18)*2
s(38) =< aux(17)*2
s(50) =< aux(17)
s(51) =< aux(17)
s(52) =< aux(17)
s(52) =< s(48)
s(50) =< s(49)
s(51) =< s(49)
s(53) =< s(50)
s(54) =< s(48)
s(55) =< s(48)*2

  with precondition: [V>=1] 

* Chain [24]: 1
  with precondition: [V2=0,V>=0] 


Closed-form bounds of start(V,V2): 
-------------------------------------
* Chain [26] with precondition: [V=0] 
    - Upper bound: 3 
    - Complexity: constant 
* Chain [25] with precondition: [V>=1] 
    - Upper bound: 24*V+12+nat(V2)*15+28*V 
    - Complexity: n 
* Chain [24] with precondition: [V2=0,V>=0] 
    - Upper bound: 1 
    - Complexity: constant 

### Maximum cost of start(V,V2): max([2,24*V+11+nat(V2)*15+28*V])+1 
Asymptotic class: n 
* Total analysis performed in 215 ms.


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

(10)
BOUNDS(1, n^1)

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

(11) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> if(ge(z0, s(s(0))), z0)
   if(false, z0) -> 0
   if(true, z0) -> s(half(p(p(z0))))
   p(0) -> 0
   p(s(z0)) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   log(0) -> 0
   log(s(z0)) -> s(log(half(s(z0))))
Tuples:
   HALF(z0) -> c(IF(ge(z0, s(s(0))), z0), GE(z0, s(s(0))))
   IF(false, z0) -> c1
   IF(true, z0) -> c2(HALF(p(p(z0))), P(p(z0)), P(z0))
   P(0) -> c3
   P(s(z0)) -> c4
   GE(z0, 0) -> c5
   GE(0, s(z0)) -> c6
   GE(s(z0), s(z1)) -> c7(GE(z0, z1))
   LOG(0) -> c8
   LOG(s(z0)) -> c9(LOG(half(s(z0))), HALF(s(z0)))
S tuples:
   HALF(z0) -> c(IF(ge(z0, s(s(0))), z0), GE(z0, s(s(0))))
   IF(false, z0) -> c1
   IF(true, z0) -> c2(HALF(p(p(z0))), P(p(z0)), P(z0))
   P(0) -> c3
   P(s(z0)) -> c4
   GE(z0, 0) -> c5
   GE(0, s(z0)) -> c6
   GE(s(z0), s(z1)) -> c7(GE(z0, z1))
   LOG(0) -> c8
   LOG(s(z0)) -> c9(LOG(half(s(z0))), HALF(s(z0)))
K tuples:none
Defined Rule Symbols:   half_1, if_2, p_1, ge_2, log_1

Defined Pair Symbols:   HALF_1, IF_2, P_1, GE_2, LOG_1

Compound Symbols:   c_2, c1, c2_3, c3, c4, c5, c6, c7_1, c8, c9_2


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

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

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


The TRS R consists of the following rules:

   HALF(z0) -> c(IF(ge(z0, s(s(0))), z0), GE(z0, s(s(0))))
   IF(false, z0) -> c1
   IF(true, z0) -> c2(HALF(p(p(z0))), P(p(z0)), P(z0))
   P(0) -> c3
   P(s(z0)) -> c4
   GE(z0, 0) -> c5
   GE(0, s(z0)) -> c6
   GE(s(z0), s(z1)) -> c7(GE(z0, z1))
   LOG(0) -> c8
   LOG(s(z0)) -> c9(LOG(half(s(z0))), HALF(s(z0)))

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

   half(z0) -> if(ge(z0, s(s(0))), z0)
   if(false, z0) -> 0
   if(true, z0) -> s(half(p(p(z0))))
   p(0) -> 0
   p(s(z0)) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   log(0) -> 0
   log(s(z0)) -> s(log(half(s(z0))))

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

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

(16)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   HALF(z0) -> c(IF(ge(z0, s(s(0'))), z0), GE(z0, s(s(0'))))
   IF(false, z0) -> c1
   IF(true, z0) -> c2(HALF(p(p(z0))), P(p(z0)), P(z0))
   P(0') -> c3
   P(s(z0)) -> c4
   GE(z0, 0') -> c5
   GE(0', s(z0)) -> c6
   GE(s(z0), s(z1)) -> c7(GE(z0, z1))
   LOG(0') -> c8
   LOG(s(z0)) -> c9(LOG(half(s(z0))), HALF(s(z0)))

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

   half(z0) -> if(ge(z0, s(s(0'))), z0)
   if(false, z0) -> 0'
   if(true, z0) -> s(half(p(p(z0))))
   p(0') -> 0'
   p(s(z0)) -> z0
   ge(z0, 0') -> true
   ge(0', s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   log(0') -> 0'
   log(s(z0)) -> s(log(half(s(z0))))

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

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

(18)
Obligation:
Innermost TRS:
Rules:
HALF(z0) -> c(IF(ge(z0, s(s(0'))), z0), GE(z0, s(s(0'))))
IF(false, z0) -> c1
IF(true, z0) -> c2(HALF(p(p(z0))), P(p(z0)), P(z0))
P(0') -> c3
P(s(z0)) -> c4
GE(z0, 0') -> c5
GE(0', s(z0)) -> c6
GE(s(z0), s(z1)) -> c7(GE(z0, z1))
LOG(0') -> c8
LOG(s(z0)) -> c9(LOG(half(s(z0))), HALF(s(z0)))
half(z0) -> if(ge(z0, s(s(0'))), z0)
if(false, z0) -> 0'
if(true, z0) -> s(half(p(p(z0))))
p(0') -> 0'
p(s(z0)) -> z0
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
log(0') -> 0'
log(s(z0)) -> s(log(half(s(z0))))

Types:
HALF :: 0':s -> c
c :: c1:c2 -> c5:c6:c7 -> c
IF :: false:true -> 0':s -> c1:c2
ge :: 0':s -> 0':s -> false:true
s :: 0':s -> 0':s
0' :: 0':s
GE :: 0':s -> 0':s -> c5:c6:c7
false :: false:true
c1 :: c1:c2
true :: false:true
c2 :: c -> c3:c4 -> c3:c4 -> c1:c2
p :: 0':s -> 0':s
P :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
c5 :: c5:c6:c7
c6 :: c5:c6:c7
c7 :: c5:c6:c7 -> c5:c6:c7
LOG :: 0':s -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c -> c8:c9
half :: 0':s -> 0':s
if :: false:true -> 0':s -> 0':s
log :: 0':s -> 0':s
hole_c1_10 :: c
hole_0':s2_10 :: 0':s
hole_c1:c23_10 :: c1:c2
hole_c5:c6:c74_10 :: c5:c6:c7
hole_false:true5_10 :: false:true
hole_c3:c46_10 :: c3:c4
hole_c8:c97_10 :: c8:c9
gen_0':s8_10 :: Nat -> 0':s
gen_c5:c6:c79_10 :: Nat -> c5:c6:c7
gen_c8:c910_10 :: Nat -> c8:c9

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
HALF, ge, GE, LOG, half, log

They will be analysed ascendingly in the following order:
ge < HALF
GE < HALF
HALF < LOG
ge < half
half < LOG
half < log

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

(20)
Obligation:
Innermost TRS:
Rules:
HALF(z0) -> c(IF(ge(z0, s(s(0'))), z0), GE(z0, s(s(0'))))
IF(false, z0) -> c1
IF(true, z0) -> c2(HALF(p(p(z0))), P(p(z0)), P(z0))
P(0') -> c3
P(s(z0)) -> c4
GE(z0, 0') -> c5
GE(0', s(z0)) -> c6
GE(s(z0), s(z1)) -> c7(GE(z0, z1))
LOG(0') -> c8
LOG(s(z0)) -> c9(LOG(half(s(z0))), HALF(s(z0)))
half(z0) -> if(ge(z0, s(s(0'))), z0)
if(false, z0) -> 0'
if(true, z0) -> s(half(p(p(z0))))
p(0') -> 0'
p(s(z0)) -> z0
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
log(0') -> 0'
log(s(z0)) -> s(log(half(s(z0))))

Types:
HALF :: 0':s -> c
c :: c1:c2 -> c5:c6:c7 -> c
IF :: false:true -> 0':s -> c1:c2
ge :: 0':s -> 0':s -> false:true
s :: 0':s -> 0':s
0' :: 0':s
GE :: 0':s -> 0':s -> c5:c6:c7
false :: false:true
c1 :: c1:c2
true :: false:true
c2 :: c -> c3:c4 -> c3:c4 -> c1:c2
p :: 0':s -> 0':s
P :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
c5 :: c5:c6:c7
c6 :: c5:c6:c7
c7 :: c5:c6:c7 -> c5:c6:c7
LOG :: 0':s -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c -> c8:c9
half :: 0':s -> 0':s
if :: false:true -> 0':s -> 0':s
log :: 0':s -> 0':s
hole_c1_10 :: c
hole_0':s2_10 :: 0':s
hole_c1:c23_10 :: c1:c2
hole_c5:c6:c74_10 :: c5:c6:c7
hole_false:true5_10 :: false:true
hole_c3:c46_10 :: c3:c4
hole_c8:c97_10 :: c8:c9
gen_0':s8_10 :: Nat -> 0':s
gen_c5:c6:c79_10 :: Nat -> c5:c6:c7
gen_c8:c910_10 :: Nat -> c8:c9


Generator Equations:
gen_0':s8_10(0) <=> 0'
gen_0':s8_10(+(x, 1)) <=> s(gen_0':s8_10(x))
gen_c5:c6:c79_10(0) <=> c5
gen_c5:c6:c79_10(+(x, 1)) <=> c7(gen_c5:c6:c79_10(x))
gen_c8:c910_10(0) <=> c8
gen_c8:c910_10(+(x, 1)) <=> c9(gen_c8:c910_10(x), c(c1, c5))


The following defined symbols remain to be analysed:
ge, HALF, GE, LOG, half, log

They will be analysed ascendingly in the following order:
ge < HALF
GE < HALF
HALF < LOG
ge < half
half < LOG
half < log

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_0':s8_10(n12_10), gen_0':s8_10(n12_10)) -> true, rt in Omega(0)

Induction Base:
ge(gen_0':s8_10(0), gen_0':s8_10(0)) ->_R^Omega(0)
true

Induction Step:
ge(gen_0':s8_10(+(n12_10, 1)), gen_0':s8_10(+(n12_10, 1))) ->_R^Omega(0)
ge(gen_0':s8_10(n12_10), gen_0':s8_10(n12_10)) ->_IH
true

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(22)
Obligation:
Innermost TRS:
Rules:
HALF(z0) -> c(IF(ge(z0, s(s(0'))), z0), GE(z0, s(s(0'))))
IF(false, z0) -> c1
IF(true, z0) -> c2(HALF(p(p(z0))), P(p(z0)), P(z0))
P(0') -> c3
P(s(z0)) -> c4
GE(z0, 0') -> c5
GE(0', s(z0)) -> c6
GE(s(z0), s(z1)) -> c7(GE(z0, z1))
LOG(0') -> c8
LOG(s(z0)) -> c9(LOG(half(s(z0))), HALF(s(z0)))
half(z0) -> if(ge(z0, s(s(0'))), z0)
if(false, z0) -> 0'
if(true, z0) -> s(half(p(p(z0))))
p(0') -> 0'
p(s(z0)) -> z0
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
log(0') -> 0'
log(s(z0)) -> s(log(half(s(z0))))

Types:
HALF :: 0':s -> c
c :: c1:c2 -> c5:c6:c7 -> c
IF :: false:true -> 0':s -> c1:c2
ge :: 0':s -> 0':s -> false:true
s :: 0':s -> 0':s
0' :: 0':s
GE :: 0':s -> 0':s -> c5:c6:c7
false :: false:true
c1 :: c1:c2
true :: false:true
c2 :: c -> c3:c4 -> c3:c4 -> c1:c2
p :: 0':s -> 0':s
P :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
c5 :: c5:c6:c7
c6 :: c5:c6:c7
c7 :: c5:c6:c7 -> c5:c6:c7
LOG :: 0':s -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c -> c8:c9
half :: 0':s -> 0':s
if :: false:true -> 0':s -> 0':s
log :: 0':s -> 0':s
hole_c1_10 :: c
hole_0':s2_10 :: 0':s
hole_c1:c23_10 :: c1:c2
hole_c5:c6:c74_10 :: c5:c6:c7
hole_false:true5_10 :: false:true
hole_c3:c46_10 :: c3:c4
hole_c8:c97_10 :: c8:c9
gen_0':s8_10 :: Nat -> 0':s
gen_c5:c6:c79_10 :: Nat -> c5:c6:c7
gen_c8:c910_10 :: Nat -> c8:c9


Lemmas:
ge(gen_0':s8_10(n12_10), gen_0':s8_10(n12_10)) -> true, rt in Omega(0)


Generator Equations:
gen_0':s8_10(0) <=> 0'
gen_0':s8_10(+(x, 1)) <=> s(gen_0':s8_10(x))
gen_c5:c6:c79_10(0) <=> c5
gen_c5:c6:c79_10(+(x, 1)) <=> c7(gen_c5:c6:c79_10(x))
gen_c8:c910_10(0) <=> c8
gen_c8:c910_10(+(x, 1)) <=> c9(gen_c8:c910_10(x), c(c1, c5))


The following defined symbols remain to be analysed:
GE, HALF, LOG, half, log

They will be analysed ascendingly in the following order:
GE < HALF
HALF < LOG
half < LOG
half < log

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
GE(gen_0':s8_10(n368_10), gen_0':s8_10(n368_10)) -> gen_c5:c6:c79_10(n368_10), rt in Omega(1 + n368_10)

Induction Base:
GE(gen_0':s8_10(0), gen_0':s8_10(0)) ->_R^Omega(1)
c5

Induction Step:
GE(gen_0':s8_10(+(n368_10, 1)), gen_0':s8_10(+(n368_10, 1))) ->_R^Omega(1)
c7(GE(gen_0':s8_10(n368_10), gen_0':s8_10(n368_10))) ->_IH
c7(gen_c5:c6:c79_10(c369_10))

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

(24)
Complex Obligation (BEST)

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

(25)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
HALF(z0) -> c(IF(ge(z0, s(s(0'))), z0), GE(z0, s(s(0'))))
IF(false, z0) -> c1
IF(true, z0) -> c2(HALF(p(p(z0))), P(p(z0)), P(z0))
P(0') -> c3
P(s(z0)) -> c4
GE(z0, 0') -> c5
GE(0', s(z0)) -> c6
GE(s(z0), s(z1)) -> c7(GE(z0, z1))
LOG(0') -> c8
LOG(s(z0)) -> c9(LOG(half(s(z0))), HALF(s(z0)))
half(z0) -> if(ge(z0, s(s(0'))), z0)
if(false, z0) -> 0'
if(true, z0) -> s(half(p(p(z0))))
p(0') -> 0'
p(s(z0)) -> z0
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
log(0') -> 0'
log(s(z0)) -> s(log(half(s(z0))))

Types:
HALF :: 0':s -> c
c :: c1:c2 -> c5:c6:c7 -> c
IF :: false:true -> 0':s -> c1:c2
ge :: 0':s -> 0':s -> false:true
s :: 0':s -> 0':s
0' :: 0':s
GE :: 0':s -> 0':s -> c5:c6:c7
false :: false:true
c1 :: c1:c2
true :: false:true
c2 :: c -> c3:c4 -> c3:c4 -> c1:c2
p :: 0':s -> 0':s
P :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
c5 :: c5:c6:c7
c6 :: c5:c6:c7
c7 :: c5:c6:c7 -> c5:c6:c7
LOG :: 0':s -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c -> c8:c9
half :: 0':s -> 0':s
if :: false:true -> 0':s -> 0':s
log :: 0':s -> 0':s
hole_c1_10 :: c
hole_0':s2_10 :: 0':s
hole_c1:c23_10 :: c1:c2
hole_c5:c6:c74_10 :: c5:c6:c7
hole_false:true5_10 :: false:true
hole_c3:c46_10 :: c3:c4
hole_c8:c97_10 :: c8:c9
gen_0':s8_10 :: Nat -> 0':s
gen_c5:c6:c79_10 :: Nat -> c5:c6:c7
gen_c8:c910_10 :: Nat -> c8:c9


Lemmas:
ge(gen_0':s8_10(n12_10), gen_0':s8_10(n12_10)) -> true, rt in Omega(0)


Generator Equations:
gen_0':s8_10(0) <=> 0'
gen_0':s8_10(+(x, 1)) <=> s(gen_0':s8_10(x))
gen_c5:c6:c79_10(0) <=> c5
gen_c5:c6:c79_10(+(x, 1)) <=> c7(gen_c5:c6:c79_10(x))
gen_c8:c910_10(0) <=> c8
gen_c8:c910_10(+(x, 1)) <=> c9(gen_c8:c910_10(x), c(c1, c5))


The following defined symbols remain to be analysed:
GE, HALF, LOG, half, log

They will be analysed ascendingly in the following order:
GE < HALF
HALF < LOG
half < LOG
half < log

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

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

(27)
BOUNDS(n^1, INF)

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

(28)
Obligation:
Innermost TRS:
Rules:
HALF(z0) -> c(IF(ge(z0, s(s(0'))), z0), GE(z0, s(s(0'))))
IF(false, z0) -> c1
IF(true, z0) -> c2(HALF(p(p(z0))), P(p(z0)), P(z0))
P(0') -> c3
P(s(z0)) -> c4
GE(z0, 0') -> c5
GE(0', s(z0)) -> c6
GE(s(z0), s(z1)) -> c7(GE(z0, z1))
LOG(0') -> c8
LOG(s(z0)) -> c9(LOG(half(s(z0))), HALF(s(z0)))
half(z0) -> if(ge(z0, s(s(0'))), z0)
if(false, z0) -> 0'
if(true, z0) -> s(half(p(p(z0))))
p(0') -> 0'
p(s(z0)) -> z0
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
log(0') -> 0'
log(s(z0)) -> s(log(half(s(z0))))

Types:
HALF :: 0':s -> c
c :: c1:c2 -> c5:c6:c7 -> c
IF :: false:true -> 0':s -> c1:c2
ge :: 0':s -> 0':s -> false:true
s :: 0':s -> 0':s
0' :: 0':s
GE :: 0':s -> 0':s -> c5:c6:c7
false :: false:true
c1 :: c1:c2
true :: false:true
c2 :: c -> c3:c4 -> c3:c4 -> c1:c2
p :: 0':s -> 0':s
P :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
c5 :: c5:c6:c7
c6 :: c5:c6:c7
c7 :: c5:c6:c7 -> c5:c6:c7
LOG :: 0':s -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c -> c8:c9
half :: 0':s -> 0':s
if :: false:true -> 0':s -> 0':s
log :: 0':s -> 0':s
hole_c1_10 :: c
hole_0':s2_10 :: 0':s
hole_c1:c23_10 :: c1:c2
hole_c5:c6:c74_10 :: c5:c6:c7
hole_false:true5_10 :: false:true
hole_c3:c46_10 :: c3:c4
hole_c8:c97_10 :: c8:c9
gen_0':s8_10 :: Nat -> 0':s
gen_c5:c6:c79_10 :: Nat -> c5:c6:c7
gen_c8:c910_10 :: Nat -> c8:c9


Lemmas:
ge(gen_0':s8_10(n12_10), gen_0':s8_10(n12_10)) -> true, rt in Omega(0)
GE(gen_0':s8_10(n368_10), gen_0':s8_10(n368_10)) -> gen_c5:c6:c79_10(n368_10), rt in Omega(1 + n368_10)


Generator Equations:
gen_0':s8_10(0) <=> 0'
gen_0':s8_10(+(x, 1)) <=> s(gen_0':s8_10(x))
gen_c5:c6:c79_10(0) <=> c5
gen_c5:c6:c79_10(+(x, 1)) <=> c7(gen_c5:c6:c79_10(x))
gen_c8:c910_10(0) <=> c8
gen_c8:c910_10(+(x, 1)) <=> c9(gen_c8:c910_10(x), c(c1, c5))


The following defined symbols remain to be analysed:
HALF, LOG, half, log

They will be analysed ascendingly in the following order:
HALF < LOG
half < LOG
half < log