WORST_CASE(Omega(n^1),O(n^2))
proof of input_9TN589wdr8.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


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

(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, 275 ms]
    (10) BOUNDS(1, n^2)
(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), 259 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 125 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^2).


The TRS R consists of the following rules:

   cond(true, x) -> cond(odd(x), p(p(p(x))))
   odd(0) -> false
   odd(s(0)) -> true
   odd(s(s(x))) -> odd(x)
   p(0) -> 0
   p(s(x)) -> 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^2).


The TRS R consists of the following rules:

   cond(true, x) -> cond(odd(x), p(p(p(x)))) [1]
   odd(0) -> false [1]
   odd(s(0)) -> true [1]
   odd(s(s(x))) -> odd(x) [1]
   p(0) -> 0 [1]
   p(s(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:

   cond(true, x) -> cond(odd(x), p(p(p(x)))) [1]
   odd(0) -> false [1]
   odd(s(0)) -> true [1]
   odd(s(s(x))) -> odd(x) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

The TRS has the following type information:
cond :: true:false -> 0:s -> cond
true :: true:false
odd :: 0:s -> true:false
p :: 0:s -> 0:s
0 :: 0:s
false :: true:false
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:

   cond(v0, v1) -> null_cond [0]

And the following fresh constants:    null_cond

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

(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:

   cond(true, x) -> cond(odd(x), p(p(p(x)))) [1]
   odd(0) -> false [1]
   odd(s(0)) -> true [1]
   odd(s(s(x))) -> odd(x) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   cond(v0, v1) -> null_cond [0]

The TRS has the following type information:
cond :: true:false -> 0:s -> null_cond
true :: true:false
odd :: 0:s -> true:false
p :: 0:s -> 0:s
0 :: 0:s
false :: true:false
s :: 0:s -> 0:s
null_cond :: null_cond

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:

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

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

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

   cond(z, z') -{ 1 }-> cond(odd(x), p(p(p(x)))) :|: z' = x, z = 1, x >= 0
   cond(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   odd(z) -{ 1 }-> odd(x) :|: x >= 0, z = 1 + (1 + x)
   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
   odd(z) -{ 1 }-> 0 :|: z = 0
   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(V1, V),0,[cond(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[odd(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[p(V1, Out)],[V1 >= 0]).
eq(cond(V1, V, Out),1,[odd(V2, Ret0),p(V2, Ret100),p(Ret100, Ret10),p(Ret10, Ret1),cond(Ret0, Ret1, Ret)],[Out = Ret,V = V2,V1 = 1,V2 >= 0]).
eq(odd(V1, Out),1,[],[Out = 0,V1 = 0]).
eq(odd(V1, Out),1,[],[Out = 1,V1 = 1]).
eq(odd(V1, Out),1,[odd(V3, Ret2)],[Out = Ret2,V3 >= 0,V1 = 2 + V3]).
eq(p(V1, Out),1,[],[Out = 0,V1 = 0]).
eq(p(V1, Out),1,[],[Out = V4,V4 >= 0,V1 = 1 + V4]).
eq(cond(V1, V, Out),0,[],[Out = 0,V6 >= 0,V5 >= 0,V1 = V6,V = V5]).
input_output_vars(cond(V1,V,Out),[V1,V],[Out]).
input_output_vars(odd(V1,Out),[V1],[Out]).
input_output_vars(p(V1,Out),[V1],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [odd/2]
1. non_recursive  : [p/2]
2. recursive  : [cond/3]
3. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into odd/2
1. SCC is partially evaluated into p/2
2. SCC is partially evaluated into cond/3
3. SCC is partially evaluated into start/2

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

### Specialization of cost equations odd/2 
* CE 8 is refined into CE [11] 
* CE 7 is refined into CE [12] 
* CE 6 is refined into CE [13] 


### Cost equations --> "Loop" of odd/2 
* CEs [12] --> Loop 9 
* CEs [13] --> Loop 10 
* CEs [11] --> Loop 11 

### Ranking functions of CR odd(V1,Out) 
* RF of phase [11]: [V1-1]

#### Partial ranking functions of CR odd(V1,Out) 
* Partial RF of phase [11]:
  - RF of loop [11:1]:
    V1-1


### Specialization of cost equations p/2 
* CE 10 is refined into CE [14] 
* CE 9 is refined into CE [15] 


### Cost equations --> "Loop" of p/2 
* CEs [14] --> Loop 12 
* CEs [15] --> Loop 13 

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

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


### Specialization of cost equations cond/3 
* CE 5 is refined into CE [16] 
* CE 4 is refined into CE [17,18,19,20,21] 


### Cost equations --> "Loop" of cond/3 
* CEs [21] --> Loop 14 
* CEs [20] --> Loop 15 
* CEs [19] --> Loop 16 
* CEs [18] --> Loop 17 
* CEs [17] --> Loop 18 
* CEs [16] --> Loop 19 

### Ranking functions of CR cond(V1,V,Out) 
* RF of phase [14]: [V/3-2/3]

#### Partial ranking functions of CR cond(V1,V,Out) 
* Partial RF of phase [14]:
  - RF of loop [14:1]:
    V/3-2/3


### Specialization of cost equations start/2 
* CE 1 is refined into CE [22,23,24,25] 
* CE 2 is refined into CE [26,27,28,29] 
* CE 3 is refined into CE [30,31] 


### Cost equations --> "Loop" of start/2 
* CEs [22] --> Loop 20 
* CEs [23,24,25,27,28,29,31] --> Loop 21 
* CEs [26,30] --> Loop 22 

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

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


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

#### Cost of chains of odd(V1,Out):
* Chain [[11],10]: 1*it(11)+1
  Such that:it(11) =< V1

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

* Chain [[11],9]: 1*it(11)+1
  Such that:it(11) =< V1

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

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

* Chain [9]: 1
  with precondition: [V1=1,Out=1] 


#### Cost of chains of p(V1,Out):
* Chain [13]: 1
  with precondition: [V1=0,Out=0] 

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


#### Cost of chains of cond(V1,V,Out):
* Chain [[14],19]: 5*it(14)+1*s(3)+0
  Such that:aux(1) =< V
it(14) =< V/3
s(3) =< it(14)*aux(1)

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

* Chain [[14],18,19]: 5*it(14)+1*s(3)+5
  Such that:aux(1) =< V
it(14) =< V/3
s(3) =< it(14)*aux(1)

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

* Chain [[14],17,19]: 5*it(14)+1*s(3)+5
  Such that:aux(1) =< V
it(14) =< V/3
s(3) =< it(14)*aux(1)

  with precondition: [V1=1,Out=0,V>=4] 

* Chain [[14],17,18,19]: 5*it(14)+1*s(3)+10
  Such that:aux(1) =< V
it(14) =< V/3
s(3) =< it(14)*aux(1)

  with precondition: [V1=1,Out=0,V>=4] 

* Chain [[14],16,19]: 5*it(14)+1*s(3)+1*s(4)+5
  Such that:s(4) =< 2
aux(1) =< V
it(14) =< V/3
s(3) =< it(14)*aux(1)

  with precondition: [V1=1,Out=0,V>=5] 

* Chain [[14],15,19]: 6*it(14)+1*s(3)+5
  Such that:aux(2) =< V
it(14) =< aux(2)
s(3) =< it(14)*aux(2)

  with precondition: [V1=1,Out=0,V>=6] 

* Chain [19]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [18,19]: 5
  with precondition: [V1=1,V=0,Out=0] 

* Chain [17,19]: 5
  with precondition: [V1=1,V=1,Out=0] 

* Chain [17,18,19]: 10
  with precondition: [V1=1,V=1,Out=0] 

* Chain [16,19]: 1*s(4)+5
  Such that:s(4) =< 2

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

* Chain [15,19]: 1*s(5)+5
  Such that:s(5) =< V

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


#### Cost of chains of start(V1,V):
* Chain [22]: 1
  with precondition: [V1=0] 

* Chain [21]: 2*s(26)+7*s(30)+25*s(31)+5*s(32)+1*s(33)+2*s(34)+10
  Such that:s(28) =< V
s(29) =< V/3
aux(5) =< 2
aux(6) =< V1
s(26) =< aux(5)
s(34) =< aux(6)
s(30) =< s(28)
s(31) =< s(29)
s(32) =< s(31)*s(28)
s(33) =< s(30)*s(28)

  with precondition: [V1>=1] 

* Chain [20]: 5
  with precondition: [V1>=0,V>=0] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [22] with precondition: [V1=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [21] with precondition: [V1>=1] 
    - Upper bound: 2*V1+14+nat(V)*7+nat(V)*nat(V)+nat(V)*5*nat(V/3)+nat(V/3)*25 
    - Complexity: n^2 
* Chain [20] with precondition: [V1>=0,V>=0] 
    - Upper bound: 5 
    - Complexity: constant 

### Maximum cost of start(V1,V): max([4,2*V1+13+nat(V)*7+nat(V)*nat(V)+nat(V)*5*nat(V/3)+nat(V/3)*25])+1 
Asymptotic class: n^2 
* Total analysis performed in 202 ms.


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

(10)
BOUNDS(1, n^2)

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

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   cond(true, z0) -> cond(odd(z0), p(p(p(z0))))
   odd(0) -> false
   odd(s(0)) -> true
   odd(s(s(z0))) -> odd(z0)
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   COND(true, z0) -> c(COND(odd(z0), p(p(p(z0)))), ODD(z0))
   COND(true, z0) -> c1(COND(odd(z0), p(p(p(z0)))), P(p(p(z0))), P(p(z0)), P(z0))
   ODD(0) -> c2
   ODD(s(0)) -> c3
   ODD(s(s(z0))) -> c4(ODD(z0))
   P(0) -> c5
   P(s(z0)) -> c6
S tuples:
   COND(true, z0) -> c(COND(odd(z0), p(p(p(z0)))), ODD(z0))
   COND(true, z0) -> c1(COND(odd(z0), p(p(p(z0)))), P(p(p(z0))), P(p(z0)), P(z0))
   ODD(0) -> c2
   ODD(s(0)) -> c3
   ODD(s(s(z0))) -> c4(ODD(z0))
   P(0) -> c5
   P(s(z0)) -> c6
K tuples:none
Defined Rule Symbols:   cond_2, odd_1, p_1

Defined Pair Symbols:   COND_2, ODD_1, P_1

Compound Symbols:   c_2, c1_4, c2, c3, c4_1, c5, c6


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

(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:

   COND(true, z0) -> c(COND(odd(z0), p(p(p(z0)))), ODD(z0))
   COND(true, z0) -> c1(COND(odd(z0), p(p(p(z0)))), P(p(p(z0))), P(p(z0)), P(z0))
   ODD(0) -> c2
   ODD(s(0)) -> c3
   ODD(s(s(z0))) -> c4(ODD(z0))
   P(0) -> c5
   P(s(z0)) -> c6

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

   cond(true, z0) -> cond(odd(z0), p(p(p(z0))))
   odd(0) -> false
   odd(s(0)) -> true
   odd(s(s(z0))) -> odd(z0)
   p(0) -> 0
   p(s(z0)) -> 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:

   COND(true, z0) -> c(COND(odd(z0), p(p(p(z0)))), ODD(z0))
   COND(true, z0) -> c1(COND(odd(z0), p(p(p(z0)))), P(p(p(z0))), P(p(z0)), P(z0))
   ODD(0') -> c2
   ODD(s(0')) -> c3
   ODD(s(s(z0))) -> c4(ODD(z0))
   P(0') -> c5
   P(s(z0)) -> c6

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

   cond(true, z0) -> cond(odd(z0), p(p(p(z0))))
   odd(0') -> false
   odd(s(0')) -> true
   odd(s(s(z0))) -> odd(z0)
   p(0') -> 0'
   p(s(z0)) -> z0

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

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

(18)
Obligation:
Innermost TRS:
Rules:
COND(true, z0) -> c(COND(odd(z0), p(p(p(z0)))), ODD(z0))
COND(true, z0) -> c1(COND(odd(z0), p(p(p(z0)))), P(p(p(z0))), P(p(z0)), P(z0))
ODD(0') -> c2
ODD(s(0')) -> c3
ODD(s(s(z0))) -> c4(ODD(z0))
P(0') -> c5
P(s(z0)) -> c6
cond(true, z0) -> cond(odd(z0), p(p(p(z0))))
odd(0') -> false
odd(s(0')) -> true
odd(s(s(z0))) -> odd(z0)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND :: true:false -> 0':s -> c:c1
true :: true:false
c :: c:c1 -> c2:c3:c4 -> c:c1
odd :: 0':s -> true:false
p :: 0':s -> 0':s
ODD :: 0':s -> c2:c3:c4
c1 :: c:c1 -> c5:c6 -> c5:c6 -> c5:c6 -> c:c1
P :: 0':s -> c5:c6
0' :: 0':s
c2 :: c2:c3:c4
s :: 0':s -> 0':s
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
c5 :: c5:c6
c6 :: c5:c6
cond :: true:false -> 0':s -> cond
false :: true:false
hole_c:c11_7 :: c:c1
hole_true:false2_7 :: true:false
hole_0':s3_7 :: 0':s
hole_c2:c3:c44_7 :: c2:c3:c4
hole_c5:c65_7 :: c5:c6
hole_cond6_7 :: cond
gen_c:c17_7 :: Nat -> c:c1
gen_0':s8_7 :: Nat -> 0':s
gen_c2:c3:c49_7 :: Nat -> c2:c3:c4

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
COND, odd, ODD, cond

They will be analysed ascendingly in the following order:
odd < COND
ODD < COND
odd < cond

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

(20)
Obligation:
Innermost TRS:
Rules:
COND(true, z0) -> c(COND(odd(z0), p(p(p(z0)))), ODD(z0))
COND(true, z0) -> c1(COND(odd(z0), p(p(p(z0)))), P(p(p(z0))), P(p(z0)), P(z0))
ODD(0') -> c2
ODD(s(0')) -> c3
ODD(s(s(z0))) -> c4(ODD(z0))
P(0') -> c5
P(s(z0)) -> c6
cond(true, z0) -> cond(odd(z0), p(p(p(z0))))
odd(0') -> false
odd(s(0')) -> true
odd(s(s(z0))) -> odd(z0)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND :: true:false -> 0':s -> c:c1
true :: true:false
c :: c:c1 -> c2:c3:c4 -> c:c1
odd :: 0':s -> true:false
p :: 0':s -> 0':s
ODD :: 0':s -> c2:c3:c4
c1 :: c:c1 -> c5:c6 -> c5:c6 -> c5:c6 -> c:c1
P :: 0':s -> c5:c6
0' :: 0':s
c2 :: c2:c3:c4
s :: 0':s -> 0':s
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
c5 :: c5:c6
c6 :: c5:c6
cond :: true:false -> 0':s -> cond
false :: true:false
hole_c:c11_7 :: c:c1
hole_true:false2_7 :: true:false
hole_0':s3_7 :: 0':s
hole_c2:c3:c44_7 :: c2:c3:c4
hole_c5:c65_7 :: c5:c6
hole_cond6_7 :: cond
gen_c:c17_7 :: Nat -> c:c1
gen_0':s8_7 :: Nat -> 0':s
gen_c2:c3:c49_7 :: Nat -> c2:c3:c4


Generator Equations:
gen_c:c17_7(0) <=> hole_c:c11_7
gen_c:c17_7(+(x, 1)) <=> c(gen_c:c17_7(x), c2)
gen_0':s8_7(0) <=> 0'
gen_0':s8_7(+(x, 1)) <=> s(gen_0':s8_7(x))
gen_c2:c3:c49_7(0) <=> c2
gen_c2:c3:c49_7(+(x, 1)) <=> c4(gen_c2:c3:c49_7(x))


The following defined symbols remain to be analysed:
odd, COND, ODD, cond

They will be analysed ascendingly in the following order:
odd < COND
ODD < COND
odd < cond

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
odd(gen_0':s8_7(*(2, n11_7))) -> false, rt in Omega(0)

Induction Base:
odd(gen_0':s8_7(*(2, 0))) ->_R^Omega(0)
false

Induction Step:
odd(gen_0':s8_7(*(2, +(n11_7, 1)))) ->_R^Omega(0)
odd(gen_0':s8_7(*(2, n11_7))) ->_IH
false

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:
COND(true, z0) -> c(COND(odd(z0), p(p(p(z0)))), ODD(z0))
COND(true, z0) -> c1(COND(odd(z0), p(p(p(z0)))), P(p(p(z0))), P(p(z0)), P(z0))
ODD(0') -> c2
ODD(s(0')) -> c3
ODD(s(s(z0))) -> c4(ODD(z0))
P(0') -> c5
P(s(z0)) -> c6
cond(true, z0) -> cond(odd(z0), p(p(p(z0))))
odd(0') -> false
odd(s(0')) -> true
odd(s(s(z0))) -> odd(z0)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND :: true:false -> 0':s -> c:c1
true :: true:false
c :: c:c1 -> c2:c3:c4 -> c:c1
odd :: 0':s -> true:false
p :: 0':s -> 0':s
ODD :: 0':s -> c2:c3:c4
c1 :: c:c1 -> c5:c6 -> c5:c6 -> c5:c6 -> c:c1
P :: 0':s -> c5:c6
0' :: 0':s
c2 :: c2:c3:c4
s :: 0':s -> 0':s
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
c5 :: c5:c6
c6 :: c5:c6
cond :: true:false -> 0':s -> cond
false :: true:false
hole_c:c11_7 :: c:c1
hole_true:false2_7 :: true:false
hole_0':s3_7 :: 0':s
hole_c2:c3:c44_7 :: c2:c3:c4
hole_c5:c65_7 :: c5:c6
hole_cond6_7 :: cond
gen_c:c17_7 :: Nat -> c:c1
gen_0':s8_7 :: Nat -> 0':s
gen_c2:c3:c49_7 :: Nat -> c2:c3:c4


Lemmas:
odd(gen_0':s8_7(*(2, n11_7))) -> false, rt in Omega(0)


Generator Equations:
gen_c:c17_7(0) <=> hole_c:c11_7
gen_c:c17_7(+(x, 1)) <=> c(gen_c:c17_7(x), c2)
gen_0':s8_7(0) <=> 0'
gen_0':s8_7(+(x, 1)) <=> s(gen_0':s8_7(x))
gen_c2:c3:c49_7(0) <=> c2
gen_c2:c3:c49_7(+(x, 1)) <=> c4(gen_c2:c3:c49_7(x))


The following defined symbols remain to be analysed:
ODD, COND, cond

They will be analysed ascendingly in the following order:
ODD < COND

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(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ODD(gen_0':s8_7(*(2, n171_7))) -> gen_c2:c3:c49_7(n171_7), rt in Omega(1 + n171_7)

Induction Base:
ODD(gen_0':s8_7(*(2, 0))) ->_R^Omega(1)
c2

Induction Step:
ODD(gen_0':s8_7(*(2, +(n171_7, 1)))) ->_R^Omega(1)
c4(ODD(gen_0':s8_7(*(2, n171_7)))) ->_IH
c4(gen_c2:c3:c49_7(c172_7))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(24)
Complex Obligation (BEST)

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

Innermost TRS:
Rules:
COND(true, z0) -> c(COND(odd(z0), p(p(p(z0)))), ODD(z0))
COND(true, z0) -> c1(COND(odd(z0), p(p(p(z0)))), P(p(p(z0))), P(p(z0)), P(z0))
ODD(0') -> c2
ODD(s(0')) -> c3
ODD(s(s(z0))) -> c4(ODD(z0))
P(0') -> c5
P(s(z0)) -> c6
cond(true, z0) -> cond(odd(z0), p(p(p(z0))))
odd(0') -> false
odd(s(0')) -> true
odd(s(s(z0))) -> odd(z0)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND :: true:false -> 0':s -> c:c1
true :: true:false
c :: c:c1 -> c2:c3:c4 -> c:c1
odd :: 0':s -> true:false
p :: 0':s -> 0':s
ODD :: 0':s -> c2:c3:c4
c1 :: c:c1 -> c5:c6 -> c5:c6 -> c5:c6 -> c:c1
P :: 0':s -> c5:c6
0' :: 0':s
c2 :: c2:c3:c4
s :: 0':s -> 0':s
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
c5 :: c5:c6
c6 :: c5:c6
cond :: true:false -> 0':s -> cond
false :: true:false
hole_c:c11_7 :: c:c1
hole_true:false2_7 :: true:false
hole_0':s3_7 :: 0':s
hole_c2:c3:c44_7 :: c2:c3:c4
hole_c5:c65_7 :: c5:c6
hole_cond6_7 :: cond
gen_c:c17_7 :: Nat -> c:c1
gen_0':s8_7 :: Nat -> 0':s
gen_c2:c3:c49_7 :: Nat -> c2:c3:c4


Lemmas:
odd(gen_0':s8_7(*(2, n11_7))) -> false, rt in Omega(0)


Generator Equations:
gen_c:c17_7(0) <=> hole_c:c11_7
gen_c:c17_7(+(x, 1)) <=> c(gen_c:c17_7(x), c2)
gen_0':s8_7(0) <=> 0'
gen_0':s8_7(+(x, 1)) <=> s(gen_0':s8_7(x))
gen_c2:c3:c49_7(0) <=> c2
gen_c2:c3:c49_7(+(x, 1)) <=> c4(gen_c2:c3:c49_7(x))


The following defined symbols remain to be analysed:
ODD, COND, cond

They will be analysed ascendingly in the following order:
ODD < COND

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(26) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(27)
BOUNDS(n^1, INF)

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(28)
Obligation:
Innermost TRS:
Rules:
COND(true, z0) -> c(COND(odd(z0), p(p(p(z0)))), ODD(z0))
COND(true, z0) -> c1(COND(odd(z0), p(p(p(z0)))), P(p(p(z0))), P(p(z0)), P(z0))
ODD(0') -> c2
ODD(s(0')) -> c3
ODD(s(s(z0))) -> c4(ODD(z0))
P(0') -> c5
P(s(z0)) -> c6
cond(true, z0) -> cond(odd(z0), p(p(p(z0))))
odd(0') -> false
odd(s(0')) -> true
odd(s(s(z0))) -> odd(z0)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND :: true:false -> 0':s -> c:c1
true :: true:false
c :: c:c1 -> c2:c3:c4 -> c:c1
odd :: 0':s -> true:false
p :: 0':s -> 0':s
ODD :: 0':s -> c2:c3:c4
c1 :: c:c1 -> c5:c6 -> c5:c6 -> c5:c6 -> c:c1
P :: 0':s -> c5:c6
0' :: 0':s
c2 :: c2:c3:c4
s :: 0':s -> 0':s
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
c5 :: c5:c6
c6 :: c5:c6
cond :: true:false -> 0':s -> cond
false :: true:false
hole_c:c11_7 :: c:c1
hole_true:false2_7 :: true:false
hole_0':s3_7 :: 0':s
hole_c2:c3:c44_7 :: c2:c3:c4
hole_c5:c65_7 :: c5:c6
hole_cond6_7 :: cond
gen_c:c17_7 :: Nat -> c:c1
gen_0':s8_7 :: Nat -> 0':s
gen_c2:c3:c49_7 :: Nat -> c2:c3:c4


Lemmas:
odd(gen_0':s8_7(*(2, n11_7))) -> false, rt in Omega(0)
ODD(gen_0':s8_7(*(2, n171_7))) -> gen_c2:c3:c49_7(n171_7), rt in Omega(1 + n171_7)


Generator Equations:
gen_c:c17_7(0) <=> hole_c:c11_7
gen_c:c17_7(+(x, 1)) <=> c(gen_c:c17_7(x), c2)
gen_0':s8_7(0) <=> 0'
gen_0':s8_7(+(x, 1)) <=> s(gen_0':s8_7(x))
gen_c2:c3:c49_7(0) <=> c2
gen_c2:c3:c49_7(+(x, 1)) <=> c4(gen_c2:c3:c49_7(x))


The following defined symbols remain to be analysed:
COND, cond