WORST_CASE(Omega(n^1),O(n^1))
proof of input_S2WQoGIISA.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 274 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), 16 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 1190 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 911 ms]
    (24) BEST
        (25) proven lower bound
            (26) LowerBoundPropagationProof [FINISHED, 0 ms]
            (27) BOUNDS(n^1, INF)
        (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 38 ms]
            (30) BOUNDS(1, INF)


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

(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:

   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))

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:

   pred(s(x)) -> x [1]
   minus(x, 0) -> x [1]
   minus(x, s(y)) -> pred(minus(x, y)) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [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:

   pred(s(x)) -> x [1]
   minus(x, 0) -> x [1]
   minus(x, s(y)) -> pred(minus(x, y)) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]

The TRS has the following type information:
pred :: s:0 -> s:0
s :: s:0 -> s:0
minus :: s:0 -> s:0 -> s:0
0 :: s:0
quot :: s:0 -> s:0 -> s:0

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:

   pred(v0) -> null_pred [0]
   quot(v0, v1) -> null_quot [0]
   minus(v0, v1) -> null_minus [0]

And the following fresh constants:    null_pred, null_quot, null_minus

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

(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:

   pred(s(x)) -> x [1]
   minus(x, 0) -> x [1]
   minus(x, s(y)) -> pred(minus(x, y)) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   pred(v0) -> null_pred [0]
   quot(v0, v1) -> null_quot [0]
   minus(v0, v1) -> null_minus [0]

The TRS has the following type information:
pred :: s:0:null_pred:null_quot:null_minus -> s:0:null_pred:null_quot:null_minus
s :: s:0:null_pred:null_quot:null_minus -> s:0:null_pred:null_quot:null_minus
minus :: s:0:null_pred:null_quot:null_minus -> s:0:null_pred:null_quot:null_minus -> s:0:null_pred:null_quot:null_minus
0 :: s:0:null_pred:null_quot:null_minus
quot :: s:0:null_pred:null_quot:null_minus -> s:0:null_pred:null_quot:null_minus -> s:0:null_pred:null_quot:null_minus
null_pred :: s:0:null_pred:null_quot:null_minus
null_quot :: s:0:null_pred:null_quot:null_minus
null_minus :: s:0:null_pred:null_quot:null_minus

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
   null_pred => 0
   null_quot => 0
   null_minus => 0

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

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

   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> pred(minus(x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   pred(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   pred(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   quot(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   quot(z, z') -{ 1 }-> 1 + quot(minus(x, y), 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x

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,[pred(V, Out)],[V >= 0]).
eq(start(V, V2),0,[minus(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(start(V, V2),0,[quot(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(pred(V, Out),1,[],[Out = V1,V1 >= 0,V = 1 + V1]).
eq(minus(V, V2, Out),1,[],[Out = V3,V3 >= 0,V = V3,V2 = 0]).
eq(minus(V, V2, Out),1,[minus(V4, V5, Ret0),pred(Ret0, Ret)],[Out = Ret,V2 = 1 + V5,V4 >= 0,V5 >= 0,V = V4]).
eq(quot(V, V2, Out),1,[],[Out = 0,V2 = 1 + V6,V6 >= 0,V = 0]).
eq(quot(V, V2, Out),1,[minus(V7, V8, Ret10),quot(Ret10, 1 + V8, Ret1)],[Out = 1 + Ret1,V2 = 1 + V8,V7 >= 0,V8 >= 0,V = 1 + V7]).
eq(pred(V, Out),0,[],[Out = 0,V9 >= 0,V = V9]).
eq(quot(V, V2, Out),0,[],[Out = 0,V11 >= 0,V10 >= 0,V = V11,V2 = V10]).
eq(minus(V, V2, Out),0,[],[Out = 0,V13 >= 0,V12 >= 0,V = V13,V2 = V12]).
input_output_vars(pred(V,Out),[V],[Out]).
input_output_vars(minus(V,V2,Out),[V,V2],[Out]).
input_output_vars(quot(V,V2,Out),[V,V2],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [pred/2]
1. recursive [non_tail] : [minus/3]
2. recursive  : [quot/3]
3. non_recursive  : [start/2]

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

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

### Specialization of cost equations pred/2 
* CE 4 is refined into CE [12] 
* CE 5 is refined into CE [13] 


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

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

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


### Specialization of cost equations minus/3 
* CE 8 is refined into CE [14] 
* CE 6 is refined into CE [15] 
* CE 7 is refined into CE [16,17] 


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

### Ranking functions of CR minus(V,V2,Out) 
* RF of phase [11]: [V2]
* RF of phase [12]: [V2]

#### Partial ranking functions of CR minus(V,V2,Out) 
* Partial RF of phase [11]:
  - RF of loop [11:1]:
    V2
* Partial RF of phase [12]:
  - RF of loop [12:1]:
    V2


### Specialization of cost equations quot/3 
* CE 9 is refined into CE [18] 
* CE 11 is refined into CE [19] 
* CE 10 is refined into CE [20,21,22] 


### Cost equations --> "Loop" of quot/3 
* CEs [22] --> Loop 15 
* CEs [21] --> Loop 16 
* CEs [20] --> Loop 17 
* CEs [18,19] --> Loop 18 

### Ranking functions of CR quot(V,V2,Out) 
* RF of phase [15]: [V-1,V-V2+1]
* RF of phase [17]: [V]

#### Partial ranking functions of CR quot(V,V2,Out) 
* Partial RF of phase [15]:
  - RF of loop [15:1]:
    V-1
    V-V2+1
* Partial RF of phase [17]:
  - RF of loop [17:1]:
    V


### Specialization of cost equations start/2 
* CE 1 is refined into CE [23,24] 
* CE 2 is refined into CE [25,26,27] 
* CE 3 is refined into CE [28,29,30,31,32] 


### Cost equations --> "Loop" of start/2 
* CEs [28] --> Loop 19 
* CEs [23,24,25,26,27,29,30,31,32] --> Loop 20 

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

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


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

#### Cost of chains of pred(V,Out):
* Chain [10]: 0
  with precondition: [Out=0,V>=0] 

* Chain [9]: 1
  with precondition: [V=Out+1,V>=1] 


#### Cost of chains of minus(V,V2,Out):
* Chain [[12],[11],14]: 3*it(11)+1
  Such that:aux(1) =< V2
it(11) =< aux(1)

  with precondition: [Out=0,V>=1,V2>=2] 

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

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

* Chain [[12],13]: 1*it(12)+0
  Such that:it(12) =< V2

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

* Chain [[11],14]: 2*it(11)+1
  Such that:it(11) =< V2

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

* Chain [14]: 1
  with precondition: [V2=0,V=Out,V>=0] 

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


#### Cost of chains of quot(V,V2,Out):
* Chain [[17],18]: 2*it(17)+1
  Such that:it(17) =< Out

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

* Chain [[17],16,18]: 2*it(17)+5*s(6)+3
  Such that:s(5) =< 1
it(17) =< Out
s(6) =< s(5)

  with precondition: [V2=1,Out>=2,V>=Out] 

* Chain [[15],18]: 2*it(15)+2*s(9)+1
  Such that:it(15) =< V-V2+1
aux(5) =< V
it(15) =< aux(5)
s(9) =< aux(5)

  with precondition: [V2>=2,Out>=1,V+2>=2*Out+V2] 

* Chain [[15],16,18]: 2*it(15)+5*s(6)+2*s(9)+3
  Such that:it(15) =< V-V2+1
s(5) =< V2
aux(6) =< V
s(6) =< s(5)
it(15) =< aux(6)
s(9) =< aux(6)

  with precondition: [V2>=2,Out>=2,V+3>=2*Out+V2] 

* Chain [18]: 1
  with precondition: [Out=0,V>=0,V2>=0] 

* Chain [16,18]: 5*s(6)+3
  Such that:s(5) =< V2
s(6) =< s(5)

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


#### Cost of chains of start(V,V2):
* Chain [20]: 17*s(15)+4*s(19)+4*s(21)+3
  Such that:aux(8) =< V
aux(9) =< V-V2+1
aux(10) =< V2
s(19) =< aux(9)
s(15) =< aux(10)
s(19) =< aux(8)
s(21) =< aux(8)

  with precondition: [V>=0] 

* Chain [19]: 4*s(29)+5*s(30)+3
  Such that:s(27) =< 1
s(28) =< V
s(29) =< s(28)
s(30) =< s(27)

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


Closed-form bounds of start(V,V2): 
-------------------------------------
* Chain [20] with precondition: [V>=0] 
    - Upper bound: 4*V+3+nat(V2)*17+nat(V-V2+1)*4 
    - Complexity: n 
* Chain [19] with precondition: [V2=1,V>=1] 
    - Upper bound: 4*V+8 
    - Complexity: n 

### Maximum cost of start(V,V2): 4*V+3+max([5,nat(V-V2+1)*4+nat(V2)*17]) 
Asymptotic class: n 
* Total analysis performed in 204 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:
   pred(s(z0)) -> z0
   minus(z0, 0) -> z0
   minus(z0, s(z1)) -> pred(minus(z0, z1))
   quot(0, s(z0)) -> 0
   quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
Tuples:
   PRED(s(z0)) -> c
   MINUS(z0, 0) -> c1
   MINUS(z0, s(z1)) -> c2(PRED(minus(z0, z1)), MINUS(z0, z1))
   QUOT(0, s(z0)) -> c3
   QUOT(s(z0), s(z1)) -> c4(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
S tuples:
   PRED(s(z0)) -> c
   MINUS(z0, 0) -> c1
   MINUS(z0, s(z1)) -> c2(PRED(minus(z0, z1)), MINUS(z0, z1))
   QUOT(0, s(z0)) -> c3
   QUOT(s(z0), s(z1)) -> c4(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:   pred_1, minus_2, quot_2

Defined Pair Symbols:   PRED_1, MINUS_2, QUOT_2

Compound Symbols:   c, c1, c2_2, c3, c4_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:

   PRED(s(z0)) -> c
   MINUS(z0, 0) -> c1
   MINUS(z0, s(z1)) -> c2(PRED(minus(z0, z1)), MINUS(z0, z1))
   QUOT(0, s(z0)) -> c3
   QUOT(s(z0), s(z1)) -> c4(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))

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

   pred(s(z0)) -> z0
   minus(z0, 0) -> z0
   minus(z0, s(z1)) -> pred(minus(z0, z1))
   quot(0, s(z0)) -> 0
   quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))

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:

   PRED(s(z0)) -> c
   MINUS(z0, 0') -> c1
   MINUS(z0, s(z1)) -> c2(PRED(minus(z0, z1)), MINUS(z0, z1))
   QUOT(0', s(z0)) -> c3
   QUOT(s(z0), s(z1)) -> c4(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))

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

   pred(s(z0)) -> z0
   minus(z0, 0') -> z0
   minus(z0, s(z1)) -> pred(minus(z0, z1))
   quot(0', s(z0)) -> 0'
   quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))

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

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

(18)
Obligation:
Innermost TRS:
Rules:
PRED(s(z0)) -> c
MINUS(z0, 0') -> c1
MINUS(z0, s(z1)) -> c2(PRED(minus(z0, z1)), MINUS(z0, z1))
QUOT(0', s(z0)) -> c3
QUOT(s(z0), s(z1)) -> c4(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
pred(s(z0)) -> z0
minus(z0, 0') -> z0
minus(z0, s(z1)) -> pred(minus(z0, z1))
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))

Types:
PRED :: s:0' -> c
s :: s:0' -> s:0'
c :: c
MINUS :: s:0' -> s:0' -> c1:c2
0' :: s:0'
c1 :: c1:c2
c2 :: c -> c1:c2 -> c1:c2
minus :: s:0' -> s:0' -> s:0'
QUOT :: s:0' -> s:0' -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c1:c2 -> c3:c4
pred :: s:0' -> s:0'
quot :: s:0' -> s:0' -> s:0'
hole_c1_5 :: c
hole_s:0'2_5 :: s:0'
hole_c1:c23_5 :: c1:c2
hole_c3:c44_5 :: c3:c4
gen_s:0'5_5 :: Nat -> s:0'
gen_c1:c26_5 :: Nat -> c1:c2
gen_c3:c47_5 :: Nat -> c3:c4

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
MINUS, minus, QUOT, quot

They will be analysed ascendingly in the following order:
minus < MINUS
MINUS < QUOT
minus < QUOT
minus < quot

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

(20)
Obligation:
Innermost TRS:
Rules:
PRED(s(z0)) -> c
MINUS(z0, 0') -> c1
MINUS(z0, s(z1)) -> c2(PRED(minus(z0, z1)), MINUS(z0, z1))
QUOT(0', s(z0)) -> c3
QUOT(s(z0), s(z1)) -> c4(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
pred(s(z0)) -> z0
minus(z0, 0') -> z0
minus(z0, s(z1)) -> pred(minus(z0, z1))
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))

Types:
PRED :: s:0' -> c
s :: s:0' -> s:0'
c :: c
MINUS :: s:0' -> s:0' -> c1:c2
0' :: s:0'
c1 :: c1:c2
c2 :: c -> c1:c2 -> c1:c2
minus :: s:0' -> s:0' -> s:0'
QUOT :: s:0' -> s:0' -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c1:c2 -> c3:c4
pred :: s:0' -> s:0'
quot :: s:0' -> s:0' -> s:0'
hole_c1_5 :: c
hole_s:0'2_5 :: s:0'
hole_c1:c23_5 :: c1:c2
hole_c3:c44_5 :: c3:c4
gen_s:0'5_5 :: Nat -> s:0'
gen_c1:c26_5 :: Nat -> c1:c2
gen_c3:c47_5 :: Nat -> c3:c4


Generator Equations:
gen_s:0'5_5(0) <=> 0'
gen_s:0'5_5(+(x, 1)) <=> s(gen_s:0'5_5(x))
gen_c1:c26_5(0) <=> c1
gen_c1:c26_5(+(x, 1)) <=> c2(c, gen_c1:c26_5(x))
gen_c3:c47_5(0) <=> c3
gen_c3:c47_5(+(x, 1)) <=> c4(gen_c3:c47_5(x), c1)


The following defined symbols remain to be analysed:
minus, MINUS, QUOT, quot

They will be analysed ascendingly in the following order:
minus < MINUS
MINUS < QUOT
minus < QUOT
minus < quot

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_s:0'5_5(a), gen_s:0'5_5(+(1, n9_5))) -> *8_5, rt in Omega(0)

Induction Base:
minus(gen_s:0'5_5(a), gen_s:0'5_5(+(1, 0)))

Induction Step:
minus(gen_s:0'5_5(a), gen_s:0'5_5(+(1, +(n9_5, 1)))) ->_R^Omega(0)
pred(minus(gen_s:0'5_5(a), gen_s:0'5_5(+(1, n9_5)))) ->_IH
pred(*8_5)

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:
PRED(s(z0)) -> c
MINUS(z0, 0') -> c1
MINUS(z0, s(z1)) -> c2(PRED(minus(z0, z1)), MINUS(z0, z1))
QUOT(0', s(z0)) -> c3
QUOT(s(z0), s(z1)) -> c4(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
pred(s(z0)) -> z0
minus(z0, 0') -> z0
minus(z0, s(z1)) -> pred(minus(z0, z1))
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))

Types:
PRED :: s:0' -> c
s :: s:0' -> s:0'
c :: c
MINUS :: s:0' -> s:0' -> c1:c2
0' :: s:0'
c1 :: c1:c2
c2 :: c -> c1:c2 -> c1:c2
minus :: s:0' -> s:0' -> s:0'
QUOT :: s:0' -> s:0' -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c1:c2 -> c3:c4
pred :: s:0' -> s:0'
quot :: s:0' -> s:0' -> s:0'
hole_c1_5 :: c
hole_s:0'2_5 :: s:0'
hole_c1:c23_5 :: c1:c2
hole_c3:c44_5 :: c3:c4
gen_s:0'5_5 :: Nat -> s:0'
gen_c1:c26_5 :: Nat -> c1:c2
gen_c3:c47_5 :: Nat -> c3:c4


Lemmas:
minus(gen_s:0'5_5(a), gen_s:0'5_5(+(1, n9_5))) -> *8_5, rt in Omega(0)


Generator Equations:
gen_s:0'5_5(0) <=> 0'
gen_s:0'5_5(+(x, 1)) <=> s(gen_s:0'5_5(x))
gen_c1:c26_5(0) <=> c1
gen_c1:c26_5(+(x, 1)) <=> c2(c, gen_c1:c26_5(x))
gen_c3:c47_5(0) <=> c3
gen_c3:c47_5(+(x, 1)) <=> c4(gen_c3:c47_5(x), c1)


The following defined symbols remain to be analysed:
MINUS, QUOT, quot

They will be analysed ascendingly in the following order:
MINUS < QUOT

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(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
QUOT(gen_s:0'5_5(n15164_5), gen_s:0'5_5(1)) -> gen_c3:c47_5(n15164_5), rt in Omega(1 + n15164_5)

Induction Base:
QUOT(gen_s:0'5_5(0), gen_s:0'5_5(1)) ->_R^Omega(1)
c3

Induction Step:
QUOT(gen_s:0'5_5(+(n15164_5, 1)), gen_s:0'5_5(1)) ->_R^Omega(1)
c4(QUOT(minus(gen_s:0'5_5(n15164_5), gen_s:0'5_5(0)), s(gen_s:0'5_5(0))), MINUS(gen_s:0'5_5(n15164_5), gen_s:0'5_5(0))) ->_R^Omega(0)
c4(QUOT(gen_s:0'5_5(n15164_5), s(gen_s:0'5_5(0))), MINUS(gen_s:0'5_5(n15164_5), gen_s:0'5_5(0))) ->_IH
c4(gen_c3:c47_5(c15165_5), MINUS(gen_s:0'5_5(n15164_5), gen_s:0'5_5(0))) ->_R^Omega(1)
c4(gen_c3:c47_5(n15164_5), c1)

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:
PRED(s(z0)) -> c
MINUS(z0, 0') -> c1
MINUS(z0, s(z1)) -> c2(PRED(minus(z0, z1)), MINUS(z0, z1))
QUOT(0', s(z0)) -> c3
QUOT(s(z0), s(z1)) -> c4(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
pred(s(z0)) -> z0
minus(z0, 0') -> z0
minus(z0, s(z1)) -> pred(minus(z0, z1))
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))

Types:
PRED :: s:0' -> c
s :: s:0' -> s:0'
c :: c
MINUS :: s:0' -> s:0' -> c1:c2
0' :: s:0'
c1 :: c1:c2
c2 :: c -> c1:c2 -> c1:c2
minus :: s:0' -> s:0' -> s:0'
QUOT :: s:0' -> s:0' -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c1:c2 -> c3:c4
pred :: s:0' -> s:0'
quot :: s:0' -> s:0' -> s:0'
hole_c1_5 :: c
hole_s:0'2_5 :: s:0'
hole_c1:c23_5 :: c1:c2
hole_c3:c44_5 :: c3:c4
gen_s:0'5_5 :: Nat -> s:0'
gen_c1:c26_5 :: Nat -> c1:c2
gen_c3:c47_5 :: Nat -> c3:c4


Lemmas:
minus(gen_s:0'5_5(a), gen_s:0'5_5(+(1, n9_5))) -> *8_5, rt in Omega(0)


Generator Equations:
gen_s:0'5_5(0) <=> 0'
gen_s:0'5_5(+(x, 1)) <=> s(gen_s:0'5_5(x))
gen_c1:c26_5(0) <=> c1
gen_c1:c26_5(+(x, 1)) <=> c2(c, gen_c1:c26_5(x))
gen_c3:c47_5(0) <=> c3
gen_c3:c47_5(+(x, 1)) <=> c4(gen_c3:c47_5(x), c1)


The following defined symbols remain to be analysed:
QUOT, quot
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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:
PRED(s(z0)) -> c
MINUS(z0, 0') -> c1
MINUS(z0, s(z1)) -> c2(PRED(minus(z0, z1)), MINUS(z0, z1))
QUOT(0', s(z0)) -> c3
QUOT(s(z0), s(z1)) -> c4(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
pred(s(z0)) -> z0
minus(z0, 0') -> z0
minus(z0, s(z1)) -> pred(minus(z0, z1))
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))

Types:
PRED :: s:0' -> c
s :: s:0' -> s:0'
c :: c
MINUS :: s:0' -> s:0' -> c1:c2
0' :: s:0'
c1 :: c1:c2
c2 :: c -> c1:c2 -> c1:c2
minus :: s:0' -> s:0' -> s:0'
QUOT :: s:0' -> s:0' -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c1:c2 -> c3:c4
pred :: s:0' -> s:0'
quot :: s:0' -> s:0' -> s:0'
hole_c1_5 :: c
hole_s:0'2_5 :: s:0'
hole_c1:c23_5 :: c1:c2
hole_c3:c44_5 :: c3:c4
gen_s:0'5_5 :: Nat -> s:0'
gen_c1:c26_5 :: Nat -> c1:c2
gen_c3:c47_5 :: Nat -> c3:c4


Lemmas:
minus(gen_s:0'5_5(a), gen_s:0'5_5(+(1, n9_5))) -> *8_5, rt in Omega(0)
QUOT(gen_s:0'5_5(n15164_5), gen_s:0'5_5(1)) -> gen_c3:c47_5(n15164_5), rt in Omega(1 + n15164_5)


Generator Equations:
gen_s:0'5_5(0) <=> 0'
gen_s:0'5_5(+(x, 1)) <=> s(gen_s:0'5_5(x))
gen_c1:c26_5(0) <=> c1
gen_c1:c26_5(+(x, 1)) <=> c2(c, gen_c1:c26_5(x))
gen_c3:c47_5(0) <=> c3
gen_c3:c47_5(+(x, 1)) <=> c4(gen_c3:c47_5(x), c1)


The following defined symbols remain to be analysed:
quot
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(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
quot(gen_s:0'5_5(n17337_5), gen_s:0'5_5(1)) -> gen_s:0'5_5(n17337_5), rt in Omega(0)

Induction Base:
quot(gen_s:0'5_5(0), gen_s:0'5_5(1)) ->_R^Omega(0)
0'

Induction Step:
quot(gen_s:0'5_5(+(n17337_5, 1)), gen_s:0'5_5(1)) ->_R^Omega(0)
s(quot(minus(gen_s:0'5_5(n17337_5), gen_s:0'5_5(0)), s(gen_s:0'5_5(0)))) ->_R^Omega(0)
s(quot(gen_s:0'5_5(n17337_5), s(gen_s:0'5_5(0)))) ->_IH
s(gen_s:0'5_5(c17338_5))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(30)
BOUNDS(1, INF)