WORST_CASE(Omega(n^1),O(n^2))
proof of input_EYLyKM8P1u.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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^2).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxWeightedTrs
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedTrs
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
    (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) CompleteCoflocoProof [FINISHED, 305 ms]
    (18) BOUNDS(1, n^2)
(19) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CdtProblem
    (21) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxRelTRS
    (23) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxRelTRS
    (25) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) typed CpxTrs
    (27) OrderProof [LOWER BOUND(ID), 9 ms]
    (28) typed CpxTrs
    (29) RewriteLemmaProof [LOWER BOUND(ID), 315 ms]
    (30) typed CpxTrs
    (31) RewriteLemmaProof [LOWER BOUND(ID), 83 ms]
    (32) BEST
        (33) proven lower bound
            (34) LowerBoundPropagationProof [FINISHED, 0 ms]
            (35) BOUNDS(n^1, INF)
        (36) 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:

   minus(x, y) -> cond(ge(x, s(y)), x, y)
   cond(false, x, y) -> 0
   cond(true, x, y) -> s(minus(x, s(y)))
   ge(u, 0) -> true
   ge(0, s(v)) -> false
   ge(s(u), s(v)) -> ge(u, v)

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

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, z1) -> cond(ge(z0, s(z1)), z0, z1)
   cond(false, z0, z1) -> 0
   cond(true, z0, z1) -> s(minus(z0, s(z1)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   MINUS(z0, z1) -> c(COND(ge(z0, s(z1)), z0, z1), GE(z0, s(z1)))
   COND(false, z0, z1) -> c1
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GE(z0, 0) -> c3
   GE(0, s(z0)) -> c4
   GE(s(z0), s(z1)) -> c5(GE(z0, z1))
S tuples:
   MINUS(z0, z1) -> c(COND(ge(z0, s(z1)), z0, z1), GE(z0, s(z1)))
   COND(false, z0, z1) -> c1
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GE(z0, 0) -> c3
   GE(0, s(z0)) -> c4
   GE(s(z0), s(z1)) -> c5(GE(z0, z1))
K tuples:none
Defined Rule Symbols:   minus_2, cond_3, ge_2

Defined Pair Symbols:   MINUS_2, COND_3, GE_2

Compound Symbols:   c_2, c1, c2_1, c3, c4, c5_1


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   GE(0, s(z0)) -> c4
   COND(false, z0, z1) -> c1
   GE(z0, 0) -> c3

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, z1) -> cond(ge(z0, s(z1)), z0, z1)
   cond(false, z0, z1) -> 0
   cond(true, z0, z1) -> s(minus(z0, s(z1)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   MINUS(z0, z1) -> c(COND(ge(z0, s(z1)), z0, z1), GE(z0, s(z1)))
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GE(s(z0), s(z1)) -> c5(GE(z0, z1))
S tuples:
   MINUS(z0, z1) -> c(COND(ge(z0, s(z1)), z0, z1), GE(z0, s(z1)))
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GE(s(z0), s(z1)) -> c5(GE(z0, z1))
K tuples:none
Defined Rule Symbols:   minus_2, cond_3, ge_2

Defined Pair Symbols:   MINUS_2, COND_3, GE_2

Compound Symbols:   c_2, c2_1, c5_1


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   minus(z0, z1) -> cond(ge(z0, s(z1)), z0, z1)
   cond(false, z0, z1) -> 0
   cond(true, z0, z1) -> s(minus(z0, s(z1)))

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   ge(z0, 0) -> true
Tuples:
   MINUS(z0, z1) -> c(COND(ge(z0, s(z1)), z0, z1), GE(z0, s(z1)))
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GE(s(z0), s(z1)) -> c5(GE(z0, z1))
S tuples:
   MINUS(z0, z1) -> c(COND(ge(z0, s(z1)), z0, z1), GE(z0, s(z1)))
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GE(s(z0), s(z1)) -> c5(GE(z0, z1))
K tuples:none
Defined Rule Symbols:   ge_2

Defined Pair Symbols:   MINUS_2, COND_3, GE_2

Compound Symbols:   c_2, c2_1, c5_1


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

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

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   MINUS(z0, z1) -> c(COND(ge(z0, s(z1)), z0, z1), GE(z0, s(z1)))
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GE(s(z0), s(z1)) -> c5(GE(z0, z1))

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

   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   ge(z0, 0) -> true

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

(9) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(10)
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:

   MINUS(z0, z1) -> c(COND(ge(z0, s(z1)), z0, z1), GE(z0, s(z1))) [1]
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1))) [1]
   GE(s(z0), s(z1)) -> c5(GE(z0, z1)) [1]
   ge(0, s(z0)) -> false [0]
   ge(s(z0), s(z1)) -> ge(z0, z1) [0]
   ge(z0, 0) -> true [0]

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

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

(12)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   MINUS(z0, z1) -> c(COND(ge(z0, s(z1)), z0, z1), GE(z0, s(z1))) [1]
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1))) [1]
   GE(s(z0), s(z1)) -> c5(GE(z0, z1)) [1]
   ge(0, s(z0)) -> false [0]
   ge(s(z0), s(z1)) -> ge(z0, z1) [0]
   ge(z0, 0) -> true [0]

The TRS has the following type information:
MINUS :: s:0 -> s:0 -> c
c :: c2 -> c5 -> c
COND :: true:false -> s:0 -> s:0 -> c2
ge :: s:0 -> s:0 -> true:false
s :: s:0 -> s:0
GE :: s:0 -> s:0 -> c5
true :: true:false
c2 :: c -> c2
c5 :: c5 -> c5
0 :: s:0
false :: true:false

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

(13) 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:

   ge(v0, v1) -> null_ge [0]
   COND(v0, v1, v2) -> null_COND [0]
   GE(v0, v1) -> null_GE [0]

And the following fresh constants:    null_ge, null_COND, null_GE, const

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

(14)
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:

   MINUS(z0, z1) -> c(COND(ge(z0, s(z1)), z0, z1), GE(z0, s(z1))) [1]
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1))) [1]
   GE(s(z0), s(z1)) -> c5(GE(z0, z1)) [1]
   ge(0, s(z0)) -> false [0]
   ge(s(z0), s(z1)) -> ge(z0, z1) [0]
   ge(z0, 0) -> true [0]
   ge(v0, v1) -> null_ge [0]
   COND(v0, v1, v2) -> null_COND [0]
   GE(v0, v1) -> null_GE [0]

The TRS has the following type information:
MINUS :: s:0 -> s:0 -> c
c :: c2:null_COND -> c5:null_GE -> c
COND :: true:false:null_ge -> s:0 -> s:0 -> c2:null_COND
ge :: s:0 -> s:0 -> true:false:null_ge
s :: s:0 -> s:0
GE :: s:0 -> s:0 -> c5:null_GE
true :: true:false:null_ge
c2 :: c -> c2:null_COND
c5 :: c5:null_GE -> c5:null_GE
0 :: s:0
false :: true:false:null_ge
null_ge :: true:false:null_ge
null_COND :: c2:null_COND
null_GE :: c5:null_GE
const :: c

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

(15) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   true => 2
   0 => 0
   false => 1
   null_ge => 0
   null_COND => 0
   null_GE => 0
   const => 0

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

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   COND(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   COND(z, z', z'') -{ 1 }-> 1 + MINUS(z0, 1 + z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   GE(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   GE(z, z') -{ 1 }-> 1 + GE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MINUS(z, z') -{ 1 }-> 1 + COND(ge(z0, 1 + z1), z0, z1) + GE(z0, 1 + z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   ge(z, z') -{ 0 }-> ge(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   ge(z, z') -{ 0 }-> 2 :|: z = z0, z0 >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z0 >= 0, z' = 1 + z0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

(17) CompleteCoflocoProof (FINISHED)
Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V1, V, V4),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V4),0,[fun1(V1, V, V4, Out)],[V1 >= 0,V >= 0,V4 >= 0]).
eq(start(V1, V, V4),0,[fun2(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V4),0,[ge(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(fun(V1, V, Out),1,[ge(V3, 1 + V2, Ret010),fun1(Ret010, V3, V2, Ret01),fun2(V3, 1 + V2, Ret1)],[Out = 1 + Ret01 + Ret1,V1 = V3,V2 >= 0,V = V2,V3 >= 0]).
eq(fun1(V1, V, V4, Out),1,[fun(V6, 1 + V5, Ret11)],[Out = 1 + Ret11,V1 = 2,V5 >= 0,V6 >= 0,V = V6,V4 = V5]).
eq(fun2(V1, V, Out),1,[fun2(V8, V7, Ret12)],[Out = 1 + Ret12,V7 >= 0,V1 = 1 + V8,V8 >= 0,V = 1 + V7]).
eq(ge(V1, V, Out),0,[],[Out = 1,V9 >= 0,V = 1 + V9,V1 = 0]).
eq(ge(V1, V, Out),0,[ge(V10, V11, Ret)],[Out = Ret,V11 >= 0,V1 = 1 + V10,V10 >= 0,V = 1 + V11]).
eq(ge(V1, V, Out),0,[],[Out = 2,V1 = V12,V12 >= 0,V = 0]).
eq(ge(V1, V, Out),0,[],[Out = 0,V14 >= 0,V13 >= 0,V1 = V14,V = V13]).
eq(fun1(V1, V, V4, Out),0,[],[Out = 0,V16 >= 0,V4 = V17,V15 >= 0,V1 = V16,V = V15,V17 >= 0]).
eq(fun2(V1, V, Out),0,[],[Out = 0,V19 >= 0,V18 >= 0,V1 = V19,V = V18]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,V4,Out),[V1,V,V4],[Out]).
input_output_vars(fun2(V1,V,Out),[V1,V],[Out]).
input_output_vars(ge(V1,V,Out),[V1,V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [fun2/3]
1. recursive  : [ge/3]
2. recursive [non_tail] : [fun/3,fun1/4]
3. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into fun2/3
1. SCC is partially evaluated into ge/3
2. SCC is partially evaluated into fun/3
3. SCC is partially evaluated into start/3

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

### Specialization of cost equations fun2/3 
* CE 9 is refined into CE [14] 
* CE 8 is refined into CE [15] 


### Cost equations --> "Loop" of fun2/3 
* CEs [15] --> Loop 10 
* CEs [14] --> Loop 11 

### Ranking functions of CR fun2(V1,V,Out) 
* RF of phase [10]: [V,V1]

#### Partial ranking functions of CR fun2(V1,V,Out) 
* Partial RF of phase [10]:
  - RF of loop [10:1]:
    V
    V1


### Specialization of cost equations ge/3 
* CE 13 is refined into CE [16] 
* CE 12 is refined into CE [17] 
* CE 10 is refined into CE [18] 
* CE 11 is refined into CE [19] 


### Cost equations --> "Loop" of ge/3 
* CEs [19] --> Loop 12 
* CEs [16] --> Loop 13 
* CEs [17] --> Loop 14 
* CEs [18] --> Loop 15 

### Ranking functions of CR ge(V1,V,Out) 
* RF of phase [12]: [V,V1]

#### Partial ranking functions of CR ge(V1,V,Out) 
* Partial RF of phase [12]:
  - RF of loop [12:1]:
    V
    V1


### Specialization of cost equations fun/3 
* CE 7 is refined into CE [20,21] 
* CE 6 is refined into CE [22,23,24,25,26,27,28] 


### Cost equations --> "Loop" of fun/3 
* CEs [28] --> Loop 16 
* CEs [24,26] --> Loop 17 
* CEs [22,23,25,27] --> Loop 18 
* CEs [21] --> Loop 19 
* CEs [20] --> Loop 20 

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

#### Partial ranking functions of CR fun(V1,V,Out) 
* Partial RF of phase [19,20]:
  - RF of loop [19:1,20:1]:
    V1-V


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


### Cost equations --> "Loop" of start/3 
* CEs [39] --> Loop 21 
* CEs [30,31,32] --> Loop 22 
* CEs [29,33,34,35,36,37,38,40,41,42] --> Loop 23 

### Ranking functions of CR start(V1,V,V4) 

#### Partial ranking functions of CR start(V1,V,V4) 


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

#### Cost of chains of fun2(V1,V,Out):
* Chain [[10],11]: 1*it(10)+0
  Such that:it(10) =< Out

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

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


#### Cost of chains of ge(V1,V,Out):
* Chain [[12],15]: 0
  with precondition: [Out=1,V1>=1,V>=V1+1] 

* Chain [[12],14]: 0
  with precondition: [Out=2,V>=1,V1>=V] 

* Chain [[12],13]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

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

* Chain [14]: 0
  with precondition: [V=0,Out=2,V1>=0] 

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


#### Cost of chains of fun(V1,V,Out):
* Chain [[19,20],18]: 4*it(19)+1*s(3)+1
  Such that:aux(1) =< V1
aux(4) =< V1-V
it(19) =< aux(4)
s(3) =< it(19)*aux(1)

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

* Chain [[19,20],17]: 4*it(19)+1*s(3)+1*s(4)+1*s(5)+1
  Such that:s(4) =< V1+1
aux(5) =< V1
aux(6) =< V1-V
s(5) =< aux(5)
it(19) =< aux(6)
s(3) =< it(19)*aux(5)

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

* Chain [[19,20],16]: 4*it(19)+1*s(3)+1*s(6)+1
  Such that:aux(7) =< V1
aux(8) =< V1-V
s(6) =< aux(7)
it(19) =< aux(8)
s(3) =< it(19)*aux(7)

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

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

* Chain [17]: 1*s(4)+1*s(5)+1
  Such that:s(5) =< V1
s(4) =< V+1

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

* Chain [16]: 1*s(6)+1
  Such that:s(6) =< V+1

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


#### Cost of chains of start(V1,V,V4):
* Chain [23]: 1*s(25)+3*s(28)+12*s(29)+3*s(30)+2*s(33)+1*s(34)+1
  Such that:s(25) =< V1+1
s(27) =< V1-V
s(34) =< V
s(32) =< V+1
aux(12) =< V1
s(28) =< aux(12)
s(29) =< s(27)
s(30) =< s(29)*aux(12)
s(33) =< s(32)

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

* Chain [22]: 1*s(35)+3*s(38)+12*s(39)+3*s(40)+2*s(43)+2
  Such that:s(35) =< V+1
s(37) =< V-V4
s(42) =< V4+2
aux(13) =< V
s(38) =< aux(13)
s(39) =< s(37)
s(40) =< s(39)*aux(13)
s(43) =< s(42)

  with precondition: [V1=2,V>=0,V4>=0] 

* Chain [21]: 0
  with precondition: [V=0,V1>=0] 


Closed-form bounds of start(V1,V,V4): 
-------------------------------------
* Chain [23] with precondition: [V1>=0,V>=0] 
    - Upper bound: 3*V1+1+3*V1*nat(V1-V)+V+(V1+1)+(2*V+2)+nat(V1-V)*12 
    - Complexity: n^2 
* Chain [22] with precondition: [V1=2,V>=0,V4>=0] 
    - Upper bound: 3*V+2+3*V*nat(V-V4)+(V+1)+(2*V4+4)+nat(V-V4)*12 
    - Complexity: n^2 
* Chain [21] with precondition: [V=0,V1>=0] 
    - Upper bound: 0 
    - Complexity: constant 

### Maximum cost of start(V1,V,V4): 2*V+2+max([2*V+1+3*V*nat(V-V4)+nat(V4+2)*2+nat(V-V4)*12,3*V1*nat(V1-V)+3*V1+(V1+1)+(V+1)+nat(V1-V)*12]) 
Asymptotic class: n^2 
* Total analysis performed in 254 ms.


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

(18)
BOUNDS(1, n^2)

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

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, z1) -> cond(ge(z0, s(z1)), z0, z1)
   cond(false, z0, z1) -> 0
   cond(true, z0, z1) -> s(minus(z0, s(z1)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   MINUS(z0, z1) -> c(COND(ge(z0, s(z1)), z0, z1), GE(z0, s(z1)))
   COND(false, z0, z1) -> c1
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GE(z0, 0) -> c3
   GE(0, s(z0)) -> c4
   GE(s(z0), s(z1)) -> c5(GE(z0, z1))
S tuples:
   MINUS(z0, z1) -> c(COND(ge(z0, s(z1)), z0, z1), GE(z0, s(z1)))
   COND(false, z0, z1) -> c1
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GE(z0, 0) -> c3
   GE(0, s(z0)) -> c4
   GE(s(z0), s(z1)) -> c5(GE(z0, z1))
K tuples:none
Defined Rule Symbols:   minus_2, cond_3, ge_2

Defined Pair Symbols:   MINUS_2, COND_3, GE_2

Compound Symbols:   c_2, c1, c2_1, c3, c4, c5_1


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

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

(22)
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:

   MINUS(z0, z1) -> c(COND(ge(z0, s(z1)), z0, z1), GE(z0, s(z1)))
   COND(false, z0, z1) -> c1
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GE(z0, 0) -> c3
   GE(0, s(z0)) -> c4
   GE(s(z0), s(z1)) -> c5(GE(z0, z1))

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

   minus(z0, z1) -> cond(ge(z0, s(z1)), z0, z1)
   cond(false, z0, z1) -> 0
   cond(true, z0, z1) -> s(minus(z0, s(z1)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)

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

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

(24)
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:

   MINUS(z0, z1) -> c(COND(ge(z0, s(z1)), z0, z1), GE(z0, s(z1)))
   COND(false, z0, z1) -> c1
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GE(z0, 0') -> c3
   GE(0', s(z0)) -> c4
   GE(s(z0), s(z1)) -> c5(GE(z0, z1))

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

   minus(z0, z1) -> cond(ge(z0, s(z1)), z0, z1)
   cond(false, z0, z1) -> 0'
   cond(true, z0, z1) -> s(minus(z0, s(z1)))
   ge(z0, 0') -> true
   ge(0', s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)

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

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

(26)
Obligation:
Innermost TRS:
Rules:
MINUS(z0, z1) -> c(COND(ge(z0, s(z1)), z0, z1), GE(z0, s(z1)))
COND(false, z0, z1) -> c1
COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
GE(z0, 0') -> c3
GE(0', s(z0)) -> c4
GE(s(z0), s(z1)) -> c5(GE(z0, z1))
minus(z0, z1) -> cond(ge(z0, s(z1)), z0, z1)
cond(false, z0, z1) -> 0'
cond(true, z0, z1) -> s(minus(z0, s(z1)))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)

Types:
MINUS :: s:0' -> s:0' -> c
c :: c1:c2 -> c3:c4:c5 -> c
COND :: false:true -> s:0' -> s:0' -> c1:c2
ge :: s:0' -> s:0' -> false:true
s :: s:0' -> s:0'
GE :: s:0' -> s:0' -> c3:c4:c5
false :: false:true
c1 :: c1:c2
true :: false:true
c2 :: c -> c1:c2
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
minus :: s:0' -> s:0' -> s:0'
cond :: false:true -> s:0' -> s:0' -> s:0'
hole_c1_6 :: c
hole_s:0'2_6 :: s:0'
hole_c1:c23_6 :: c1:c2
hole_c3:c4:c54_6 :: c3:c4:c5
hole_false:true5_6 :: false:true
gen_s:0'6_6 :: Nat -> s:0'
gen_c3:c4:c57_6 :: Nat -> c3:c4:c5

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

(27) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
MINUS, ge, GE, minus

They will be analysed ascendingly in the following order:
ge < MINUS
GE < MINUS
ge < minus

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

(28)
Obligation:
Innermost TRS:
Rules:
MINUS(z0, z1) -> c(COND(ge(z0, s(z1)), z0, z1), GE(z0, s(z1)))
COND(false, z0, z1) -> c1
COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
GE(z0, 0') -> c3
GE(0', s(z0)) -> c4
GE(s(z0), s(z1)) -> c5(GE(z0, z1))
minus(z0, z1) -> cond(ge(z0, s(z1)), z0, z1)
cond(false, z0, z1) -> 0'
cond(true, z0, z1) -> s(minus(z0, s(z1)))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)

Types:
MINUS :: s:0' -> s:0' -> c
c :: c1:c2 -> c3:c4:c5 -> c
COND :: false:true -> s:0' -> s:0' -> c1:c2
ge :: s:0' -> s:0' -> false:true
s :: s:0' -> s:0'
GE :: s:0' -> s:0' -> c3:c4:c5
false :: false:true
c1 :: c1:c2
true :: false:true
c2 :: c -> c1:c2
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
minus :: s:0' -> s:0' -> s:0'
cond :: false:true -> s:0' -> s:0' -> s:0'
hole_c1_6 :: c
hole_s:0'2_6 :: s:0'
hole_c1:c23_6 :: c1:c2
hole_c3:c4:c54_6 :: c3:c4:c5
hole_false:true5_6 :: false:true
gen_s:0'6_6 :: Nat -> s:0'
gen_c3:c4:c57_6 :: Nat -> c3:c4:c5


Generator Equations:
gen_s:0'6_6(0) <=> 0'
gen_s:0'6_6(+(x, 1)) <=> s(gen_s:0'6_6(x))
gen_c3:c4:c57_6(0) <=> c3
gen_c3:c4:c57_6(+(x, 1)) <=> c5(gen_c3:c4:c57_6(x))


The following defined symbols remain to be analysed:
ge, MINUS, GE, minus

They will be analysed ascendingly in the following order:
ge < MINUS
GE < MINUS
ge < minus

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_s:0'6_6(n9_6), gen_s:0'6_6(n9_6)) -> true, rt in Omega(0)

Induction Base:
ge(gen_s:0'6_6(0), gen_s:0'6_6(0)) ->_R^Omega(0)
true

Induction Step:
ge(gen_s:0'6_6(+(n9_6, 1)), gen_s:0'6_6(+(n9_6, 1))) ->_R^Omega(0)
ge(gen_s:0'6_6(n9_6), gen_s:0'6_6(n9_6)) ->_IH
true

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

(30)
Obligation:
Innermost TRS:
Rules:
MINUS(z0, z1) -> c(COND(ge(z0, s(z1)), z0, z1), GE(z0, s(z1)))
COND(false, z0, z1) -> c1
COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
GE(z0, 0') -> c3
GE(0', s(z0)) -> c4
GE(s(z0), s(z1)) -> c5(GE(z0, z1))
minus(z0, z1) -> cond(ge(z0, s(z1)), z0, z1)
cond(false, z0, z1) -> 0'
cond(true, z0, z1) -> s(minus(z0, s(z1)))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)

Types:
MINUS :: s:0' -> s:0' -> c
c :: c1:c2 -> c3:c4:c5 -> c
COND :: false:true -> s:0' -> s:0' -> c1:c2
ge :: s:0' -> s:0' -> false:true
s :: s:0' -> s:0'
GE :: s:0' -> s:0' -> c3:c4:c5
false :: false:true
c1 :: c1:c2
true :: false:true
c2 :: c -> c1:c2
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
minus :: s:0' -> s:0' -> s:0'
cond :: false:true -> s:0' -> s:0' -> s:0'
hole_c1_6 :: c
hole_s:0'2_6 :: s:0'
hole_c1:c23_6 :: c1:c2
hole_c3:c4:c54_6 :: c3:c4:c5
hole_false:true5_6 :: false:true
gen_s:0'6_6 :: Nat -> s:0'
gen_c3:c4:c57_6 :: Nat -> c3:c4:c5


Lemmas:
ge(gen_s:0'6_6(n9_6), gen_s:0'6_6(n9_6)) -> true, rt in Omega(0)


Generator Equations:
gen_s:0'6_6(0) <=> 0'
gen_s:0'6_6(+(x, 1)) <=> s(gen_s:0'6_6(x))
gen_c3:c4:c57_6(0) <=> c3
gen_c3:c4:c57_6(+(x, 1)) <=> c5(gen_c3:c4:c57_6(x))


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

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

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
GE(gen_s:0'6_6(n289_6), gen_s:0'6_6(n289_6)) -> gen_c3:c4:c57_6(n289_6), rt in Omega(1 + n289_6)

Induction Base:
GE(gen_s:0'6_6(0), gen_s:0'6_6(0)) ->_R^Omega(1)
c3

Induction Step:
GE(gen_s:0'6_6(+(n289_6, 1)), gen_s:0'6_6(+(n289_6, 1))) ->_R^Omega(1)
c5(GE(gen_s:0'6_6(n289_6), gen_s:0'6_6(n289_6))) ->_IH
c5(gen_c3:c4:c57_6(c290_6))

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

(32)
Complex Obligation (BEST)

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

(33)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
MINUS(z0, z1) -> c(COND(ge(z0, s(z1)), z0, z1), GE(z0, s(z1)))
COND(false, z0, z1) -> c1
COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
GE(z0, 0') -> c3
GE(0', s(z0)) -> c4
GE(s(z0), s(z1)) -> c5(GE(z0, z1))
minus(z0, z1) -> cond(ge(z0, s(z1)), z0, z1)
cond(false, z0, z1) -> 0'
cond(true, z0, z1) -> s(minus(z0, s(z1)))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)

Types:
MINUS :: s:0' -> s:0' -> c
c :: c1:c2 -> c3:c4:c5 -> c
COND :: false:true -> s:0' -> s:0' -> c1:c2
ge :: s:0' -> s:0' -> false:true
s :: s:0' -> s:0'
GE :: s:0' -> s:0' -> c3:c4:c5
false :: false:true
c1 :: c1:c2
true :: false:true
c2 :: c -> c1:c2
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
minus :: s:0' -> s:0' -> s:0'
cond :: false:true -> s:0' -> s:0' -> s:0'
hole_c1_6 :: c
hole_s:0'2_6 :: s:0'
hole_c1:c23_6 :: c1:c2
hole_c3:c4:c54_6 :: c3:c4:c5
hole_false:true5_6 :: false:true
gen_s:0'6_6 :: Nat -> s:0'
gen_c3:c4:c57_6 :: Nat -> c3:c4:c5


Lemmas:
ge(gen_s:0'6_6(n9_6), gen_s:0'6_6(n9_6)) -> true, rt in Omega(0)


Generator Equations:
gen_s:0'6_6(0) <=> 0'
gen_s:0'6_6(+(x, 1)) <=> s(gen_s:0'6_6(x))
gen_c3:c4:c57_6(0) <=> c3
gen_c3:c4:c57_6(+(x, 1)) <=> c5(gen_c3:c4:c57_6(x))


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

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

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

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

(35)
BOUNDS(n^1, INF)

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

(36)
Obligation:
Innermost TRS:
Rules:
MINUS(z0, z1) -> c(COND(ge(z0, s(z1)), z0, z1), GE(z0, s(z1)))
COND(false, z0, z1) -> c1
COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
GE(z0, 0') -> c3
GE(0', s(z0)) -> c4
GE(s(z0), s(z1)) -> c5(GE(z0, z1))
minus(z0, z1) -> cond(ge(z0, s(z1)), z0, z1)
cond(false, z0, z1) -> 0'
cond(true, z0, z1) -> s(minus(z0, s(z1)))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)

Types:
MINUS :: s:0' -> s:0' -> c
c :: c1:c2 -> c3:c4:c5 -> c
COND :: false:true -> s:0' -> s:0' -> c1:c2
ge :: s:0' -> s:0' -> false:true
s :: s:0' -> s:0'
GE :: s:0' -> s:0' -> c3:c4:c5
false :: false:true
c1 :: c1:c2
true :: false:true
c2 :: c -> c1:c2
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
minus :: s:0' -> s:0' -> s:0'
cond :: false:true -> s:0' -> s:0' -> s:0'
hole_c1_6 :: c
hole_s:0'2_6 :: s:0'
hole_c1:c23_6 :: c1:c2
hole_c3:c4:c54_6 :: c3:c4:c5
hole_false:true5_6 :: false:true
gen_s:0'6_6 :: Nat -> s:0'
gen_c3:c4:c57_6 :: Nat -> c3:c4:c5


Lemmas:
ge(gen_s:0'6_6(n9_6), gen_s:0'6_6(n9_6)) -> true, rt in Omega(0)
GE(gen_s:0'6_6(n289_6), gen_s:0'6_6(n289_6)) -> gen_c3:c4:c57_6(n289_6), rt in Omega(1 + n289_6)


Generator Equations:
gen_s:0'6_6(0) <=> 0'
gen_s:0'6_6(+(x, 1)) <=> s(gen_s:0'6_6(x))
gen_c3:c4:c57_6(0) <=> c3
gen_c3:c4:c57_6(+(x, 1)) <=> c5(gen_c3:c4:c57_6(x))


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