WORST_CASE(Omega(n^1),O(n^1))
proof of input_GPlgMHbwDg.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxWeightedTrs
    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxTypedWeightedTrs
    (15) CompletionProof [UPPER BOUND(ID), 0 ms]
    (16) CpxTypedWeightedCompleteTrs
    (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) CompleteCoflocoProof [FINISHED, 197 ms]
    (20) BOUNDS(1, n^1)
(21) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) CdtProblem
    (23) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxRelTRS
    (25) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CpxRelTRS
    (27) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) typed CpxTrs
    (29) OrderProof [LOWER BOUND(ID), 11 ms]
    (30) typed CpxTrs
    (31) RewriteLemmaProof [LOWER BOUND(ID), 302 ms]
    (32) BEST
        (33) proven lower bound
            (34) LowerBoundPropagationProof [FINISHED, 0 ms]
            (35) BOUNDS(n^1, INF)
        (36) typed CpxTrs
            (37) RewriteLemmaProof [LOWER BOUND(ID), 135 ms]
            (38) typed CpxTrs
            (39) RewriteLemmaProof [LOWER BOUND(ID), 20 ms]
            (40) typed CpxTrs
            (41) RewriteLemmaProof [LOWER BOUND(ID), 64 ms]
            (42) 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:

   minus(0, Y) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   geq(X, 0) -> true
   geq(0, s(Y)) -> false
   geq(s(X), s(Y)) -> geq(X, Y)
   div(0, s(Y)) -> 0
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

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(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   geq(z0, 0) -> true
   geq(0, s(z0)) -> false
   geq(s(z0), s(z1)) -> geq(z0, z1)
   div(0, s(z0)) -> 0
   div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
Tuples:
   MINUS(0, z0) -> c
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   GEQ(z0, 0) -> c2
   GEQ(0, s(z0)) -> c3
   GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
   DIV(0, s(z0)) -> c5
   DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF(true, z0, z1) -> c8
   IF(false, z0, z1) -> c9
S tuples:
   MINUS(0, z0) -> c
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   GEQ(z0, 0) -> c2
   GEQ(0, s(z0)) -> c3
   GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
   DIV(0, s(z0)) -> c5
   DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF(true, z0, z1) -> c8
   IF(false, z0, z1) -> c9
K tuples:none
Defined Rule Symbols:   minus_2, geq_2, div_2, if_3

Defined Pair Symbols:   MINUS_2, GEQ_2, DIV_2, IF_3

Compound Symbols:   c, c1_1, c2, c3, c4_1, c5, c6_2, c7_3, c8, c9


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   DIV(0, s(z0)) -> c5
   GEQ(0, s(z0)) -> c3
   GEQ(z0, 0) -> c2
   MINUS(0, z0) -> c
   IF(false, z0, z1) -> c9
   IF(true, z0, z1) -> c8

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   geq(z0, 0) -> true
   geq(0, s(z0)) -> false
   geq(s(z0), s(z1)) -> geq(z0, z1)
   div(0, s(z0)) -> 0
   div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:   minus_2, geq_2, div_2, if_3

Defined Pair Symbols:   MINUS_2, GEQ_2, DIV_2

Compound Symbols:   c1_1, c4_1, c6_2, c7_3


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

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   geq(z0, 0) -> true
   geq(0, s(z0)) -> false
   geq(s(z0), s(z1)) -> geq(z0, z1)
   div(0, s(z0)) -> 0
   div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c6(GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c7(DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c6(GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c7(DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:   minus_2, geq_2, div_2, if_3

Defined Pair Symbols:   MINUS_2, GEQ_2, DIV_2

Compound Symbols:   c1_1, c4_1, c6_1, c7_2


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   geq(z0, 0) -> true
   geq(0, s(z0)) -> false
   geq(s(z0), s(z1)) -> geq(z0, z1)
   div(0, s(z0)) -> 0
   div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c6(GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c7(DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c6(GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c7(DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:   minus_2

Defined Pair Symbols:   MINUS_2, GEQ_2, DIV_2

Compound Symbols:   c1_1, c4_1, c6_1, c7_2


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

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

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c6(GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c7(DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))

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

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

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

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

(12)
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:

   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) [1]
   GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1)) [1]
   DIV(s(z0), s(z1)) -> c6(GEQ(z0, z1)) [1]
   DIV(s(z0), s(z1)) -> c7(DIV(minus(z0, z1), s(z1)), MINUS(z0, z1)) [1]
   minus(0, z0) -> 0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]

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

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

(14)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) [1]
   GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1)) [1]
   DIV(s(z0), s(z1)) -> c6(GEQ(z0, z1)) [1]
   DIV(s(z0), s(z1)) -> c7(DIV(minus(z0, z1), s(z1)), MINUS(z0, z1)) [1]
   minus(0, z0) -> 0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]

The TRS has the following type information:
MINUS :: s:0 -> s:0 -> c1
s :: s:0 -> s:0
c1 :: c1 -> c1
GEQ :: s:0 -> s:0 -> c4
c4 :: c4 -> c4
DIV :: s:0 -> s:0 -> c6:c7
c6 :: c4 -> c6:c7
c7 :: c6:c7 -> c1 -> c6:c7
minus :: s:0 -> s:0 -> s:0
0 :: s:0

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

(15) 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:

   minus(v0, v1) -> null_minus [0]
   MINUS(v0, v1) -> null_MINUS [0]
   GEQ(v0, v1) -> null_GEQ [0]
   DIV(v0, v1) -> null_DIV [0]

And the following fresh constants:    null_minus, null_MINUS, null_GEQ, null_DIV

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

(16)
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(s(z0), s(z1)) -> c1(MINUS(z0, z1)) [1]
   GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1)) [1]
   DIV(s(z0), s(z1)) -> c6(GEQ(z0, z1)) [1]
   DIV(s(z0), s(z1)) -> c7(DIV(minus(z0, z1), s(z1)), MINUS(z0, z1)) [1]
   minus(0, z0) -> 0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   minus(v0, v1) -> null_minus [0]
   MINUS(v0, v1) -> null_MINUS [0]
   GEQ(v0, v1) -> null_GEQ [0]
   DIV(v0, v1) -> null_DIV [0]

The TRS has the following type information:
MINUS :: s:0:null_minus -> s:0:null_minus -> c1:null_MINUS
s :: s:0:null_minus -> s:0:null_minus
c1 :: c1:null_MINUS -> c1:null_MINUS
GEQ :: s:0:null_minus -> s:0:null_minus -> c4:null_GEQ
c4 :: c4:null_GEQ -> c4:null_GEQ
DIV :: s:0:null_minus -> s:0:null_minus -> c6:c7:null_DIV
c6 :: c4:null_GEQ -> c6:c7:null_DIV
c7 :: c6:c7:null_DIV -> c1:null_MINUS -> c6:c7:null_DIV
minus :: s:0:null_minus -> s:0:null_minus -> s:0:null_minus
0 :: s:0:null_minus
null_minus :: s:0:null_minus
null_MINUS :: c1:null_MINUS
null_GEQ :: c4:null_GEQ
null_DIV :: c6:c7:null_DIV

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

(17) 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_minus => 0
   null_MINUS => 0
   null_GEQ => 0
   null_DIV => 0

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

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   DIV(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   DIV(z, z') -{ 1 }-> 1 + GEQ(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   DIV(z, z') -{ 1 }-> 1 + DIV(minus(z0, z1), 1 + z1) + MINUS(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   GEQ(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   GEQ(z, z') -{ 1 }-> 1 + GEQ(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MINUS(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> minus(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[fun1(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[fun2(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(fun(V1, V, Out),1,[fun(V3, V2, Ret1)],[Out = 1 + Ret1,V2 >= 0,V1 = 1 + V3,V3 >= 0,V = 1 + V2]).
eq(fun1(V1, V, Out),1,[fun1(V5, V4, Ret11)],[Out = 1 + Ret11,V4 >= 0,V1 = 1 + V5,V5 >= 0,V = 1 + V4]).
eq(fun2(V1, V, Out),1,[fun1(V7, V6, Ret12)],[Out = 1 + Ret12,V6 >= 0,V1 = 1 + V7,V7 >= 0,V = 1 + V6]).
eq(fun2(V1, V, Out),1,[minus(V8, V9, Ret010),fun2(Ret010, 1 + V9, Ret01),fun(V8, V9, Ret13)],[Out = 1 + Ret01 + Ret13,V9 >= 0,V1 = 1 + V8,V8 >= 0,V = 1 + V9]).
eq(minus(V1, V, Out),0,[],[Out = 0,V10 >= 0,V1 = 0,V = V10]).
eq(minus(V1, V, Out),0,[minus(V12, V11, Ret)],[Out = Ret,V11 >= 0,V1 = 1 + V12,V12 >= 0,V = 1 + V11]).
eq(minus(V1, V, Out),0,[],[Out = 0,V14 >= 0,V13 >= 0,V1 = V14,V = V13]).
eq(fun(V1, V, Out),0,[],[Out = 0,V16 >= 0,V15 >= 0,V1 = V16,V = V15]).
eq(fun1(V1, V, Out),0,[],[Out = 0,V18 >= 0,V17 >= 0,V1 = V18,V = V17]).
eq(fun2(V1, V, Out),0,[],[Out = 0,V19 >= 0,V20 >= 0,V1 = V19,V = V20]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun2(V1,V,Out),[V1,V],[Out]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).


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

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

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into fun/3
1. SCC is partially evaluated into fun1/3
2. SCC is partially evaluated into minus/3
3. SCC is partially evaluated into fun2/3
4. SCC is partially evaluated into start/2

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

### Specialization of cost equations fun/3 
* CE 6 is refined into CE [14] 
* CE 5 is refined into CE [15] 


### Cost equations --> "Loop" of fun/3 
* CEs [15] --> Loop 11 
* CEs [14] --> Loop 12 

### Ranking functions of CR fun(V1,V,Out) 
* RF of phase [11]: [V,V1]

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


### Specialization of cost equations fun1/3 
* CE 8 is refined into CE [16] 
* CE 7 is refined into CE [17] 


### Cost equations --> "Loop" of fun1/3 
* CEs [17] --> Loop 13 
* CEs [16] --> Loop 14 

### Ranking functions of CR fun1(V1,V,Out) 
* RF of phase [13]: [V,V1]

#### Partial ranking functions of CR fun1(V1,V,Out) 
* Partial RF of phase [13]:
  - RF of loop [13:1]:
    V
    V1


### Specialization of cost equations minus/3 
* CE 12 is refined into CE [18] 
* CE 13 is refined into CE [19] 


### Cost equations --> "Loop" of minus/3 
* CEs [19] --> Loop 15 
* CEs [18] --> Loop 16 

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [15]: [V,V1]

#### Partial ranking functions of CR minus(V1,V,Out) 
* Partial RF of phase [15]:
  - RF of loop [15:1]:
    V
    V1


### Specialization of cost equations fun2/3 
* CE 9 is refined into CE [20,21] 
* CE 11 is refined into CE [22] 
* CE 10 is refined into CE [23,24] 


### Cost equations --> "Loop" of fun2/3 
* CEs [24] --> Loop 17 
* CEs [23] --> Loop 18 
* CEs [21] --> Loop 19 
* CEs [20] --> Loop 20 
* CEs [22] --> Loop 21 

### Ranking functions of CR fun2(V1,V,Out) 

#### Partial ranking functions of CR fun2(V1,V,Out) 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [25,26] 
* CE 2 is refined into CE [27,28] 
* CE 3 is refined into CE [29,30,31] 
* CE 4 is refined into CE [32] 


### Cost equations --> "Loop" of start/2 
* CEs [25,26,27,28,29,30,31,32] --> Loop 22 

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

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


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

#### Cost of chains of fun(V1,V,Out):
* Chain [[11],12]: 1*it(11)+0
  Such that:it(11) =< Out

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

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


#### Cost of chains of fun1(V1,V,Out):
* Chain [[13],14]: 1*it(13)+0
  Such that:it(13) =< Out

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

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


#### Cost of chains of minus(V1,V,Out):
* Chain [[15],16]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

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


#### Cost of chains of fun2(V1,V,Out):
* Chain [21]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [20]: 1
  with precondition: [Out=1,V1>=1,V>=1] 

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

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

* Chain [18,21]: 1
  with precondition: [Out=1,V1>=1,V>=1] 

* Chain [17,21]: 1*s(2)+1
  Such that:s(2) =< V1

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


#### Cost of chains of start(V1,V):
* Chain [22]: 3*s(5)+1*s(7)+1
  Such that:s(7) =< V1
aux(1) =< V
s(5) =< aux(1)

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


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [22] with precondition: [V1>=0,V>=0] 
    - Upper bound: V1+3*V+1 
    - Complexity: n 

### Maximum cost of start(V1,V): V1+3*V+1 
Asymptotic class: n 
* Total analysis performed in 164 ms.


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

(20)
BOUNDS(1, n^1)

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

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   geq(z0, 0) -> true
   geq(0, s(z0)) -> false
   geq(s(z0), s(z1)) -> geq(z0, z1)
   div(0, s(z0)) -> 0
   div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
Tuples:
   MINUS(0, z0) -> c
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   GEQ(z0, 0) -> c2
   GEQ(0, s(z0)) -> c3
   GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
   DIV(0, s(z0)) -> c5
   DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF(true, z0, z1) -> c8
   IF(false, z0, z1) -> c9
S tuples:
   MINUS(0, z0) -> c
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   GEQ(z0, 0) -> c2
   GEQ(0, s(z0)) -> c3
   GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
   DIV(0, s(z0)) -> c5
   DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF(true, z0, z1) -> c8
   IF(false, z0, z1) -> c9
K tuples:none
Defined Rule Symbols:   minus_2, geq_2, div_2, if_3

Defined Pair Symbols:   MINUS_2, GEQ_2, DIV_2, IF_3

Compound Symbols:   c, c1_1, c2, c3, c4_1, c5, c6_2, c7_3, c8, c9


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

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

(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(0, z0) -> c
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   GEQ(z0, 0) -> c2
   GEQ(0, s(z0)) -> c3
   GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
   DIV(0, s(z0)) -> c5
   DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF(true, z0, z1) -> c8
   IF(false, z0, z1) -> c9

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

   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   geq(z0, 0) -> true
   geq(0, s(z0)) -> false
   geq(s(z0), s(z1)) -> geq(z0, z1)
   div(0, s(z0)) -> 0
   div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1

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

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

(26)
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(0', z0) -> c
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   GEQ(z0, 0') -> c2
   GEQ(0', s(z0)) -> c3
   GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
   DIV(0', s(z0)) -> c5
   DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF(true, z0, z1) -> c8
   IF(false, z0, z1) -> c9

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

   minus(0', z0) -> 0'
   minus(s(z0), s(z1)) -> minus(z0, z1)
   geq(z0, 0') -> true
   geq(0', s(z0)) -> false
   geq(s(z0), s(z1)) -> geq(z0, z1)
   div(0', s(z0)) -> 0'
   div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0')
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1

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

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

(28)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
GEQ(z0, 0') -> c2
GEQ(0', s(z0)) -> c3
GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
DIV(0', s(z0)) -> c5
DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), GEQ(z0, z1))
DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(true, z0, z1) -> c8
IF(false, z0, z1) -> c9
minus(0', z0) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
geq(z0, 0') -> true
geq(0', s(z0)) -> false
geq(s(z0), s(z1)) -> geq(z0, z1)
div(0', s(z0)) -> 0'
div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0')
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
GEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
DIV :: 0':s -> 0':s -> c5:c6:c7
c5 :: c5:c6:c7
c6 :: c8:c9 -> c2:c3:c4 -> c5:c6:c7
IF :: true:false -> 0':s -> 0':s -> c8:c9
geq :: 0':s -> 0':s -> true:false
div :: 0':s -> 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
c7 :: c8:c9 -> c5:c6:c7 -> c:c1 -> c5:c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c8:c9
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_10 :: c:c1
hole_0':s2_10 :: 0':s
hole_c2:c3:c43_10 :: c2:c3:c4
hole_c5:c6:c74_10 :: c5:c6:c7
hole_c8:c95_10 :: c8:c9
hole_true:false6_10 :: true:false
gen_c:c17_10 :: Nat -> c:c1
gen_0':s8_10 :: Nat -> 0':s
gen_c2:c3:c49_10 :: Nat -> c2:c3:c4
gen_c5:c6:c710_10 :: Nat -> c5:c6:c7

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

(29) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
MINUS, GEQ, DIV, geq, div, minus

They will be analysed ascendingly in the following order:
MINUS < DIV
GEQ < DIV
geq < DIV
div < DIV
minus < DIV
geq < div
minus < div

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

(30)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
GEQ(z0, 0') -> c2
GEQ(0', s(z0)) -> c3
GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
DIV(0', s(z0)) -> c5
DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), GEQ(z0, z1))
DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(true, z0, z1) -> c8
IF(false, z0, z1) -> c9
minus(0', z0) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
geq(z0, 0') -> true
geq(0', s(z0)) -> false
geq(s(z0), s(z1)) -> geq(z0, z1)
div(0', s(z0)) -> 0'
div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0')
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
GEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
DIV :: 0':s -> 0':s -> c5:c6:c7
c5 :: c5:c6:c7
c6 :: c8:c9 -> c2:c3:c4 -> c5:c6:c7
IF :: true:false -> 0':s -> 0':s -> c8:c9
geq :: 0':s -> 0':s -> true:false
div :: 0':s -> 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
c7 :: c8:c9 -> c5:c6:c7 -> c:c1 -> c5:c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c8:c9
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_10 :: c:c1
hole_0':s2_10 :: 0':s
hole_c2:c3:c43_10 :: c2:c3:c4
hole_c5:c6:c74_10 :: c5:c6:c7
hole_c8:c95_10 :: c8:c9
hole_true:false6_10 :: true:false
gen_c:c17_10 :: Nat -> c:c1
gen_0':s8_10 :: Nat -> 0':s
gen_c2:c3:c49_10 :: Nat -> c2:c3:c4
gen_c5:c6:c710_10 :: Nat -> c5:c6:c7


Generator Equations:
gen_c:c17_10(0) <=> c
gen_c:c17_10(+(x, 1)) <=> c1(gen_c:c17_10(x))
gen_0':s8_10(0) <=> 0'
gen_0':s8_10(+(x, 1)) <=> s(gen_0':s8_10(x))
gen_c2:c3:c49_10(0) <=> c2
gen_c2:c3:c49_10(+(x, 1)) <=> c4(gen_c2:c3:c49_10(x))
gen_c5:c6:c710_10(0) <=> c5
gen_c5:c6:c710_10(+(x, 1)) <=> c7(c8, gen_c5:c6:c710_10(x), c)


The following defined symbols remain to be analysed:
MINUS, GEQ, DIV, geq, div, minus

They will be analysed ascendingly in the following order:
MINUS < DIV
GEQ < DIV
geq < DIV
div < DIV
minus < DIV
geq < div
minus < div

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MINUS(gen_0':s8_10(n12_10), gen_0':s8_10(n12_10)) -> gen_c:c17_10(n12_10), rt in Omega(1 + n12_10)

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

Induction Step:
MINUS(gen_0':s8_10(+(n12_10, 1)), gen_0':s8_10(+(n12_10, 1))) ->_R^Omega(1)
c1(MINUS(gen_0':s8_10(n12_10), gen_0':s8_10(n12_10))) ->_IH
c1(gen_c:c17_10(c13_10))

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(0', z0) -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
GEQ(z0, 0') -> c2
GEQ(0', s(z0)) -> c3
GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
DIV(0', s(z0)) -> c5
DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), GEQ(z0, z1))
DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(true, z0, z1) -> c8
IF(false, z0, z1) -> c9
minus(0', z0) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
geq(z0, 0') -> true
geq(0', s(z0)) -> false
geq(s(z0), s(z1)) -> geq(z0, z1)
div(0', s(z0)) -> 0'
div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0')
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
GEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
DIV :: 0':s -> 0':s -> c5:c6:c7
c5 :: c5:c6:c7
c6 :: c8:c9 -> c2:c3:c4 -> c5:c6:c7
IF :: true:false -> 0':s -> 0':s -> c8:c9
geq :: 0':s -> 0':s -> true:false
div :: 0':s -> 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
c7 :: c8:c9 -> c5:c6:c7 -> c:c1 -> c5:c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c8:c9
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_10 :: c:c1
hole_0':s2_10 :: 0':s
hole_c2:c3:c43_10 :: c2:c3:c4
hole_c5:c6:c74_10 :: c5:c6:c7
hole_c8:c95_10 :: c8:c9
hole_true:false6_10 :: true:false
gen_c:c17_10 :: Nat -> c:c1
gen_0':s8_10 :: Nat -> 0':s
gen_c2:c3:c49_10 :: Nat -> c2:c3:c4
gen_c5:c6:c710_10 :: Nat -> c5:c6:c7


Generator Equations:
gen_c:c17_10(0) <=> c
gen_c:c17_10(+(x, 1)) <=> c1(gen_c:c17_10(x))
gen_0':s8_10(0) <=> 0'
gen_0':s8_10(+(x, 1)) <=> s(gen_0':s8_10(x))
gen_c2:c3:c49_10(0) <=> c2
gen_c2:c3:c49_10(+(x, 1)) <=> c4(gen_c2:c3:c49_10(x))
gen_c5:c6:c710_10(0) <=> c5
gen_c5:c6:c710_10(+(x, 1)) <=> c7(c8, gen_c5:c6:c710_10(x), c)


The following defined symbols remain to be analysed:
MINUS, GEQ, DIV, geq, div, minus

They will be analysed ascendingly in the following order:
MINUS < DIV
GEQ < DIV
geq < DIV
div < DIV
minus < DIV
geq < div
minus < div

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

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

(35)
BOUNDS(n^1, INF)

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

(36)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
GEQ(z0, 0') -> c2
GEQ(0', s(z0)) -> c3
GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
DIV(0', s(z0)) -> c5
DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), GEQ(z0, z1))
DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(true, z0, z1) -> c8
IF(false, z0, z1) -> c9
minus(0', z0) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
geq(z0, 0') -> true
geq(0', s(z0)) -> false
geq(s(z0), s(z1)) -> geq(z0, z1)
div(0', s(z0)) -> 0'
div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0')
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
GEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
DIV :: 0':s -> 0':s -> c5:c6:c7
c5 :: c5:c6:c7
c6 :: c8:c9 -> c2:c3:c4 -> c5:c6:c7
IF :: true:false -> 0':s -> 0':s -> c8:c9
geq :: 0':s -> 0':s -> true:false
div :: 0':s -> 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
c7 :: c8:c9 -> c5:c6:c7 -> c:c1 -> c5:c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c8:c9
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_10 :: c:c1
hole_0':s2_10 :: 0':s
hole_c2:c3:c43_10 :: c2:c3:c4
hole_c5:c6:c74_10 :: c5:c6:c7
hole_c8:c95_10 :: c8:c9
hole_true:false6_10 :: true:false
gen_c:c17_10 :: Nat -> c:c1
gen_0':s8_10 :: Nat -> 0':s
gen_c2:c3:c49_10 :: Nat -> c2:c3:c4
gen_c5:c6:c710_10 :: Nat -> c5:c6:c7


Lemmas:
MINUS(gen_0':s8_10(n12_10), gen_0':s8_10(n12_10)) -> gen_c:c17_10(n12_10), rt in Omega(1 + n12_10)


Generator Equations:
gen_c:c17_10(0) <=> c
gen_c:c17_10(+(x, 1)) <=> c1(gen_c:c17_10(x))
gen_0':s8_10(0) <=> 0'
gen_0':s8_10(+(x, 1)) <=> s(gen_0':s8_10(x))
gen_c2:c3:c49_10(0) <=> c2
gen_c2:c3:c49_10(+(x, 1)) <=> c4(gen_c2:c3:c49_10(x))
gen_c5:c6:c710_10(0) <=> c5
gen_c5:c6:c710_10(+(x, 1)) <=> c7(c8, gen_c5:c6:c710_10(x), c)


The following defined symbols remain to be analysed:
GEQ, DIV, geq, div, minus

They will be analysed ascendingly in the following order:
GEQ < DIV
geq < DIV
div < DIV
minus < DIV
geq < div
minus < div

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

(37) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
GEQ(gen_0':s8_10(n402_10), gen_0':s8_10(n402_10)) -> gen_c2:c3:c49_10(n402_10), rt in Omega(1 + n402_10)

Induction Base:
GEQ(gen_0':s8_10(0), gen_0':s8_10(0)) ->_R^Omega(1)
c2

Induction Step:
GEQ(gen_0':s8_10(+(n402_10, 1)), gen_0':s8_10(+(n402_10, 1))) ->_R^Omega(1)
c4(GEQ(gen_0':s8_10(n402_10), gen_0':s8_10(n402_10))) ->_IH
c4(gen_c2:c3:c49_10(c403_10))

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

(38)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
GEQ(z0, 0') -> c2
GEQ(0', s(z0)) -> c3
GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
DIV(0', s(z0)) -> c5
DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), GEQ(z0, z1))
DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(true, z0, z1) -> c8
IF(false, z0, z1) -> c9
minus(0', z0) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
geq(z0, 0') -> true
geq(0', s(z0)) -> false
geq(s(z0), s(z1)) -> geq(z0, z1)
div(0', s(z0)) -> 0'
div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0')
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
GEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
DIV :: 0':s -> 0':s -> c5:c6:c7
c5 :: c5:c6:c7
c6 :: c8:c9 -> c2:c3:c4 -> c5:c6:c7
IF :: true:false -> 0':s -> 0':s -> c8:c9
geq :: 0':s -> 0':s -> true:false
div :: 0':s -> 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
c7 :: c8:c9 -> c5:c6:c7 -> c:c1 -> c5:c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c8:c9
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_10 :: c:c1
hole_0':s2_10 :: 0':s
hole_c2:c3:c43_10 :: c2:c3:c4
hole_c5:c6:c74_10 :: c5:c6:c7
hole_c8:c95_10 :: c8:c9
hole_true:false6_10 :: true:false
gen_c:c17_10 :: Nat -> c:c1
gen_0':s8_10 :: Nat -> 0':s
gen_c2:c3:c49_10 :: Nat -> c2:c3:c4
gen_c5:c6:c710_10 :: Nat -> c5:c6:c7


Lemmas:
MINUS(gen_0':s8_10(n12_10), gen_0':s8_10(n12_10)) -> gen_c:c17_10(n12_10), rt in Omega(1 + n12_10)
GEQ(gen_0':s8_10(n402_10), gen_0':s8_10(n402_10)) -> gen_c2:c3:c49_10(n402_10), rt in Omega(1 + n402_10)


Generator Equations:
gen_c:c17_10(0) <=> c
gen_c:c17_10(+(x, 1)) <=> c1(gen_c:c17_10(x))
gen_0':s8_10(0) <=> 0'
gen_0':s8_10(+(x, 1)) <=> s(gen_0':s8_10(x))
gen_c2:c3:c49_10(0) <=> c2
gen_c2:c3:c49_10(+(x, 1)) <=> c4(gen_c2:c3:c49_10(x))
gen_c5:c6:c710_10(0) <=> c5
gen_c5:c6:c710_10(+(x, 1)) <=> c7(c8, gen_c5:c6:c710_10(x), c)


The following defined symbols remain to be analysed:
geq, DIV, div, minus

They will be analysed ascendingly in the following order:
geq < DIV
div < DIV
minus < DIV
geq < div
minus < div

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

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
geq(gen_0':s8_10(n1051_10), gen_0':s8_10(n1051_10)) -> true, rt in Omega(0)

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

Induction Step:
geq(gen_0':s8_10(+(n1051_10, 1)), gen_0':s8_10(+(n1051_10, 1))) ->_R^Omega(0)
geq(gen_0':s8_10(n1051_10), gen_0':s8_10(n1051_10)) ->_IH
true

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

(40)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
GEQ(z0, 0') -> c2
GEQ(0', s(z0)) -> c3
GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
DIV(0', s(z0)) -> c5
DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), GEQ(z0, z1))
DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(true, z0, z1) -> c8
IF(false, z0, z1) -> c9
minus(0', z0) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
geq(z0, 0') -> true
geq(0', s(z0)) -> false
geq(s(z0), s(z1)) -> geq(z0, z1)
div(0', s(z0)) -> 0'
div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0')
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
GEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
DIV :: 0':s -> 0':s -> c5:c6:c7
c5 :: c5:c6:c7
c6 :: c8:c9 -> c2:c3:c4 -> c5:c6:c7
IF :: true:false -> 0':s -> 0':s -> c8:c9
geq :: 0':s -> 0':s -> true:false
div :: 0':s -> 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
c7 :: c8:c9 -> c5:c6:c7 -> c:c1 -> c5:c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c8:c9
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_10 :: c:c1
hole_0':s2_10 :: 0':s
hole_c2:c3:c43_10 :: c2:c3:c4
hole_c5:c6:c74_10 :: c5:c6:c7
hole_c8:c95_10 :: c8:c9
hole_true:false6_10 :: true:false
gen_c:c17_10 :: Nat -> c:c1
gen_0':s8_10 :: Nat -> 0':s
gen_c2:c3:c49_10 :: Nat -> c2:c3:c4
gen_c5:c6:c710_10 :: Nat -> c5:c6:c7


Lemmas:
MINUS(gen_0':s8_10(n12_10), gen_0':s8_10(n12_10)) -> gen_c:c17_10(n12_10), rt in Omega(1 + n12_10)
GEQ(gen_0':s8_10(n402_10), gen_0':s8_10(n402_10)) -> gen_c2:c3:c49_10(n402_10), rt in Omega(1 + n402_10)
geq(gen_0':s8_10(n1051_10), gen_0':s8_10(n1051_10)) -> true, rt in Omega(0)


Generator Equations:
gen_c:c17_10(0) <=> c
gen_c:c17_10(+(x, 1)) <=> c1(gen_c:c17_10(x))
gen_0':s8_10(0) <=> 0'
gen_0':s8_10(+(x, 1)) <=> s(gen_0':s8_10(x))
gen_c2:c3:c49_10(0) <=> c2
gen_c2:c3:c49_10(+(x, 1)) <=> c4(gen_c2:c3:c49_10(x))
gen_c5:c6:c710_10(0) <=> c5
gen_c5:c6:c710_10(+(x, 1)) <=> c7(c8, gen_c5:c6:c710_10(x), c)


The following defined symbols remain to be analysed:
minus, DIV, div

They will be analysed ascendingly in the following order:
div < DIV
minus < DIV
minus < div

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

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s8_10(n1411_10), gen_0':s8_10(n1411_10)) -> gen_0':s8_10(0), rt in Omega(0)

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

Induction Step:
minus(gen_0':s8_10(+(n1411_10, 1)), gen_0':s8_10(+(n1411_10, 1))) ->_R^Omega(0)
minus(gen_0':s8_10(n1411_10), gen_0':s8_10(n1411_10)) ->_IH
gen_0':s8_10(0)

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

(42)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
GEQ(z0, 0') -> c2
GEQ(0', s(z0)) -> c3
GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
DIV(0', s(z0)) -> c5
DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), GEQ(z0, z1))
DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(true, z0, z1) -> c8
IF(false, z0, z1) -> c9
minus(0', z0) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
geq(z0, 0') -> true
geq(0', s(z0)) -> false
geq(s(z0), s(z1)) -> geq(z0, z1)
div(0', s(z0)) -> 0'
div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0')
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
GEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
DIV :: 0':s -> 0':s -> c5:c6:c7
c5 :: c5:c6:c7
c6 :: c8:c9 -> c2:c3:c4 -> c5:c6:c7
IF :: true:false -> 0':s -> 0':s -> c8:c9
geq :: 0':s -> 0':s -> true:false
div :: 0':s -> 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
c7 :: c8:c9 -> c5:c6:c7 -> c:c1 -> c5:c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c8:c9
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_10 :: c:c1
hole_0':s2_10 :: 0':s
hole_c2:c3:c43_10 :: c2:c3:c4
hole_c5:c6:c74_10 :: c5:c6:c7
hole_c8:c95_10 :: c8:c9
hole_true:false6_10 :: true:false
gen_c:c17_10 :: Nat -> c:c1
gen_0':s8_10 :: Nat -> 0':s
gen_c2:c3:c49_10 :: Nat -> c2:c3:c4
gen_c5:c6:c710_10 :: Nat -> c5:c6:c7


Lemmas:
MINUS(gen_0':s8_10(n12_10), gen_0':s8_10(n12_10)) -> gen_c:c17_10(n12_10), rt in Omega(1 + n12_10)
GEQ(gen_0':s8_10(n402_10), gen_0':s8_10(n402_10)) -> gen_c2:c3:c49_10(n402_10), rt in Omega(1 + n402_10)
geq(gen_0':s8_10(n1051_10), gen_0':s8_10(n1051_10)) -> true, rt in Omega(0)
minus(gen_0':s8_10(n1411_10), gen_0':s8_10(n1411_10)) -> gen_0':s8_10(0), rt in Omega(0)


Generator Equations:
gen_c:c17_10(0) <=> c
gen_c:c17_10(+(x, 1)) <=> c1(gen_c:c17_10(x))
gen_0':s8_10(0) <=> 0'
gen_0':s8_10(+(x, 1)) <=> s(gen_0':s8_10(x))
gen_c2:c3:c49_10(0) <=> c2
gen_c2:c3:c49_10(+(x, 1)) <=> c4(gen_c2:c3:c49_10(x))
gen_c5:c6:c710_10(0) <=> c5
gen_c5:c6:c710_10(+(x, 1)) <=> c7(c8, gen_c5:c6:c710_10(x), c)


The following defined symbols remain to be analysed:
div, DIV

They will be analysed ascendingly in the following order:
div < DIV