WORST_CASE(Omega(n^1),O(n^2))
proof of input_zSOMMWB3If.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# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished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) 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, 298 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), 11 ms]
    (28) typed CpxTrs
    (29) RewriteLemmaProof [LOWER BOUND(ID), 457 ms]
    (30) BEST
        (31) proven lower bound
            (32) LowerBoundPropagationProof [FINISHED, 0 ms]
            (33) BOUNDS(n^1, INF)
        (34) typed CpxTrs
            (35) RewriteLemmaProof [LOWER BOUND(ID), 81 ms]
            (36) typed CpxTrs
            (37) RewriteLemmaProof [LOWER BOUND(ID), 621 ms]
            (38) typed CpxTrs
            (39) RewriteLemmaProof [LOWER BOUND(ID), 96 ms]
            (40) BOUNDS(1, INF)


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

(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:

   sub(0, 0) -> 0
   sub(s(x), 0) -> s(x)
   sub(0, s(x)) -> 0
   sub(s(x), s(y)) -> sub(x, y)
   zero(nil) -> zero2(0, nil)
   zero(cons(x, xs)) -> zero2(sub(x, x), cons(x, xs))
   zero2(0, nil) -> nil
   zero2(0, cons(x, xs)) -> cons(sub(x, x), zero(xs))
   zero2(s(y), nil) -> zero(nil)
   zero2(s(y), cons(x, xs)) -> zero(cons(x, xs))

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:
   sub(0, 0) -> 0
   sub(s(z0), 0) -> s(z0)
   sub(0, s(z0)) -> 0
   sub(s(z0), s(z1)) -> sub(z0, z1)
   zero(nil) -> zero2(0, nil)
   zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
   zero2(0, nil) -> nil
   zero2(0, cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
   zero2(s(z0), nil) -> zero(nil)
   zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))
Tuples:
   SUB(0, 0) -> c
   SUB(s(z0), 0) -> c1
   SUB(0, s(z0)) -> c2
   SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
   ZERO(nil) -> c4(ZERO2(0, nil))
   ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
   ZERO2(0, nil) -> c6
   ZERO2(0, cons(z0, z1)) -> c7(SUB(z0, z0))
   ZERO2(0, cons(z0, z1)) -> c8(ZERO(z1))
   ZERO2(s(z0), nil) -> c9(ZERO(nil))
   ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))
S tuples:
   SUB(0, 0) -> c
   SUB(s(z0), 0) -> c1
   SUB(0, s(z0)) -> c2
   SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
   ZERO(nil) -> c4(ZERO2(0, nil))
   ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
   ZERO2(0, nil) -> c6
   ZERO2(0, cons(z0, z1)) -> c7(SUB(z0, z0))
   ZERO2(0, cons(z0, z1)) -> c8(ZERO(z1))
   ZERO2(s(z0), nil) -> c9(ZERO(nil))
   ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))
K tuples:none
Defined Rule Symbols:   sub_2, zero_1, zero2_2

Defined Pair Symbols:   SUB_2, ZERO_1, ZERO2_2

Compound Symbols:   c, c1, c2, c3_1, c4_1, c5_2, c6, c7_1, c8_1, c9_1, c10_1


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   SUB(0, 0) -> c
   ZERO2(s(z0), nil) -> c9(ZERO(nil))
   ZERO(nil) -> c4(ZERO2(0, nil))
   SUB(0, s(z0)) -> c2
   SUB(s(z0), 0) -> c1
   ZERO2(0, nil) -> c6

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sub(0, 0) -> 0
   sub(s(z0), 0) -> s(z0)
   sub(0, s(z0)) -> 0
   sub(s(z0), s(z1)) -> sub(z0, z1)
   zero(nil) -> zero2(0, nil)
   zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
   zero2(0, nil) -> nil
   zero2(0, cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
   zero2(s(z0), nil) -> zero(nil)
   zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))
Tuples:
   SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
   ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
   ZERO2(0, cons(z0, z1)) -> c7(SUB(z0, z0))
   ZERO2(0, cons(z0, z1)) -> c8(ZERO(z1))
   ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))
S tuples:
   SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
   ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
   ZERO2(0, cons(z0, z1)) -> c7(SUB(z0, z0))
   ZERO2(0, cons(z0, z1)) -> c8(ZERO(z1))
   ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))
K tuples:none
Defined Rule Symbols:   sub_2, zero_1, zero2_2

Defined Pair Symbols:   SUB_2, ZERO_1, ZERO2_2

Compound Symbols:   c3_1, c5_2, c7_1, c8_1, c10_1


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   zero(nil) -> zero2(0, nil)
   zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
   zero2(0, nil) -> nil
   zero2(0, cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
   zero2(s(z0), nil) -> zero(nil)
   zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sub(0, 0) -> 0
   sub(s(z0), s(z1)) -> sub(z0, z1)
   sub(s(z0), 0) -> s(z0)
   sub(0, s(z0)) -> 0
Tuples:
   SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
   ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
   ZERO2(0, cons(z0, z1)) -> c7(SUB(z0, z0))
   ZERO2(0, cons(z0, z1)) -> c8(ZERO(z1))
   ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))
S tuples:
   SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
   ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
   ZERO2(0, cons(z0, z1)) -> c7(SUB(z0, z0))
   ZERO2(0, cons(z0, z1)) -> c8(ZERO(z1))
   ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))
K tuples:none
Defined Rule Symbols:   sub_2

Defined Pair Symbols:   SUB_2, ZERO_1, ZERO2_2

Compound Symbols:   c3_1, c5_2, c7_1, c8_1, c10_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:

   SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
   ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
   ZERO2(0, cons(z0, z1)) -> c7(SUB(z0, z0))
   ZERO2(0, cons(z0, z1)) -> c8(ZERO(z1))
   ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))

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

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

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:

   SUB(s(z0), s(z1)) -> c3(SUB(z0, z1)) [1]
   ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0)) [1]
   ZERO2(0, cons(z0, z1)) -> c7(SUB(z0, z0)) [1]
   ZERO2(0, cons(z0, z1)) -> c8(ZERO(z1)) [1]
   ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2))) [1]
   sub(0, 0) -> 0 [0]
   sub(s(z0), s(z1)) -> sub(z0, z1) [0]
   sub(s(z0), 0) -> s(z0) [0]
   sub(0, s(z0)) -> 0 [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:

   SUB(s(z0), s(z1)) -> c3(SUB(z0, z1)) [1]
   ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0)) [1]
   ZERO2(0, cons(z0, z1)) -> c7(SUB(z0, z0)) [1]
   ZERO2(0, cons(z0, z1)) -> c8(ZERO(z1)) [1]
   ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2))) [1]
   sub(0, 0) -> 0 [0]
   sub(s(z0), s(z1)) -> sub(z0, z1) [0]
   sub(s(z0), 0) -> s(z0) [0]
   sub(0, s(z0)) -> 0 [0]

The TRS has the following type information:
SUB :: s:0 -> s:0 -> c3
s :: s:0 -> s:0
c3 :: c3 -> c3
ZERO :: cons -> c5
cons :: s:0 -> cons -> cons
c5 :: c7:c8:c10 -> c3 -> c5
ZERO2 :: s:0 -> cons -> c7:c8:c10
sub :: s:0 -> s:0 -> s:0
0 :: s:0
c7 :: c3 -> c7:c8:c10
c8 :: c5 -> c7:c8:c10
c10 :: c5 -> c7:c8:c10

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:

   sub(v0, v1) -> null_sub [0]
   SUB(v0, v1) -> null_SUB [0]
   ZERO(v0) -> null_ZERO [0]
   ZERO2(v0, v1) -> null_ZERO2 [0]

And the following fresh constants:    null_sub, null_SUB, null_ZERO, null_ZERO2, 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:

   SUB(s(z0), s(z1)) -> c3(SUB(z0, z1)) [1]
   ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0)) [1]
   ZERO2(0, cons(z0, z1)) -> c7(SUB(z0, z0)) [1]
   ZERO2(0, cons(z0, z1)) -> c8(ZERO(z1)) [1]
   ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2))) [1]
   sub(0, 0) -> 0 [0]
   sub(s(z0), s(z1)) -> sub(z0, z1) [0]
   sub(s(z0), 0) -> s(z0) [0]
   sub(0, s(z0)) -> 0 [0]
   sub(v0, v1) -> null_sub [0]
   SUB(v0, v1) -> null_SUB [0]
   ZERO(v0) -> null_ZERO [0]
   ZERO2(v0, v1) -> null_ZERO2 [0]

The TRS has the following type information:
SUB :: s:0:null_sub -> s:0:null_sub -> c3:null_SUB
s :: s:0:null_sub -> s:0:null_sub
c3 :: c3:null_SUB -> c3:null_SUB
ZERO :: cons -> c5:null_ZERO
cons :: s:0:null_sub -> cons -> cons
c5 :: c7:c8:c10:null_ZERO2 -> c3:null_SUB -> c5:null_ZERO
ZERO2 :: s:0:null_sub -> cons -> c7:c8:c10:null_ZERO2
sub :: s:0:null_sub -> s:0:null_sub -> s:0:null_sub
0 :: s:0:null_sub
c7 :: c3:null_SUB -> c7:c8:c10:null_ZERO2
c8 :: c5:null_ZERO -> c7:c8:c10:null_ZERO2
c10 :: c5:null_ZERO -> c7:c8:c10:null_ZERO2
null_sub :: s:0:null_sub
null_SUB :: c3:null_SUB
null_ZERO :: c5:null_ZERO
null_ZERO2 :: c7:c8:c10:null_ZERO2
const :: cons

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:

   0 => 0
   null_sub => 0
   null_SUB => 0
   null_ZERO => 0
   null_ZERO2 => 0
   const => 0

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

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

   SUB(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   SUB(z, z') -{ 1 }-> 1 + SUB(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   ZERO(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   ZERO(z) -{ 1 }-> 1 + ZERO2(sub(z0, z0), 1 + z0 + z1) + SUB(z0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ZERO2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   ZERO2(z, z') -{ 1 }-> 1 + ZERO(z1) :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 0
   ZERO2(z, z') -{ 1 }-> 1 + ZERO(1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 1 + z0, z0 >= 0, z2 >= 0
   ZERO2(z, z') -{ 1 }-> 1 + SUB(z0, z0) :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 0
   sub(z, z') -{ 0 }-> sub(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   sub(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
   sub(z, z') -{ 0 }-> 0 :|: z0 >= 0, z' = 1 + z0, z = 0
   sub(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   sub(z, z') -{ 0 }-> 1 + z0 :|: z = 1 + z0, z0 >= 0, z' = 0

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),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[fun1(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[fun2(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[sub(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, Out),1,[sub(V5, V5, Ret010),fun2(Ret010, 1 + V5 + V4, Ret01),fun(V5, V5, Ret11)],[Out = 1 + Ret01 + Ret11,V4 >= 0,V5 >= 0,V1 = 1 + V4 + V5]).
eq(fun2(V1, V, Out),1,[fun(V7, V7, Ret12)],[Out = 1 + Ret12,V = 1 + V6 + V7,V6 >= 0,V7 >= 0,V1 = 0]).
eq(fun2(V1, V, Out),1,[fun1(V9, Ret13)],[Out = 1 + Ret13,V = 1 + V8 + V9,V9 >= 0,V8 >= 0,V1 = 0]).
eq(fun2(V1, V, Out),1,[fun1(1 + V10 + V12, Ret14)],[Out = 1 + Ret14,V10 >= 0,V = 1 + V10 + V12,V1 = 1 + V11,V11 >= 0,V12 >= 0]).
eq(sub(V1, V, Out),0,[],[Out = 0,V1 = 0,V = 0]).
eq(sub(V1, V, Out),0,[sub(V14, V13, Ret)],[Out = Ret,V13 >= 0,V1 = 1 + V14,V14 >= 0,V = 1 + V13]).
eq(sub(V1, V, Out),0,[],[Out = 1 + V15,V1 = 1 + V15,V15 >= 0,V = 0]).
eq(sub(V1, V, Out),0,[],[Out = 0,V16 >= 0,V = 1 + V16,V1 = 0]).
eq(sub(V1, V, Out),0,[],[Out = 0,V18 >= 0,V17 >= 0,V1 = V18,V = V17]).
eq(fun(V1, V, Out),0,[],[Out = 0,V20 >= 0,V19 >= 0,V1 = V20,V = V19]).
eq(fun1(V1, Out),0,[],[Out = 0,V21 >= 0,V1 = V21]).
eq(fun2(V1, V, Out),0,[],[Out = 0,V22 >= 0,V23 >= 0,V1 = V22,V = V23]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,Out),[V1],[Out]).
input_output_vars(fun2(V1,V,Out),[V1,V],[Out]).
input_output_vars(sub(V1,V,Out),[V1,V],[Out]).


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

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

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

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

### Specialization of cost equations fun/3 
* CE 14 is refined into CE [19] 
* CE 13 is refined into CE [20] 


### Cost equations --> "Loop" of fun/3 
* CEs [20] --> Loop 11 
* CEs [19] --> 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 sub/3 
* CE 17 is refined into CE [21] 
* CE 15 is refined into CE [22] 
* CE 18 is refined into CE [23] 
* CE 16 is refined into CE [24] 


### Cost equations --> "Loop" of sub/3 
* CEs [24] --> Loop 13 
* CEs [21] --> Loop 14 
* CEs [22,23] --> Loop 15 

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

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


### Specialization of cost equations fun1/2 
* CE 8 is refined into CE [25,26] 
* CE 11 is refined into CE [27,28,29,30] 
* CE 12 is refined into CE [31] 
* CE 10 is refined into CE [32,33] 
* CE 9 is discarded (unfeasible) 


### Cost equations --> "Loop" of fun1/2 
* CEs [33] --> Loop 16 
* CEs [32] --> Loop 17 
* CEs [30] --> Loop 18 
* CEs [28,29] --> Loop 19 
* CEs [26] --> Loop 20 
* CEs [27] --> Loop 21 
* CEs [25] --> Loop 22 
* CEs [31] --> Loop 23 

### Ranking functions of CR fun1(V1,Out) 
* RF of phase [16,17]: [V1]

#### Partial ranking functions of CR fun1(V1,Out) 
* Partial RF of phase [16,17]:
  - RF of loop [16:1]:
    V1-1
  - RF of loop [17:1]:
    V1


### Specialization of cost equations start/2 
* CE 1 is refined into CE [34] 
* CE 2 is refined into CE [35,36,37] 
* CE 3 is refined into CE [38,39,40] 
* CE 4 is refined into CE [41,42] 
* CE 5 is refined into CE [43,44] 
* CE 6 is refined into CE [45,46,47] 
* CE 7 is refined into CE [48,49,50] 


### Cost equations --> "Loop" of start/2 
* CEs [48] --> Loop 24 
* CEs [34,35,36,37,38,39,40,41,42,43,44,45,46,47,49,50] --> Loop 25 

### 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 sub(V1,V,Out):
* Chain [[13],15]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

* Chain [[13],14]: 0
  with precondition: [V1=Out+V,V>=1,V1>=V+1] 

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

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


#### Cost of chains of fun1(V1,Out):
* Chain [[16,17],23]: 4*it(16)+1*s(3)+0
  Such that:aux(4) =< V1
it(16) =< aux(4)
s(3) =< it(16)*aux(4)

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

* Chain [[16,17],22]: 4*it(16)+1*s(3)+1
  Such that:aux(5) =< V1
it(16) =< aux(5)
s(3) =< it(16)*aux(5)

  with precondition: [V1>=2,Out>=3] 

* Chain [[16,17],21]: 4*it(16)+1*s(3)+2
  Such that:aux(6) =< V1
it(16) =< aux(6)
s(3) =< it(16)*aux(6)

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

* Chain [[16,17],20]: 5*it(16)+1*s(3)+1
  Such that:aux(7) =< V1
it(16) =< aux(7)
s(3) =< it(16)*aux(7)

  with precondition: [V1>=3,Out>=4] 

* Chain [[16,17],19]: 6*it(16)+1*s(3)+2
  Such that:aux(9) =< V1
it(16) =< aux(9)
s(3) =< it(16)*aux(9)

  with precondition: [V1>=3,Out>=5] 

* Chain [[16,17],18]: 6*it(16)+1*s(3)+2
  Such that:aux(11) =< V1
it(16) =< aux(11)
s(3) =< it(16)*aux(11)

  with precondition: [V1>=3,Out>=6] 

* Chain [23]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [22]: 1
  with precondition: [Out=1,V1>=1] 

* Chain [21]: 2
  with precondition: [Out=2,V1>=1] 

* Chain [20]: 1*s(4)+1
  Such that:s(4) =< V1

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

* Chain [19]: 2*s(5)+2
  Such that:aux(8) =< V1
s(5) =< aux(8)

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

* Chain [18]: 2*s(7)+2
  Such that:aux(10) =< V1
s(7) =< aux(10)

  with precondition: [Out>=4,2*V1>=Out] 


#### Cost of chains of start(V1,V):
* Chain [25]: 70*s(33)+12*s(34)+34*s(41)+6*s(42)+3
  Such that:s(40) =< V1
aux(13) =< V
s(33) =< aux(13)
s(41) =< s(40)
s(42) =< s(41)*s(40)
s(34) =< s(33)*aux(13)

  with precondition: [V1>=0] 

* Chain [24]: 0
  with precondition: [V=0,V1>=1] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [25] with precondition: [V1>=0] 
    - Upper bound: 34*V1+3+6*V1*V1+nat(V)*70+nat(V)*12*nat(V) 
    - Complexity: n^2 
* Chain [24] with precondition: [V=0,V1>=1] 
    - Upper bound: 0 
    - Complexity: constant 

### Maximum cost of start(V1,V): 34*V1+3+6*V1*V1+nat(V)*70+nat(V)*12*nat(V) 
Asymptotic class: n^2 
* Total analysis performed in 261 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:
   sub(0, 0) -> 0
   sub(s(z0), 0) -> s(z0)
   sub(0, s(z0)) -> 0
   sub(s(z0), s(z1)) -> sub(z0, z1)
   zero(nil) -> zero2(0, nil)
   zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
   zero2(0, nil) -> nil
   zero2(0, cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
   zero2(s(z0), nil) -> zero(nil)
   zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))
Tuples:
   SUB(0, 0) -> c
   SUB(s(z0), 0) -> c1
   SUB(0, s(z0)) -> c2
   SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
   ZERO(nil) -> c4(ZERO2(0, nil))
   ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
   ZERO2(0, nil) -> c6
   ZERO2(0, cons(z0, z1)) -> c7(SUB(z0, z0))
   ZERO2(0, cons(z0, z1)) -> c8(ZERO(z1))
   ZERO2(s(z0), nil) -> c9(ZERO(nil))
   ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))
S tuples:
   SUB(0, 0) -> c
   SUB(s(z0), 0) -> c1
   SUB(0, s(z0)) -> c2
   SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
   ZERO(nil) -> c4(ZERO2(0, nil))
   ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
   ZERO2(0, nil) -> c6
   ZERO2(0, cons(z0, z1)) -> c7(SUB(z0, z0))
   ZERO2(0, cons(z0, z1)) -> c8(ZERO(z1))
   ZERO2(s(z0), nil) -> c9(ZERO(nil))
   ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))
K tuples:none
Defined Rule Symbols:   sub_2, zero_1, zero2_2

Defined Pair Symbols:   SUB_2, ZERO_1, ZERO2_2

Compound Symbols:   c, c1, c2, c3_1, c4_1, c5_2, c6, c7_1, c8_1, c9_1, c10_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:

   SUB(0, 0) -> c
   SUB(s(z0), 0) -> c1
   SUB(0, s(z0)) -> c2
   SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
   ZERO(nil) -> c4(ZERO2(0, nil))
   ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
   ZERO2(0, nil) -> c6
   ZERO2(0, cons(z0, z1)) -> c7(SUB(z0, z0))
   ZERO2(0, cons(z0, z1)) -> c8(ZERO(z1))
   ZERO2(s(z0), nil) -> c9(ZERO(nil))
   ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))

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

   sub(0, 0) -> 0
   sub(s(z0), 0) -> s(z0)
   sub(0, s(z0)) -> 0
   sub(s(z0), s(z1)) -> sub(z0, z1)
   zero(nil) -> zero2(0, nil)
   zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
   zero2(0, nil) -> nil
   zero2(0, cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
   zero2(s(z0), nil) -> zero(nil)
   zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))

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:

   SUB(0', 0') -> c
   SUB(s(z0), 0') -> c1
   SUB(0', s(z0)) -> c2
   SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
   ZERO(nil) -> c4(ZERO2(0', nil))
   ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
   ZERO2(0', nil) -> c6
   ZERO2(0', cons(z0, z1)) -> c7(SUB(z0, z0))
   ZERO2(0', cons(z0, z1)) -> c8(ZERO(z1))
   ZERO2(s(z0), nil) -> c9(ZERO(nil))
   ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))

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

   sub(0', 0') -> 0'
   sub(s(z0), 0') -> s(z0)
   sub(0', s(z0)) -> 0'
   sub(s(z0), s(z1)) -> sub(z0, z1)
   zero(nil) -> zero2(0', nil)
   zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
   zero2(0', nil) -> nil
   zero2(0', cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
   zero2(s(z0), nil) -> zero(nil)
   zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))

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

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

(26)
Obligation:
Innermost TRS:
Rules:
SUB(0', 0') -> c
SUB(s(z0), 0') -> c1
SUB(0', s(z0)) -> c2
SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
ZERO(nil) -> c4(ZERO2(0', nil))
ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
ZERO2(0', nil) -> c6
ZERO2(0', cons(z0, z1)) -> c7(SUB(z0, z0))
ZERO2(0', cons(z0, z1)) -> c8(ZERO(z1))
ZERO2(s(z0), nil) -> c9(ZERO(nil))
ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))
sub(0', 0') -> 0'
sub(s(z0), 0') -> s(z0)
sub(0', s(z0)) -> 0'
sub(s(z0), s(z1)) -> sub(z0, z1)
zero(nil) -> zero2(0', nil)
zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
zero2(0', nil) -> nil
zero2(0', cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
zero2(s(z0), nil) -> zero(nil)
zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))

Types:
SUB :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
ZERO :: nil:cons -> c4:c5
nil :: nil:cons
c4 :: c6:c7:c8:c9:c10 -> c4:c5
ZERO2 :: 0':s -> nil:cons -> c6:c7:c8:c9:c10
cons :: 0':s -> nil:cons -> nil:cons
c5 :: c6:c7:c8:c9:c10 -> c:c1:c2:c3 -> c4:c5
sub :: 0':s -> 0':s -> 0':s
c6 :: c6:c7:c8:c9:c10
c7 :: c:c1:c2:c3 -> c6:c7:c8:c9:c10
c8 :: c4:c5 -> c6:c7:c8:c9:c10
c9 :: c4:c5 -> c6:c7:c8:c9:c10
c10 :: c4:c5 -> c6:c7:c8:c9:c10
zero :: nil:cons -> nil:cons
zero2 :: 0':s -> nil:cons -> nil:cons
hole_c:c1:c2:c31_11 :: c:c1:c2:c3
hole_0':s2_11 :: 0':s
hole_c4:c53_11 :: c4:c5
hole_nil:cons4_11 :: nil:cons
hole_c6:c7:c8:c9:c105_11 :: c6:c7:c8:c9:c10
gen_c:c1:c2:c36_11 :: Nat -> c:c1:c2:c3
gen_0':s7_11 :: Nat -> 0':s
gen_nil:cons8_11 :: Nat -> nil:cons

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

(27) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
SUB, ZERO, sub, zero

They will be analysed ascendingly in the following order:
SUB < ZERO
sub < ZERO
sub < zero

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

(28)
Obligation:
Innermost TRS:
Rules:
SUB(0', 0') -> c
SUB(s(z0), 0') -> c1
SUB(0', s(z0)) -> c2
SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
ZERO(nil) -> c4(ZERO2(0', nil))
ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
ZERO2(0', nil) -> c6
ZERO2(0', cons(z0, z1)) -> c7(SUB(z0, z0))
ZERO2(0', cons(z0, z1)) -> c8(ZERO(z1))
ZERO2(s(z0), nil) -> c9(ZERO(nil))
ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))
sub(0', 0') -> 0'
sub(s(z0), 0') -> s(z0)
sub(0', s(z0)) -> 0'
sub(s(z0), s(z1)) -> sub(z0, z1)
zero(nil) -> zero2(0', nil)
zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
zero2(0', nil) -> nil
zero2(0', cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
zero2(s(z0), nil) -> zero(nil)
zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))

Types:
SUB :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
ZERO :: nil:cons -> c4:c5
nil :: nil:cons
c4 :: c6:c7:c8:c9:c10 -> c4:c5
ZERO2 :: 0':s -> nil:cons -> c6:c7:c8:c9:c10
cons :: 0':s -> nil:cons -> nil:cons
c5 :: c6:c7:c8:c9:c10 -> c:c1:c2:c3 -> c4:c5
sub :: 0':s -> 0':s -> 0':s
c6 :: c6:c7:c8:c9:c10
c7 :: c:c1:c2:c3 -> c6:c7:c8:c9:c10
c8 :: c4:c5 -> c6:c7:c8:c9:c10
c9 :: c4:c5 -> c6:c7:c8:c9:c10
c10 :: c4:c5 -> c6:c7:c8:c9:c10
zero :: nil:cons -> nil:cons
zero2 :: 0':s -> nil:cons -> nil:cons
hole_c:c1:c2:c31_11 :: c:c1:c2:c3
hole_0':s2_11 :: 0':s
hole_c4:c53_11 :: c4:c5
hole_nil:cons4_11 :: nil:cons
hole_c6:c7:c8:c9:c105_11 :: c6:c7:c8:c9:c10
gen_c:c1:c2:c36_11 :: Nat -> c:c1:c2:c3
gen_0':s7_11 :: Nat -> 0':s
gen_nil:cons8_11 :: Nat -> nil:cons


Generator Equations:
gen_c:c1:c2:c36_11(0) <=> c
gen_c:c1:c2:c36_11(+(x, 1)) <=> c3(gen_c:c1:c2:c36_11(x))
gen_0':s7_11(0) <=> 0'
gen_0':s7_11(+(x, 1)) <=> s(gen_0':s7_11(x))
gen_nil:cons8_11(0) <=> nil
gen_nil:cons8_11(+(x, 1)) <=> cons(0', gen_nil:cons8_11(x))


The following defined symbols remain to be analysed:
SUB, ZERO, sub, zero

They will be analysed ascendingly in the following order:
SUB < ZERO
sub < ZERO
sub < zero

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
SUB(gen_0':s7_11(n10_11), gen_0':s7_11(n10_11)) -> gen_c:c1:c2:c36_11(n10_11), rt in Omega(1 + n10_11)

Induction Base:
SUB(gen_0':s7_11(0), gen_0':s7_11(0)) ->_R^Omega(1)
c

Induction Step:
SUB(gen_0':s7_11(+(n10_11, 1)), gen_0':s7_11(+(n10_11, 1))) ->_R^Omega(1)
c3(SUB(gen_0':s7_11(n10_11), gen_0':s7_11(n10_11))) ->_IH
c3(gen_c:c1:c2:c36_11(c11_11))

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

(30)
Complex Obligation (BEST)

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

(31)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
SUB(0', 0') -> c
SUB(s(z0), 0') -> c1
SUB(0', s(z0)) -> c2
SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
ZERO(nil) -> c4(ZERO2(0', nil))
ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
ZERO2(0', nil) -> c6
ZERO2(0', cons(z0, z1)) -> c7(SUB(z0, z0))
ZERO2(0', cons(z0, z1)) -> c8(ZERO(z1))
ZERO2(s(z0), nil) -> c9(ZERO(nil))
ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))
sub(0', 0') -> 0'
sub(s(z0), 0') -> s(z0)
sub(0', s(z0)) -> 0'
sub(s(z0), s(z1)) -> sub(z0, z1)
zero(nil) -> zero2(0', nil)
zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
zero2(0', nil) -> nil
zero2(0', cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
zero2(s(z0), nil) -> zero(nil)
zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))

Types:
SUB :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
ZERO :: nil:cons -> c4:c5
nil :: nil:cons
c4 :: c6:c7:c8:c9:c10 -> c4:c5
ZERO2 :: 0':s -> nil:cons -> c6:c7:c8:c9:c10
cons :: 0':s -> nil:cons -> nil:cons
c5 :: c6:c7:c8:c9:c10 -> c:c1:c2:c3 -> c4:c5
sub :: 0':s -> 0':s -> 0':s
c6 :: c6:c7:c8:c9:c10
c7 :: c:c1:c2:c3 -> c6:c7:c8:c9:c10
c8 :: c4:c5 -> c6:c7:c8:c9:c10
c9 :: c4:c5 -> c6:c7:c8:c9:c10
c10 :: c4:c5 -> c6:c7:c8:c9:c10
zero :: nil:cons -> nil:cons
zero2 :: 0':s -> nil:cons -> nil:cons
hole_c:c1:c2:c31_11 :: c:c1:c2:c3
hole_0':s2_11 :: 0':s
hole_c4:c53_11 :: c4:c5
hole_nil:cons4_11 :: nil:cons
hole_c6:c7:c8:c9:c105_11 :: c6:c7:c8:c9:c10
gen_c:c1:c2:c36_11 :: Nat -> c:c1:c2:c3
gen_0':s7_11 :: Nat -> 0':s
gen_nil:cons8_11 :: Nat -> nil:cons


Generator Equations:
gen_c:c1:c2:c36_11(0) <=> c
gen_c:c1:c2:c36_11(+(x, 1)) <=> c3(gen_c:c1:c2:c36_11(x))
gen_0':s7_11(0) <=> 0'
gen_0':s7_11(+(x, 1)) <=> s(gen_0':s7_11(x))
gen_nil:cons8_11(0) <=> nil
gen_nil:cons8_11(+(x, 1)) <=> cons(0', gen_nil:cons8_11(x))


The following defined symbols remain to be analysed:
SUB, ZERO, sub, zero

They will be analysed ascendingly in the following order:
SUB < ZERO
sub < ZERO
sub < zero

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

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

(33)
BOUNDS(n^1, INF)

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

(34)
Obligation:
Innermost TRS:
Rules:
SUB(0', 0') -> c
SUB(s(z0), 0') -> c1
SUB(0', s(z0)) -> c2
SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
ZERO(nil) -> c4(ZERO2(0', nil))
ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
ZERO2(0', nil) -> c6
ZERO2(0', cons(z0, z1)) -> c7(SUB(z0, z0))
ZERO2(0', cons(z0, z1)) -> c8(ZERO(z1))
ZERO2(s(z0), nil) -> c9(ZERO(nil))
ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))
sub(0', 0') -> 0'
sub(s(z0), 0') -> s(z0)
sub(0', s(z0)) -> 0'
sub(s(z0), s(z1)) -> sub(z0, z1)
zero(nil) -> zero2(0', nil)
zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
zero2(0', nil) -> nil
zero2(0', cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
zero2(s(z0), nil) -> zero(nil)
zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))

Types:
SUB :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
ZERO :: nil:cons -> c4:c5
nil :: nil:cons
c4 :: c6:c7:c8:c9:c10 -> c4:c5
ZERO2 :: 0':s -> nil:cons -> c6:c7:c8:c9:c10
cons :: 0':s -> nil:cons -> nil:cons
c5 :: c6:c7:c8:c9:c10 -> c:c1:c2:c3 -> c4:c5
sub :: 0':s -> 0':s -> 0':s
c6 :: c6:c7:c8:c9:c10
c7 :: c:c1:c2:c3 -> c6:c7:c8:c9:c10
c8 :: c4:c5 -> c6:c7:c8:c9:c10
c9 :: c4:c5 -> c6:c7:c8:c9:c10
c10 :: c4:c5 -> c6:c7:c8:c9:c10
zero :: nil:cons -> nil:cons
zero2 :: 0':s -> nil:cons -> nil:cons
hole_c:c1:c2:c31_11 :: c:c1:c2:c3
hole_0':s2_11 :: 0':s
hole_c4:c53_11 :: c4:c5
hole_nil:cons4_11 :: nil:cons
hole_c6:c7:c8:c9:c105_11 :: c6:c7:c8:c9:c10
gen_c:c1:c2:c36_11 :: Nat -> c:c1:c2:c3
gen_0':s7_11 :: Nat -> 0':s
gen_nil:cons8_11 :: Nat -> nil:cons


Lemmas:
SUB(gen_0':s7_11(n10_11), gen_0':s7_11(n10_11)) -> gen_c:c1:c2:c36_11(n10_11), rt in Omega(1 + n10_11)


Generator Equations:
gen_c:c1:c2:c36_11(0) <=> c
gen_c:c1:c2:c36_11(+(x, 1)) <=> c3(gen_c:c1:c2:c36_11(x))
gen_0':s7_11(0) <=> 0'
gen_0':s7_11(+(x, 1)) <=> s(gen_0':s7_11(x))
gen_nil:cons8_11(0) <=> nil
gen_nil:cons8_11(+(x, 1)) <=> cons(0', gen_nil:cons8_11(x))


The following defined symbols remain to be analysed:
sub, ZERO, zero

They will be analysed ascendingly in the following order:
sub < ZERO
sub < zero

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

(35) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sub(gen_0':s7_11(n878_11), gen_0':s7_11(n878_11)) -> gen_0':s7_11(0), rt in Omega(0)

Induction Base:
sub(gen_0':s7_11(0), gen_0':s7_11(0)) ->_R^Omega(0)
0'

Induction Step:
sub(gen_0':s7_11(+(n878_11, 1)), gen_0':s7_11(+(n878_11, 1))) ->_R^Omega(0)
sub(gen_0':s7_11(n878_11), gen_0':s7_11(n878_11)) ->_IH
gen_0':s7_11(0)

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

(36)
Obligation:
Innermost TRS:
Rules:
SUB(0', 0') -> c
SUB(s(z0), 0') -> c1
SUB(0', s(z0)) -> c2
SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
ZERO(nil) -> c4(ZERO2(0', nil))
ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
ZERO2(0', nil) -> c6
ZERO2(0', cons(z0, z1)) -> c7(SUB(z0, z0))
ZERO2(0', cons(z0, z1)) -> c8(ZERO(z1))
ZERO2(s(z0), nil) -> c9(ZERO(nil))
ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))
sub(0', 0') -> 0'
sub(s(z0), 0') -> s(z0)
sub(0', s(z0)) -> 0'
sub(s(z0), s(z1)) -> sub(z0, z1)
zero(nil) -> zero2(0', nil)
zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
zero2(0', nil) -> nil
zero2(0', cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
zero2(s(z0), nil) -> zero(nil)
zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))

Types:
SUB :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
ZERO :: nil:cons -> c4:c5
nil :: nil:cons
c4 :: c6:c7:c8:c9:c10 -> c4:c5
ZERO2 :: 0':s -> nil:cons -> c6:c7:c8:c9:c10
cons :: 0':s -> nil:cons -> nil:cons
c5 :: c6:c7:c8:c9:c10 -> c:c1:c2:c3 -> c4:c5
sub :: 0':s -> 0':s -> 0':s
c6 :: c6:c7:c8:c9:c10
c7 :: c:c1:c2:c3 -> c6:c7:c8:c9:c10
c8 :: c4:c5 -> c6:c7:c8:c9:c10
c9 :: c4:c5 -> c6:c7:c8:c9:c10
c10 :: c4:c5 -> c6:c7:c8:c9:c10
zero :: nil:cons -> nil:cons
zero2 :: 0':s -> nil:cons -> nil:cons
hole_c:c1:c2:c31_11 :: c:c1:c2:c3
hole_0':s2_11 :: 0':s
hole_c4:c53_11 :: c4:c5
hole_nil:cons4_11 :: nil:cons
hole_c6:c7:c8:c9:c105_11 :: c6:c7:c8:c9:c10
gen_c:c1:c2:c36_11 :: Nat -> c:c1:c2:c3
gen_0':s7_11 :: Nat -> 0':s
gen_nil:cons8_11 :: Nat -> nil:cons


Lemmas:
SUB(gen_0':s7_11(n10_11), gen_0':s7_11(n10_11)) -> gen_c:c1:c2:c36_11(n10_11), rt in Omega(1 + n10_11)
sub(gen_0':s7_11(n878_11), gen_0':s7_11(n878_11)) -> gen_0':s7_11(0), rt in Omega(0)


Generator Equations:
gen_c:c1:c2:c36_11(0) <=> c
gen_c:c1:c2:c36_11(+(x, 1)) <=> c3(gen_c:c1:c2:c36_11(x))
gen_0':s7_11(0) <=> 0'
gen_0':s7_11(+(x, 1)) <=> s(gen_0':s7_11(x))
gen_nil:cons8_11(0) <=> nil
gen_nil:cons8_11(+(x, 1)) <=> cons(0', gen_nil:cons8_11(x))


The following defined symbols remain to be analysed:
ZERO, zero
----------------------------------------

(37) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ZERO(gen_nil:cons8_11(n1828_11)) -> *9_11, rt in Omega(n1828_11)

Induction Base:
ZERO(gen_nil:cons8_11(0))

Induction Step:
ZERO(gen_nil:cons8_11(+(n1828_11, 1))) ->_R^Omega(1)
c5(ZERO2(sub(0', 0'), cons(0', gen_nil:cons8_11(n1828_11))), SUB(0', 0')) ->_L^Omega(0)
c5(ZERO2(gen_0':s7_11(0), cons(0', gen_nil:cons8_11(n1828_11))), SUB(0', 0')) ->_R^Omega(1)
c5(c8(ZERO(gen_nil:cons8_11(n1828_11))), SUB(0', 0')) ->_IH
c5(c8(*9_11), SUB(0', 0')) ->_L^Omega(1)
c5(c8(*9_11), gen_c:c1:c2:c36_11(0))

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:
SUB(0', 0') -> c
SUB(s(z0), 0') -> c1
SUB(0', s(z0)) -> c2
SUB(s(z0), s(z1)) -> c3(SUB(z0, z1))
ZERO(nil) -> c4(ZERO2(0', nil))
ZERO(cons(z0, z1)) -> c5(ZERO2(sub(z0, z0), cons(z0, z1)), SUB(z0, z0))
ZERO2(0', nil) -> c6
ZERO2(0', cons(z0, z1)) -> c7(SUB(z0, z0))
ZERO2(0', cons(z0, z1)) -> c8(ZERO(z1))
ZERO2(s(z0), nil) -> c9(ZERO(nil))
ZERO2(s(z0), cons(z1, z2)) -> c10(ZERO(cons(z1, z2)))
sub(0', 0') -> 0'
sub(s(z0), 0') -> s(z0)
sub(0', s(z0)) -> 0'
sub(s(z0), s(z1)) -> sub(z0, z1)
zero(nil) -> zero2(0', nil)
zero(cons(z0, z1)) -> zero2(sub(z0, z0), cons(z0, z1))
zero2(0', nil) -> nil
zero2(0', cons(z0, z1)) -> cons(sub(z0, z0), zero(z1))
zero2(s(z0), nil) -> zero(nil)
zero2(s(z0), cons(z1, z2)) -> zero(cons(z1, z2))

Types:
SUB :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
ZERO :: nil:cons -> c4:c5
nil :: nil:cons
c4 :: c6:c7:c8:c9:c10 -> c4:c5
ZERO2 :: 0':s -> nil:cons -> c6:c7:c8:c9:c10
cons :: 0':s -> nil:cons -> nil:cons
c5 :: c6:c7:c8:c9:c10 -> c:c1:c2:c3 -> c4:c5
sub :: 0':s -> 0':s -> 0':s
c6 :: c6:c7:c8:c9:c10
c7 :: c:c1:c2:c3 -> c6:c7:c8:c9:c10
c8 :: c4:c5 -> c6:c7:c8:c9:c10
c9 :: c4:c5 -> c6:c7:c8:c9:c10
c10 :: c4:c5 -> c6:c7:c8:c9:c10
zero :: nil:cons -> nil:cons
zero2 :: 0':s -> nil:cons -> nil:cons
hole_c:c1:c2:c31_11 :: c:c1:c2:c3
hole_0':s2_11 :: 0':s
hole_c4:c53_11 :: c4:c5
hole_nil:cons4_11 :: nil:cons
hole_c6:c7:c8:c9:c105_11 :: c6:c7:c8:c9:c10
gen_c:c1:c2:c36_11 :: Nat -> c:c1:c2:c3
gen_0':s7_11 :: Nat -> 0':s
gen_nil:cons8_11 :: Nat -> nil:cons


Lemmas:
SUB(gen_0':s7_11(n10_11), gen_0':s7_11(n10_11)) -> gen_c:c1:c2:c36_11(n10_11), rt in Omega(1 + n10_11)
sub(gen_0':s7_11(n878_11), gen_0':s7_11(n878_11)) -> gen_0':s7_11(0), rt in Omega(0)
ZERO(gen_nil:cons8_11(n1828_11)) -> *9_11, rt in Omega(n1828_11)


Generator Equations:
gen_c:c1:c2:c36_11(0) <=> c
gen_c:c1:c2:c36_11(+(x, 1)) <=> c3(gen_c:c1:c2:c36_11(x))
gen_0':s7_11(0) <=> 0'
gen_0':s7_11(+(x, 1)) <=> s(gen_0':s7_11(x))
gen_nil:cons8_11(0) <=> nil
gen_nil:cons8_11(+(x, 1)) <=> cons(0', gen_nil:cons8_11(x))


The following defined symbols remain to be analysed:
zero
----------------------------------------

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
zero(gen_nil:cons8_11(n7888_11)) -> gen_nil:cons8_11(n7888_11), rt in Omega(0)

Induction Base:
zero(gen_nil:cons8_11(0)) ->_R^Omega(0)
zero2(0', nil) ->_R^Omega(0)
nil

Induction Step:
zero(gen_nil:cons8_11(+(n7888_11, 1))) ->_R^Omega(0)
zero2(sub(0', 0'), cons(0', gen_nil:cons8_11(n7888_11))) ->_L^Omega(0)
zero2(gen_0':s7_11(0), cons(0', gen_nil:cons8_11(n7888_11))) ->_R^Omega(0)
cons(sub(0', 0'), zero(gen_nil:cons8_11(n7888_11))) ->_L^Omega(0)
cons(gen_0':s7_11(0), zero(gen_nil:cons8_11(n7888_11))) ->_IH
cons(gen_0':s7_11(0), gen_nil:cons8_11(c7889_11))

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

(40)
BOUNDS(1, INF)