WORST_CASE(Omega(n^1),O(n^2))
proof of input_jpwfsULal9.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 216 ms]
    (10) BOUNDS(1, n^2)
(11) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CdtProblem
    (13) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRelTRS
    (15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRelTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 0 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 387 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 119 ms]
            (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 583 ms]
            (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 100 ms]
            (32) 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) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

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


The TRS R consists of the following rules:

   sub(0, 0) -> 0 [1]
   sub(s(x), 0) -> s(x) [1]
   sub(0, s(x)) -> 0 [1]
   sub(s(x), s(y)) -> sub(x, y) [1]
   zero(nil) -> zero2(0, nil) [1]
   zero(cons(x, xs)) -> zero2(sub(x, x), cons(x, xs)) [1]
   zero2(0, nil) -> nil [1]
   zero2(0, cons(x, xs)) -> cons(sub(x, x), zero(xs)) [1]
   zero2(s(y), nil) -> zero(nil) [1]
   zero2(s(y), cons(x, xs)) -> zero(cons(x, xs)) [1]

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

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

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   sub(0, 0) -> 0 [1]
   sub(s(x), 0) -> s(x) [1]
   sub(0, s(x)) -> 0 [1]
   sub(s(x), s(y)) -> sub(x, y) [1]
   zero(nil) -> zero2(0, nil) [1]
   zero(cons(x, xs)) -> zero2(sub(x, x), cons(x, xs)) [1]
   zero2(0, nil) -> nil [1]
   zero2(0, cons(x, xs)) -> cons(sub(x, x), zero(xs)) [1]
   zero2(s(y), nil) -> zero(nil) [1]
   zero2(s(y), cons(x, xs)) -> zero(cons(x, xs)) [1]

The TRS has the following type information:
sub :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
zero :: nil:cons -> nil:cons
nil :: nil:cons
zero2 :: 0:s -> nil:cons -> nil:cons
cons :: 0:s -> nil:cons -> nil:cons

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

(5) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
none

And the following fresh constants: none

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

(6)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   sub(0, 0) -> 0 [1]
   sub(s(x), 0) -> s(x) [1]
   sub(0, s(x)) -> 0 [1]
   sub(s(x), s(y)) -> sub(x, y) [1]
   zero(nil) -> zero2(0, nil) [1]
   zero(cons(x, xs)) -> zero2(sub(x, x), cons(x, xs)) [1]
   zero2(0, nil) -> nil [1]
   zero2(0, cons(x, xs)) -> cons(sub(x, x), zero(xs)) [1]
   zero2(s(y), nil) -> zero(nil) [1]
   zero2(s(y), cons(x, xs)) -> zero(cons(x, xs)) [1]

The TRS has the following type information:
sub :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
zero :: nil:cons -> nil:cons
nil :: nil:cons
zero2 :: 0:s -> nil:cons -> nil:cons
cons :: 0:s -> nil:cons -> nil:cons

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

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

   0 => 0
   nil => 0

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

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

   sub(z, z') -{ 1 }-> sub(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   sub(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   sub(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0
   sub(z, z') -{ 1 }-> 1 + x :|: x >= 0, z = 1 + x, z' = 0
   zero(z) -{ 1 }-> zero2(sub(x, x), 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   zero(z) -{ 1 }-> zero2(0, 0) :|: z = 0
   zero2(z, z') -{ 1 }-> zero(0) :|: y >= 0, z = 1 + y, z' = 0
   zero2(z, z') -{ 1 }-> zero(1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, y >= 0, x >= 0, z = 1 + y
   zero2(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   zero2(z, z') -{ 1 }-> 1 + sub(x, x) + zero(xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V),0,[sub(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[zero(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[zero2(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(sub(V1, V, Out),1,[],[Out = 0,V1 = 0,V = 0]).
eq(sub(V1, V, Out),1,[],[Out = 1 + V2,V2 >= 0,V1 = 1 + V2,V = 0]).
eq(sub(V1, V, Out),1,[],[Out = 0,V = 1 + V3,V3 >= 0,V1 = 0]).
eq(sub(V1, V, Out),1,[sub(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]).
eq(zero(V1, Out),1,[zero2(0, 0, Ret1)],[Out = Ret1,V1 = 0]).
eq(zero(V1, Out),1,[sub(V6, V6, Ret0),zero2(Ret0, 1 + V6 + V7, Ret2)],[Out = Ret2,V1 = 1 + V6 + V7,V7 >= 0,V6 >= 0]).
eq(zero2(V1, V, Out),1,[],[Out = 0,V1 = 0,V = 0]).
eq(zero2(V1, V, Out),1,[sub(V8, V8, Ret01),zero(V9, Ret11)],[Out = 1 + Ret01 + Ret11,V9 >= 0,V = 1 + V8 + V9,V8 >= 0,V1 = 0]).
eq(zero2(V1, V, Out),1,[zero(0, Ret3)],[Out = Ret3,V10 >= 0,V1 = 1 + V10,V = 0]).
eq(zero2(V1, V, Out),1,[zero(1 + V12 + V13, Ret4)],[Out = Ret4,V13 >= 0,V = 1 + V12 + V13,V11 >= 0,V12 >= 0,V1 = 1 + V11]).
input_output_vars(sub(V1,V,Out),[V1,V],[Out]).
input_output_vars(zero(V1,Out),[V1],[Out]).
input_output_vars(zero2(V1,V,Out),[V1,V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [sub/3]
1. recursive  : [zero/2,zero2/3]
2. non_recursive  : [start/2]

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

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

### Specialization of cost equations sub/3 
* CE 13 is refined into CE [14] 
* CE 11 is refined into CE [15] 
* CE 12 is refined into CE [16] 
* CE 10 is refined into CE [17] 


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

### Ranking functions of CR sub(V1,V,Out) 
* RF of phase [14]: [V,V1]

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


### Specialization of cost equations zero/2 
* CE 9 is refined into CE [18] 
* CE 8 is refined into CE [19,20,21,22] 
* CE 7 is discarded (unfeasible) 


### Cost equations --> "Loop" of zero/2 
* CEs [20,22] --> Loop 15 
* CEs [19,21] --> Loop 16 
* CEs [18] --> Loop 17 

### Ranking functions of CR zero(V1,Out) 
* RF of phase [15,16]: [V1]

#### Partial ranking functions of CR zero(V1,Out) 
* Partial RF of phase [15,16]:
  - RF of loop [15:1]:
    V1-1
  - RF of loop [16:1]:
    V1


### Specialization of cost equations start/2 
* CE 2 is refined into CE [23] 
* CE 3 is refined into CE [24,25,26,27] 
* CE 1 is refined into CE [28] 
* CE 4 is refined into CE [29] 
* CE 5 is refined into CE [30,31,32,33,34,35] 
* CE 6 is refined into CE [36,37] 


### Cost equations --> "Loop" of start/2 
* CEs [33] --> Loop 18 
* CEs [23,28,32,34,35,37] --> Loop 19 
* CEs [24,25,26,27,29,30,31,36] --> Loop 20 

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

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


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

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

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

* Chain [[14],12]: 1*it(14)+1
  Such that:it(14) =< V1

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

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

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

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

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

* Chain [11]: 1
  with precondition: [V=0,V1=Out,V1>=1] 


#### Cost of chains of zero(V1,Out):
* Chain [[15,16],17]: 10*it(15)+1*s(10)+1*s(12)+2
  Such that:aux(6) =< V1
it(15) =< aux(6)
aux(3) =< aux(6)-1
s(10) =< it(15)*aux(6)
s(12) =< it(15)*aux(3)

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

* Chain [17]: 2
  with precondition: [V1=0,Out=0] 


#### Cost of chains of start(V1,V):
* Chain [20]: 22*s(14)+2*s(16)+2*s(17)+4
  Such that:aux(8) =< V
s(14) =< aux(8)
s(15) =< aux(8)-1
s(16) =< s(14)*aux(8)
s(17) =< s(14)*s(15)

  with precondition: [V1=0] 

* Chain [19]: 11*s(26)+1*s(28)+1*s(29)+11*s(30)+1*s(35)+1*s(36)+3
  Such that:aux(9) =< V1
aux(10) =< V
s(30) =< aux(9)
s(26) =< aux(10)
s(34) =< aux(9)-1
s(35) =< s(30)*aux(9)
s(36) =< s(30)*s(34)
s(27) =< aux(10)-1
s(28) =< s(26)*aux(10)
s(29) =< s(26)*s(27)

  with precondition: [V1>=1] 

* Chain [18]: 1*s(37)+1
  Such that:s(37) =< V

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


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [20] with precondition: [V1=0] 
    - Upper bound: nat(V)*22+4+nat(V)*2*nat(V)+nat(V)*2*nat(nat(V)+ -1) 
    - Complexity: n^2 
* Chain [19] with precondition: [V1>=1] 
    - Upper bound: 11*V1+3+V1*V1+(V1-1)*V1+nat(V)*11+nat(V)*nat(V)+nat(nat(V)+ -1)*nat(V) 
    - Complexity: n^2 
* Chain [18] with precondition: [V1=V,V1>=1] 
    - Upper bound: V+1 
    - Complexity: n 

### Maximum cost of start(V1,V): nat(V)*10+2+nat(V)*nat(V)+nat(nat(V)+ -1)*nat(V)+max([11*V1+V1*V1+nat(V1-1)*V1,nat(V)*11+1+nat(V)*nat(V)+nat(nat(V)+ -1)*nat(V)])+(nat(V)+1) 
Asymptotic class: n^2 
* Total analysis performed in 192 ms.


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

(10)
BOUNDS(1, n^2)

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

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   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


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

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

(14)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

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

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

(16)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

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

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

(18)
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

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

(19) 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

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

(20)
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

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

(21) 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).
----------------------------------------

(22)
Complex Obligation (BEST)

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

(23)
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

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

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

(25)
BOUNDS(n^1, INF)

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

(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


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

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

(27) 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).
----------------------------------------

(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


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

(29) 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).
----------------------------------------

(30)
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
----------------------------------------

(31) 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).
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(32)
BOUNDS(1, INF)