WORST_CASE(Omega(n^1),O(n^2))
proof of input_1uI4G28ozx.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxWeightedTrs
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedTrs
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
    (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) CompleteCoflocoProof [FINISHED, 225 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), 0 ms]
    (28) typed CpxTrs
    (29) RewriteLemmaProof [LOWER BOUND(ID), 307 ms]
    (30) typed CpxTrs
    (31) RewriteLemmaProof [LOWER BOUND(ID), 88 ms]
    (32) BEST
        (33) proven lower bound
            (34) LowerBoundPropagationProof [FINISHED, 0 ms]
            (35) BOUNDS(n^1, INF)
        (36) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   f(t, x, y) -> f(g(x, y), x, s(y))
   g(s(x), 0) -> t
   g(s(x), s(y)) -> g(x, 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:
   f(t, z0, z1) -> f(g(z0, z1), z0, s(z1))
   g(s(z0), 0) -> t
   g(s(z0), s(z1)) -> g(z0, z1)
Tuples:
   F(t, z0, z1) -> c(F(g(z0, z1), z0, s(z1)), G(z0, z1))
   G(s(z0), 0) -> c1
   G(s(z0), s(z1)) -> c2(G(z0, z1))
S tuples:
   F(t, z0, z1) -> c(F(g(z0, z1), z0, s(z1)), G(z0, z1))
   G(s(z0), 0) -> c1
   G(s(z0), s(z1)) -> c2(G(z0, z1))
K tuples:none
Defined Rule Symbols:   f_3, g_2

Defined Pair Symbols:   F_3, G_2

Compound Symbols:   c_2, c1, c2_1


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   G(s(z0), 0) -> c1

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(t, z0, z1) -> f(g(z0, z1), z0, s(z1))
   g(s(z0), 0) -> t
   g(s(z0), s(z1)) -> g(z0, z1)
Tuples:
   F(t, z0, z1) -> c(F(g(z0, z1), z0, s(z1)), G(z0, z1))
   G(s(z0), s(z1)) -> c2(G(z0, z1))
S tuples:
   F(t, z0, z1) -> c(F(g(z0, z1), z0, s(z1)), G(z0, z1))
   G(s(z0), s(z1)) -> c2(G(z0, z1))
K tuples:none
Defined Rule Symbols:   f_3, g_2

Defined Pair Symbols:   F_3, G_2

Compound Symbols:   c_2, c2_1


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(t, z0, z1) -> f(g(z0, z1), z0, s(z1))

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(s(z0), 0) -> t
   g(s(z0), s(z1)) -> g(z0, z1)
Tuples:
   F(t, z0, z1) -> c(F(g(z0, z1), z0, s(z1)), G(z0, z1))
   G(s(z0), s(z1)) -> c2(G(z0, z1))
S tuples:
   F(t, z0, z1) -> c(F(g(z0, z1), z0, s(z1)), G(z0, z1))
   G(s(z0), s(z1)) -> c2(G(z0, z1))
K tuples:none
Defined Rule Symbols:   g_2

Defined Pair Symbols:   F_3, G_2

Compound Symbols:   c_2, c2_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:

   F(t, z0, z1) -> c(F(g(z0, z1), z0, s(z1)), G(z0, z1))
   G(s(z0), s(z1)) -> c2(G(z0, z1))

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

   g(s(z0), 0) -> t
   g(s(z0), s(z1)) -> g(z0, z1)

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:

   F(t, z0, z1) -> c(F(g(z0, z1), z0, s(z1)), G(z0, z1)) [1]
   G(s(z0), s(z1)) -> c2(G(z0, z1)) [1]
   g(s(z0), 0) -> t [0]
   g(s(z0), s(z1)) -> g(z0, z1) [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:

   F(t, z0, z1) -> c(F(g(z0, z1), z0, s(z1)), G(z0, z1)) [1]
   G(s(z0), s(z1)) -> c2(G(z0, z1)) [1]
   g(s(z0), 0) -> t [0]
   g(s(z0), s(z1)) -> g(z0, z1) [0]

The TRS has the following type information:
F :: t -> s:0 -> s:0 -> c
t :: t
c :: c -> c2 -> c
g :: s:0 -> s:0 -> t
s :: s:0 -> s:0
G :: s:0 -> s:0 -> c2
c2 :: c2 -> c2
0 :: s:0

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:

   g(v0, v1) -> null_g [0]
   F(v0, v1, v2) -> null_F [0]
   G(v0, v1) -> null_G [0]

And the following fresh constants:    null_g, null_F, null_G

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

(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:

   F(t, z0, z1) -> c(F(g(z0, z1), z0, s(z1)), G(z0, z1)) [1]
   G(s(z0), s(z1)) -> c2(G(z0, z1)) [1]
   g(s(z0), 0) -> t [0]
   g(s(z0), s(z1)) -> g(z0, z1) [0]
   g(v0, v1) -> null_g [0]
   F(v0, v1, v2) -> null_F [0]
   G(v0, v1) -> null_G [0]

The TRS has the following type information:
F :: t:null_g -> s:0 -> s:0 -> c:null_F
t :: t:null_g
c :: c:null_F -> c2:null_G -> c:null_F
g :: s:0 -> s:0 -> t:null_g
s :: s:0 -> s:0
G :: s:0 -> s:0 -> c2:null_G
c2 :: c2:null_G -> c2:null_G
0 :: s:0
null_g :: t:null_g
null_F :: c:null_F
null_G :: c2:null_G

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:

   t => 1
   0 => 0
   null_g => 0
   null_F => 0
   null_G => 0

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

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

   F(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   F(z, z', z'') -{ 1 }-> 1 + F(g(z0, z1), z0, 1 + z1) + G(z0, z1) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1
   G(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   G(z, z') -{ 1 }-> 1 + G(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   g(z, z') -{ 0 }-> g(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   g(z, z') -{ 0 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

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

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


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

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

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

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

### Specialization of cost equations fun1/3 
* CE 7 is refined into CE [11] 
* CE 6 is refined into CE [12] 


### Cost equations --> "Loop" of fun1/3 
* CEs [12] --> Loop 9 
* CEs [11] --> Loop 10 

### Ranking functions of CR fun1(V1,V,Out) 
* RF of phase [9]: [V,V1]

#### Partial ranking functions of CR fun1(V1,V,Out) 
* Partial RF of phase [9]:
  - RF of loop [9:1]:
    V
    V1


### Specialization of cost equations g/3 
* CE 10 is refined into CE [13] 
* CE 8 is refined into CE [14] 
* CE 9 is refined into CE [15] 


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

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

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


### Specialization of cost equations fun/4 
* CE 5 is refined into CE [16] 
* CE 4 is refined into CE [17,18,19,20,21] 


### Cost equations --> "Loop" of fun/4 
* CEs [20] --> Loop 14 
* CEs [21] --> Loop 15 
* CEs [19] --> Loop 16 
* CEs [18] --> Loop 17 
* CEs [17] --> Loop 18 
* CEs [16] --> Loop 19 

### Ranking functions of CR fun(V1,V,V2,Out) 
* RF of phase [14,15]: [V-V2]

#### Partial ranking functions of CR fun(V1,V,V2,Out) 
* Partial RF of phase [14,15]:
  - RF of loop [14:1,15:1]:
    V-V2


### Specialization of cost equations start/3 
* CE 1 is refined into CE [22,23,24,25,26,27,28] 
* CE 2 is refined into CE [29,30] 
* CE 3 is refined into CE [31,32,33] 


### Cost equations --> "Loop" of start/3 
* CEs [31] --> Loop 20 
* CEs [22,23,24,25,26,27,28,29,30,32,33] --> Loop 21 

### Ranking functions of CR start(V1,V,V2) 

#### Partial ranking functions of CR start(V1,V,V2) 


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

#### Cost of chains of fun1(V1,V,Out):
* Chain [[9],10]: 1*it(9)+0
  Such that:it(9) =< Out

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

* Chain [10]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of g(V1,V,Out):
* Chain [[11],13]: 0
  with precondition: [Out=1,V>=1,V1>=V+1] 

* Chain [[11],12]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

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

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


#### Cost of chains of fun(V1,V,V2,Out):
* Chain [[14,15],19]: 2*it(14)+1*s(3)+0
  Such that:aux(1) =< V
aux(4) =< V-V2
it(14) =< aux(4)
s(3) =< it(14)*aux(1)

  with precondition: [V1=1,V2>=1,Out>=1,V>=V2+1] 

* Chain [[14,15],17,19]: 2*it(14)+1*s(3)+1
  Such that:aux(1) =< V
aux(5) =< V-V2
it(14) =< aux(5)
s(3) =< it(14)*aux(1)

  with precondition: [V1=1,V2>=1,Out>=2,V>=V2+1] 

* Chain [[14,15],16,19]: 2*it(14)+1*s(3)+1*s(4)+1
  Such that:aux(1) =< V
s(4) =< V+1
aux(6) =< V-V2
it(14) =< aux(6)
s(3) =< it(14)*aux(1)

  with precondition: [V1=1,V2>=1,Out>=3,V>=V2+1] 

* Chain [19]: 0
  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 

* Chain [18,[14,15],19]: 2*it(14)+1*s(3)+1
  Such that:aux(7) =< V
it(14) =< aux(7)
s(3) =< it(14)*aux(7)

  with precondition: [V1=1,V2=0,V>=2,Out>=2] 

* Chain [18,[14,15],17,19]: 2*it(14)+1*s(3)+2
  Such that:aux(8) =< V
it(14) =< aux(8)
s(3) =< it(14)*aux(8)

  with precondition: [V1=1,V2=0,V>=2,Out>=3] 

* Chain [18,[14,15],16,19]: 2*it(14)+1*s(3)+1*s(4)+2
  Such that:s(4) =< V+1
aux(9) =< V
it(14) =< aux(9)
s(3) =< it(14)*aux(9)

  with precondition: [V1=1,V2=0,V>=2,Out>=4] 

* Chain [18,19]: 1
  with precondition: [V1=1,V2=0,Out=1,V>=1] 

* Chain [18,17,19]: 2
  with precondition: [V1=1,V2=0,Out=2,V>=1] 

* Chain [18,16,19]: 1*s(4)+2
  Such that:s(4) =< 2

  with precondition: [V1=1,V2=0,Out=3,V>=1] 

* Chain [17,19]: 1
  with precondition: [V1=1,Out=1,V>=0,V2>=0] 

* Chain [16,19]: 1*s(4)+1
  Such that:s(4) =< V2+1

  with precondition: [V1=1,Out>=2,V+1>=Out,V2+1>=Out] 


#### Cost of chains of start(V1,V,V2):
* Chain [21]: 1*s(28)+2*s(29)+7*s(31)+3*s(32)+6*s(36)+3*s(37)+1*s(38)+2
  Such that:s(28) =< 2
s(35) =< V-V2
s(38) =< V2+1
aux(13) =< V
aux(14) =< V+1
s(31) =< aux(13)
s(29) =< aux(14)
s(32) =< s(31)*aux(13)
s(36) =< s(35)
s(37) =< s(36)*aux(13)

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

* Chain [20]: 0
  with precondition: [V=0,V1>=1] 


Closed-form bounds of start(V1,V,V2): 
-------------------------------------
* Chain [21] with precondition: [V1>=0,V>=0] 
    - Upper bound: 7*V+4+3*V*V+3*V*nat(V-V2)+(2*V+2)+nat(V2+1)+nat(V-V2)*6 
    - Complexity: n^2 
* Chain [20] with precondition: [V=0,V1>=1] 
    - Upper bound: 0 
    - Complexity: constant 

### Maximum cost of start(V1,V,V2): 7*V+4+3*V*V+3*V*nat(V-V2)+(2*V+2)+nat(V2+1)+nat(V-V2)*6 
Asymptotic class: n^2 
* Total analysis performed in 302 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:
   f(t, z0, z1) -> f(g(z0, z1), z0, s(z1))
   g(s(z0), 0) -> t
   g(s(z0), s(z1)) -> g(z0, z1)
Tuples:
   F(t, z0, z1) -> c(F(g(z0, z1), z0, s(z1)), G(z0, z1))
   G(s(z0), 0) -> c1
   G(s(z0), s(z1)) -> c2(G(z0, z1))
S tuples:
   F(t, z0, z1) -> c(F(g(z0, z1), z0, s(z1)), G(z0, z1))
   G(s(z0), 0) -> c1
   G(s(z0), s(z1)) -> c2(G(z0, z1))
K tuples:none
Defined Rule Symbols:   f_3, g_2

Defined Pair Symbols:   F_3, G_2

Compound Symbols:   c_2, c1, c2_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:

   F(t, z0, z1) -> c(F(g(z0, z1), z0, s(z1)), G(z0, z1))
   G(s(z0), 0) -> c1
   G(s(z0), s(z1)) -> c2(G(z0, z1))

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

   f(t, z0, z1) -> f(g(z0, z1), z0, s(z1))
   g(s(z0), 0) -> t
   g(s(z0), s(z1)) -> g(z0, z1)

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

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

(24)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   F(t, z0, z1) -> c(F(g(z0, z1), z0, s(z1)), G(z0, z1))
   G(s(z0), 0') -> c1
   G(s(z0), s(z1)) -> c2(G(z0, z1))

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

   f(t, z0, z1) -> f(g(z0, z1), z0, s(z1))
   g(s(z0), 0') -> t
   g(s(z0), s(z1)) -> g(z0, z1)

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

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

(26)
Obligation:
Innermost TRS:
Rules:
F(t, z0, z1) -> c(F(g(z0, z1), z0, s(z1)), G(z0, z1))
G(s(z0), 0') -> c1
G(s(z0), s(z1)) -> c2(G(z0, z1))
f(t, z0, z1) -> f(g(z0, z1), z0, s(z1))
g(s(z0), 0') -> t
g(s(z0), s(z1)) -> g(z0, z1)

Types:
F :: t -> s:0' -> s:0' -> c
t :: t
c :: c -> c1:c2 -> c
g :: s:0' -> s:0' -> t
s :: s:0' -> s:0'
G :: s:0' -> s:0' -> c1:c2
0' :: s:0'
c1 :: c1:c2
c2 :: c1:c2 -> c1:c2
f :: t -> s:0' -> s:0' -> f
hole_c1_3 :: c
hole_t2_3 :: t
hole_s:0'3_3 :: s:0'
hole_c1:c24_3 :: c1:c2
hole_f5_3 :: f
gen_c6_3 :: Nat -> c
gen_s:0'7_3 :: Nat -> s:0'
gen_c1:c28_3 :: Nat -> c1:c2

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

(27) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
F, g, G, f

They will be analysed ascendingly in the following order:
g < F
G < F
g < f

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

(28)
Obligation:
Innermost TRS:
Rules:
F(t, z0, z1) -> c(F(g(z0, z1), z0, s(z1)), G(z0, z1))
G(s(z0), 0') -> c1
G(s(z0), s(z1)) -> c2(G(z0, z1))
f(t, z0, z1) -> f(g(z0, z1), z0, s(z1))
g(s(z0), 0') -> t
g(s(z0), s(z1)) -> g(z0, z1)

Types:
F :: t -> s:0' -> s:0' -> c
t :: t
c :: c -> c1:c2 -> c
g :: s:0' -> s:0' -> t
s :: s:0' -> s:0'
G :: s:0' -> s:0' -> c1:c2
0' :: s:0'
c1 :: c1:c2
c2 :: c1:c2 -> c1:c2
f :: t -> s:0' -> s:0' -> f
hole_c1_3 :: c
hole_t2_3 :: t
hole_s:0'3_3 :: s:0'
hole_c1:c24_3 :: c1:c2
hole_f5_3 :: f
gen_c6_3 :: Nat -> c
gen_s:0'7_3 :: Nat -> s:0'
gen_c1:c28_3 :: Nat -> c1:c2


Generator Equations:
gen_c6_3(0) <=> hole_c1_3
gen_c6_3(+(x, 1)) <=> c(gen_c6_3(x), c1)
gen_s:0'7_3(0) <=> 0'
gen_s:0'7_3(+(x, 1)) <=> s(gen_s:0'7_3(x))
gen_c1:c28_3(0) <=> c1
gen_c1:c28_3(+(x, 1)) <=> c2(gen_c1:c28_3(x))


The following defined symbols remain to be analysed:
g, F, G, f

They will be analysed ascendingly in the following order:
g < F
G < F
g < f

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_s:0'7_3(+(1, n10_3)), gen_s:0'7_3(n10_3)) -> t, rt in Omega(0)

Induction Base:
g(gen_s:0'7_3(+(1, 0)), gen_s:0'7_3(0)) ->_R^Omega(0)
t

Induction Step:
g(gen_s:0'7_3(+(1, +(n10_3, 1))), gen_s:0'7_3(+(n10_3, 1))) ->_R^Omega(0)
g(gen_s:0'7_3(+(1, n10_3)), gen_s:0'7_3(n10_3)) ->_IH
t

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

(30)
Obligation:
Innermost TRS:
Rules:
F(t, z0, z1) -> c(F(g(z0, z1), z0, s(z1)), G(z0, z1))
G(s(z0), 0') -> c1
G(s(z0), s(z1)) -> c2(G(z0, z1))
f(t, z0, z1) -> f(g(z0, z1), z0, s(z1))
g(s(z0), 0') -> t
g(s(z0), s(z1)) -> g(z0, z1)

Types:
F :: t -> s:0' -> s:0' -> c
t :: t
c :: c -> c1:c2 -> c
g :: s:0' -> s:0' -> t
s :: s:0' -> s:0'
G :: s:0' -> s:0' -> c1:c2
0' :: s:0'
c1 :: c1:c2
c2 :: c1:c2 -> c1:c2
f :: t -> s:0' -> s:0' -> f
hole_c1_3 :: c
hole_t2_3 :: t
hole_s:0'3_3 :: s:0'
hole_c1:c24_3 :: c1:c2
hole_f5_3 :: f
gen_c6_3 :: Nat -> c
gen_s:0'7_3 :: Nat -> s:0'
gen_c1:c28_3 :: Nat -> c1:c2


Lemmas:
g(gen_s:0'7_3(+(1, n10_3)), gen_s:0'7_3(n10_3)) -> t, rt in Omega(0)


Generator Equations:
gen_c6_3(0) <=> hole_c1_3
gen_c6_3(+(x, 1)) <=> c(gen_c6_3(x), c1)
gen_s:0'7_3(0) <=> 0'
gen_s:0'7_3(+(x, 1)) <=> s(gen_s:0'7_3(x))
gen_c1:c28_3(0) <=> c1
gen_c1:c28_3(+(x, 1)) <=> c2(gen_c1:c28_3(x))


The following defined symbols remain to be analysed:
G, F, f

They will be analysed ascendingly in the following order:
G < F

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
G(gen_s:0'7_3(+(1, n155_3)), gen_s:0'7_3(n155_3)) -> gen_c1:c28_3(n155_3), rt in Omega(1 + n155_3)

Induction Base:
G(gen_s:0'7_3(+(1, 0)), gen_s:0'7_3(0)) ->_R^Omega(1)
c1

Induction Step:
G(gen_s:0'7_3(+(1, +(n155_3, 1))), gen_s:0'7_3(+(n155_3, 1))) ->_R^Omega(1)
c2(G(gen_s:0'7_3(+(1, n155_3)), gen_s:0'7_3(n155_3))) ->_IH
c2(gen_c1:c28_3(c156_3))

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:
F(t, z0, z1) -> c(F(g(z0, z1), z0, s(z1)), G(z0, z1))
G(s(z0), 0') -> c1
G(s(z0), s(z1)) -> c2(G(z0, z1))
f(t, z0, z1) -> f(g(z0, z1), z0, s(z1))
g(s(z0), 0') -> t
g(s(z0), s(z1)) -> g(z0, z1)

Types:
F :: t -> s:0' -> s:0' -> c
t :: t
c :: c -> c1:c2 -> c
g :: s:0' -> s:0' -> t
s :: s:0' -> s:0'
G :: s:0' -> s:0' -> c1:c2
0' :: s:0'
c1 :: c1:c2
c2 :: c1:c2 -> c1:c2
f :: t -> s:0' -> s:0' -> f
hole_c1_3 :: c
hole_t2_3 :: t
hole_s:0'3_3 :: s:0'
hole_c1:c24_3 :: c1:c2
hole_f5_3 :: f
gen_c6_3 :: Nat -> c
gen_s:0'7_3 :: Nat -> s:0'
gen_c1:c28_3 :: Nat -> c1:c2


Lemmas:
g(gen_s:0'7_3(+(1, n10_3)), gen_s:0'7_3(n10_3)) -> t, rt in Omega(0)


Generator Equations:
gen_c6_3(0) <=> hole_c1_3
gen_c6_3(+(x, 1)) <=> c(gen_c6_3(x), c1)
gen_s:0'7_3(0) <=> 0'
gen_s:0'7_3(+(x, 1)) <=> s(gen_s:0'7_3(x))
gen_c1:c28_3(0) <=> c1
gen_c1:c28_3(+(x, 1)) <=> c2(gen_c1:c28_3(x))


The following defined symbols remain to be analysed:
G, F, f

They will be analysed ascendingly in the following order:
G < F

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

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

(35)
BOUNDS(n^1, INF)

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

(36)
Obligation:
Innermost TRS:
Rules:
F(t, z0, z1) -> c(F(g(z0, z1), z0, s(z1)), G(z0, z1))
G(s(z0), 0') -> c1
G(s(z0), s(z1)) -> c2(G(z0, z1))
f(t, z0, z1) -> f(g(z0, z1), z0, s(z1))
g(s(z0), 0') -> t
g(s(z0), s(z1)) -> g(z0, z1)

Types:
F :: t -> s:0' -> s:0' -> c
t :: t
c :: c -> c1:c2 -> c
g :: s:0' -> s:0' -> t
s :: s:0' -> s:0'
G :: s:0' -> s:0' -> c1:c2
0' :: s:0'
c1 :: c1:c2
c2 :: c1:c2 -> c1:c2
f :: t -> s:0' -> s:0' -> f
hole_c1_3 :: c
hole_t2_3 :: t
hole_s:0'3_3 :: s:0'
hole_c1:c24_3 :: c1:c2
hole_f5_3 :: f
gen_c6_3 :: Nat -> c
gen_s:0'7_3 :: Nat -> s:0'
gen_c1:c28_3 :: Nat -> c1:c2


Lemmas:
g(gen_s:0'7_3(+(1, n10_3)), gen_s:0'7_3(n10_3)) -> t, rt in Omega(0)
G(gen_s:0'7_3(+(1, n155_3)), gen_s:0'7_3(n155_3)) -> gen_c1:c28_3(n155_3), rt in Omega(1 + n155_3)


Generator Equations:
gen_c6_3(0) <=> hole_c1_3
gen_c6_3(+(x, 1)) <=> c(gen_c6_3(x), c1)
gen_s:0'7_3(0) <=> 0'
gen_s:0'7_3(+(x, 1)) <=> s(gen_s:0'7_3(x))
gen_c1:c28_3(0) <=> c1
gen_c1:c28_3(+(x, 1)) <=> c2(gen_c1:c28_3(x))


The following defined symbols remain to be analysed:
F, f