/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^2), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 294 ms]
    (12) BOUNDS(1, n^2)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) typed CpxTrs
    (17) OrderProof [LOWER BOUND(ID), 0 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 296 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 814 ms]
            (26) proven lower bound
            (27) LowerBoundPropagationProof [FINISHED, 0 ms]
            (28) BOUNDS(n^2, INF)


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

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


The TRS R consists of the following rules:

   f(0) -> 1
   f(s(x)) -> g(x, s(x))
   g(0, y) -> y
   g(s(x), y) -> g(x, +(y, s(x)))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   g(s(x), y) -> g(x, s(+(y, x)))

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

(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, 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:

   f(0) -> 1 [1]
   f(s(x)) -> g(x, s(x)) [1]
   g(0, y) -> y [1]
   g(s(x), y) -> g(x, +(y, s(x))) [1]
   +(x, 0) -> x [1]
   +(x, s(y)) -> s(+(x, y)) [1]
   g(s(x), y) -> g(x, s(+(y, x))) [1]

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

(3) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

+ => plus

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

(4)
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(0) -> 1 [1]
   f(s(x)) -> g(x, s(x)) [1]
   g(0, y) -> y [1]
   g(s(x), y) -> g(x, plus(y, s(x))) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   g(s(x), y) -> g(x, s(plus(y, x))) [1]

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

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(0) -> 1 [1]
   f(s(x)) -> g(x, s(x)) [1]
   g(0, y) -> y [1]
   g(s(x), y) -> g(x, plus(y, s(x))) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   g(s(x), y) -> g(x, s(plus(y, x))) [1]

The TRS has the following type information:
f :: 0:1:s -> 0:1:s
0 :: 0:1:s
1 :: 0:1:s
s :: 0:1:s -> 0:1:s
g :: 0:1:s -> 0:1:s -> 0:1:s
plus :: 0:1:s -> 0:1:s -> 0:1:s

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

(7) 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:

   f(v0) -> null_f [0]
   g(v0, v1) -> null_g [0]
   plus(v0, v1) -> null_plus [0]

And the following fresh constants:    null_f, null_g, null_plus

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

(8)
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(0) -> 1 [1]
   f(s(x)) -> g(x, s(x)) [1]
   g(0, y) -> y [1]
   g(s(x), y) -> g(x, plus(y, s(x))) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   g(s(x), y) -> g(x, s(plus(y, x))) [1]
   f(v0) -> null_f [0]
   g(v0, v1) -> null_g [0]
   plus(v0, v1) -> null_plus [0]

The TRS has the following type information:
f :: 0:1:s:null_f:null_g:null_plus -> 0:1:s:null_f:null_g:null_plus
0 :: 0:1:s:null_f:null_g:null_plus
1 :: 0:1:s:null_f:null_g:null_plus
s :: 0:1:s:null_f:null_g:null_plus -> 0:1:s:null_f:null_g:null_plus
g :: 0:1:s:null_f:null_g:null_plus -> 0:1:s:null_f:null_g:null_plus -> 0:1:s:null_f:null_g:null_plus
plus :: 0:1:s:null_f:null_g:null_plus -> 0:1:s:null_f:null_g:null_plus -> 0:1:s:null_f:null_g:null_plus
null_f :: 0:1:s:null_f:null_g:null_plus
null_g :: 0:1:s:null_f:null_g:null_plus
null_plus :: 0:1:s:null_f:null_g:null_plus

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

(9) 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
   1 => 1
   null_f => 0
   null_g => 0
   null_plus => 0

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z) -{ 1 }-> g(x, 1 + x) :|: x >= 0, z = 1 + x
   f(z) -{ 1 }-> 1 :|: z = 0
   f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   g(z, z') -{ 1 }-> g(x, plus(y, 1 + x)) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   g(z, z') -{ 1 }-> g(x, 1 + plus(y, x)) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   g(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V, V2),0,[f(V, Out)],[V >= 0]).
eq(start(V, V2),0,[g(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(start(V, V2),0,[plus(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(f(V, Out),1,[],[Out = 1,V = 0]).
eq(f(V, Out),1,[g(V1, 1 + V1, Ret)],[Out = Ret,V1 >= 0,V = 1 + V1]).
eq(g(V, V2, Out),1,[],[Out = V3,V3 >= 0,V = 0,V2 = V3]).
eq(g(V, V2, Out),1,[plus(V5, 1 + V4, Ret1),g(V4, Ret1, Ret2)],[Out = Ret2,V4 >= 0,V5 >= 0,V = 1 + V4,V2 = V5]).
eq(plus(V, V2, Out),1,[],[Out = V6,V6 >= 0,V = V6,V2 = 0]).
eq(plus(V, V2, Out),1,[plus(V7, V8, Ret11)],[Out = 1 + Ret11,V2 = 1 + V8,V7 >= 0,V8 >= 0,V = V7]).
eq(g(V, V2, Out),1,[plus(V10, V9, Ret111),g(V9, 1 + Ret111, Ret3)],[Out = Ret3,V9 >= 0,V10 >= 0,V = 1 + V9,V2 = V10]).
eq(f(V, Out),0,[],[Out = 0,V11 >= 0,V = V11]).
eq(g(V, V2, Out),0,[],[Out = 0,V13 >= 0,V12 >= 0,V = V13,V2 = V12]).
eq(plus(V, V2, Out),0,[],[Out = 0,V15 >= 0,V14 >= 0,V = V15,V2 = V14]).
input_output_vars(f(V,Out),[V],[Out]).
input_output_vars(g(V,V2,Out),[V,V2],[Out]).
input_output_vars(plus(V,V2,Out),[V,V2],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [plus/3]
1. recursive  : [g/3]
2. non_recursive  : [f/2]
3. non_recursive  : [start/2]

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

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

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


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

### Ranking functions of CR plus(V,V2,Out) 
* RF of phase [11]: [V2]

#### Partial ranking functions of CR plus(V,V2,Out) 
* Partial RF of phase [11]:
  - RF of loop [11:1]:
    V2


### Specialization of cost equations g/3 
* CE 10 is refined into CE [17] 
* CE 7 is refined into CE [18] 
* CE 8 is refined into CE [19,20,21] 
* CE 9 is refined into CE [22,23,24,25] 


### Cost equations --> "Loop" of g/3 
* CEs [21,23,25] --> Loop 14 
* CEs [19] --> Loop 15 
* CEs [20,22,24] --> Loop 16 
* CEs [17] --> Loop 17 
* CEs [18] --> Loop 18 

### Ranking functions of CR g(V,V2,Out) 
* RF of phase [14,15,16]: [V]

#### Partial ranking functions of CR g(V,V2,Out) 
* Partial RF of phase [14,15,16]:
  - RF of loop [14:1,15:1,16:1]:
    V


### Specialization of cost equations f/2 
* CE 5 is refined into CE [26,27,28] 
* CE 6 is refined into CE [29] 
* CE 4 is refined into CE [30] 


### Cost equations --> "Loop" of f/2 
* CEs [28] --> Loop 19 
* CEs [27,29] --> Loop 20 
* CEs [26] --> Loop 21 
* CEs [30] --> Loop 22 

### Ranking functions of CR f(V,Out) 

#### Partial ranking functions of CR f(V,Out) 


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


### Cost equations --> "Loop" of start/2 
* CEs [38] --> Loop 23 
* CEs [32] --> Loop 24 
* CEs [31,33,34,35,36,37,39,40,41] --> Loop 25 

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

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


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

#### Cost of chains of plus(V,V2,Out):
* Chain [[11],13]: 1*it(11)+1
  Such that:it(11) =< V2

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

* Chain [[11],12]: 1*it(11)+0
  Such that:it(11) =< Out

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

* Chain [13]: 1
  with precondition: [V2=0,V=Out,V>=0] 

* Chain [12]: 0
  with precondition: [Out=0,V>=0,V2>=0] 


#### Cost of chains of g(V,V2,Out):
* Chain [[14,15,16],18]: 4*it(14)+2*s(9)+1*s(11)+1*s(12)+1
  Such that:aux(7) =< V
it(14) =< aux(7)
aux(4) =< aux(7)-1
aux(3) =< aux(7)
s(10) =< it(14)*aux(7)
s(12) =< it(14)*aux(4)
s(11) =< it(14)*aux(3)
s(9) =< s(10)

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

* Chain [[14,15,16],17]: 4*it(14)+2*s(9)+1*s(11)+1*s(12)+0
  Such that:aux(8) =< V
it(14) =< aux(8)
aux(4) =< aux(8)-1
aux(3) =< aux(8)
s(10) =< it(14)*aux(8)
s(12) =< it(14)*aux(4)
s(11) =< it(14)*aux(3)
s(9) =< s(10)

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

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

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


#### Cost of chains of f(V,Out):
* Chain [22]: 1
  with precondition: [V=0,Out=1] 

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

* Chain [20]: 4*s(22)+1*s(26)+1*s(27)+2*s(28)+1
  Such that:s(21) =< V
s(22) =< s(21)
s(23) =< s(21)-1
s(24) =< s(21)
s(25) =< s(22)*s(21)
s(26) =< s(22)*s(23)
s(27) =< s(22)*s(24)
s(28) =< s(25)

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

* Chain [19]: 4*s(30)+1*s(34)+1*s(35)+2*s(36)+2
  Such that:s(29) =< V
s(30) =< s(29)
s(31) =< s(29)-1
s(32) =< s(29)
s(33) =< s(30)*s(29)
s(34) =< s(30)*s(31)
s(35) =< s(30)*s(32)
s(36) =< s(33)

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


#### Cost of chains of start(V,V2):
* Chain [25]: 16*s(38)+4*s(42)+4*s(43)+8*s(44)+2*s(69)+2
  Such that:aux(9) =< V
aux(10) =< V2
s(69) =< aux(10)
s(38) =< aux(9)
s(39) =< aux(9)-1
s(40) =< aux(9)
s(41) =< s(38)*aux(9)
s(42) =< s(38)*s(39)
s(43) =< s(38)*s(40)
s(44) =< s(41)

  with precondition: [V>=0] 

* Chain [24]: 2
  with precondition: [V=1] 

* Chain [23]: 1
  with precondition: [V2=0,V>=0] 


Closed-form bounds of start(V,V2): 
-------------------------------------
* Chain [25] with precondition: [V>=0] 
    - Upper bound: 16*V+2+12*V*V+4*V*nat(V-1)+nat(V2)*2 
    - Complexity: n^2 
* Chain [24] with precondition: [V=1] 
    - Upper bound: 2 
    - Complexity: constant 
* Chain [23] with precondition: [V2=0,V>=0] 
    - Upper bound: 1 
    - Complexity: constant 

### Maximum cost of start(V,V2): max([1,16*V+1+12*V*V+4*V*nat(V-1)+nat(V2)*2])+1 
Asymptotic class: n^2 
* Total analysis performed in 206 ms.


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

(12)
BOUNDS(1, n^2)

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

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

(14)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   f(0') -> 1'
   f(s(x)) -> g(x, s(x))
   g(0', y) -> y
   g(s(x), y) -> g(x, +'(y, s(x)))
   +'(x, 0') -> x
   +'(x, s(y)) -> s(+'(x, y))
   g(s(x), y) -> g(x, s(+'(y, x)))

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

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

(16)
Obligation:
Innermost TRS:
Rules:
f(0') -> 1'
f(s(x)) -> g(x, s(x))
g(0', y) -> y
g(s(x), y) -> g(x, +'(y, s(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
g(s(x), y) -> g(x, s(+'(y, x)))

Types:
f :: 0':1':s -> 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s -> 0':1':s
g :: 0':1':s -> 0':1':s -> 0':1':s
+' :: 0':1':s -> 0':1':s -> 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat -> 0':1':s

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

(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
g, +'

They will be analysed ascendingly in the following order:
+' < g

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

(18)
Obligation:
Innermost TRS:
Rules:
f(0') -> 1'
f(s(x)) -> g(x, s(x))
g(0', y) -> y
g(s(x), y) -> g(x, +'(y, s(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
g(s(x), y) -> g(x, s(+'(y, x)))

Types:
f :: 0':1':s -> 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s -> 0':1':s
g :: 0':1':s -> 0':1':s -> 0':1':s
+' :: 0':1':s -> 0':1':s -> 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat -> 0':1':s


Generator Equations:
gen_0':1':s2_0(0) <=> 0'
gen_0':1':s2_0(+(x, 1)) <=> s(gen_0':1':s2_0(x))


The following defined symbols remain to be analysed:
+', g

They will be analysed ascendingly in the following order:
+' < g

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(n4_0)) -> gen_0':1':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)

Induction Base:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(0)) ->_R^Omega(1)
gen_0':1':s2_0(a)

Induction Step:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(+(n4_0, 1))) ->_R^Omega(1)
s(+'(gen_0':1':s2_0(a), gen_0':1':s2_0(n4_0))) ->_IH
s(gen_0':1':s2_0(+(a, c5_0)))

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

(20)
Complex Obligation (BEST)

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

(21)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
f(0') -> 1'
f(s(x)) -> g(x, s(x))
g(0', y) -> y
g(s(x), y) -> g(x, +'(y, s(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
g(s(x), y) -> g(x, s(+'(y, x)))

Types:
f :: 0':1':s -> 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s -> 0':1':s
g :: 0':1':s -> 0':1':s -> 0':1':s
+' :: 0':1':s -> 0':1':s -> 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat -> 0':1':s


Generator Equations:
gen_0':1':s2_0(0) <=> 0'
gen_0':1':s2_0(+(x, 1)) <=> s(gen_0':1':s2_0(x))


The following defined symbols remain to be analysed:
+', g

They will be analysed ascendingly in the following order:
+' < g

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
Innermost TRS:
Rules:
f(0') -> 1'
f(s(x)) -> g(x, s(x))
g(0', y) -> y
g(s(x), y) -> g(x, +'(y, s(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
g(s(x), y) -> g(x, s(+'(y, x)))

Types:
f :: 0':1':s -> 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s -> 0':1':s
g :: 0':1':s -> 0':1':s -> 0':1':s
+' :: 0':1':s -> 0':1':s -> 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat -> 0':1':s


Lemmas:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(n4_0)) -> gen_0':1':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':1':s2_0(0) <=> 0'
gen_0':1':s2_0(+(x, 1)) <=> s(gen_0':1':s2_0(x))


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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_0':1':s2_0(n513_0), gen_0':1':s2_0(b)) -> *3_0, rt in Omega(n513_0 + n513_0^2)

Induction Base:
g(gen_0':1':s2_0(0), gen_0':1':s2_0(b))

Induction Step:
g(gen_0':1':s2_0(+(n513_0, 1)), gen_0':1':s2_0(b)) ->_R^Omega(1)
g(gen_0':1':s2_0(n513_0), +'(gen_0':1':s2_0(b), s(gen_0':1':s2_0(n513_0)))) ->_L^Omega(2 + n513_0)
g(gen_0':1':s2_0(n513_0), gen_0':1':s2_0(+(+(n513_0, 1), b))) ->_IH
*3_0

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
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(26)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
f(0') -> 1'
f(s(x)) -> g(x, s(x))
g(0', y) -> y
g(s(x), y) -> g(x, +'(y, s(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
g(s(x), y) -> g(x, s(+'(y, x)))

Types:
f :: 0':1':s -> 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s -> 0':1':s
g :: 0':1':s -> 0':1':s -> 0':1':s
+' :: 0':1':s -> 0':1':s -> 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat -> 0':1':s


Lemmas:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(n4_0)) -> gen_0':1':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':1':s2_0(0) <=> 0'
gen_0':1':s2_0(+(x, 1)) <=> s(gen_0':1':s2_0(x))


The following defined symbols remain to be analysed:
g
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(27) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(28)
BOUNDS(n^2, INF)