WORST_CASE(Omega(n^2),O(n^2))
proof of input_AXQod7XSju.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, 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, 266 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), 8 ms]
    (28) typed CpxTrs
    (29) RewriteLemmaProof [LOWER BOUND(ID), 281 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), 98 ms]
            (36) typed CpxTrs
            (37) RewriteLemmaProof [LOWER BOUND(ID), 58 ms]
            (38) typed CpxTrs
            (39) RewriteLemmaProof [LOWER BOUND(ID), 664 ms]
            (40) proven lower bound
            (41) LowerBoundPropagationProof [FINISHED, 0 ms]
            (42) BOUNDS(n^2, INF)


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

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


The TRS R consists of the following rules:

   and(tt, X) -> activate(X)
   plus(N, 0) -> N
   plus(N, s(M)) -> s(plus(N, M))
   x(N, 0) -> 0
   x(N, s(M)) -> plus(x(N, M), N)
   activate(X) -> X

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:
   and(tt, z0) -> activate(z0)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   x(z0, 0) -> 0
   x(z0, s(z1)) -> plus(x(z0, z1), z0)
   activate(z0) -> z0
Tuples:
   AND(tt, z0) -> c(ACTIVATE(z0))
   PLUS(z0, 0) -> c1
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, 0) -> c3
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
   ACTIVATE(z0) -> c5
S tuples:
   AND(tt, z0) -> c(ACTIVATE(z0))
   PLUS(z0, 0) -> c1
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, 0) -> c3
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
   ACTIVATE(z0) -> c5
K tuples:none
Defined Rule Symbols:   and_2, plus_2, x_2, activate_1

Defined Pair Symbols:   AND_2, PLUS_2, X_2, ACTIVATE_1

Compound Symbols:   c_1, c1, c2_1, c3, c4_2, c5


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   PLUS(z0, 0) -> c1
   AND(tt, z0) -> c(ACTIVATE(z0))
   ACTIVATE(z0) -> c5
   X(z0, 0) -> c3

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   and(tt, z0) -> activate(z0)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   x(z0, 0) -> 0
   x(z0, s(z1)) -> plus(x(z0, z1), z0)
   activate(z0) -> z0
Tuples:
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
S tuples:
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
K tuples:none
Defined Rule Symbols:   and_2, plus_2, x_2, activate_1

Defined Pair Symbols:   PLUS_2, X_2

Compound Symbols:   c2_1, c4_2


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   and(tt, z0) -> activate(z0)
   activate(z0) -> z0

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   x(z0, 0) -> 0
   x(z0, s(z1)) -> plus(x(z0, z1), z0)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
S tuples:
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
K tuples:none
Defined Rule Symbols:   x_2, plus_2

Defined Pair Symbols:   PLUS_2, X_2

Compound Symbols:   c2_1, c4_2


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

(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:

   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))

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

   x(z0, 0) -> 0
   x(z0, s(z1)) -> plus(x(z0, z1), z0)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(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:

   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1)) [1]
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1)) [1]
   x(z0, 0) -> 0 [0]
   x(z0, s(z1)) -> plus(x(z0, z1), z0) [0]
   plus(z0, 0) -> z0 [0]
   plus(z0, s(z1)) -> s(plus(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:

   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1)) [1]
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1)) [1]
   x(z0, 0) -> 0 [0]
   x(z0, s(z1)) -> plus(x(z0, z1), z0) [0]
   plus(z0, 0) -> z0 [0]
   plus(z0, s(z1)) -> s(plus(z0, z1)) [0]

The TRS has the following type information:
PLUS :: s:0 -> s:0 -> c2
s :: s:0 -> s:0
c2 :: c2 -> c2
X :: s:0 -> s:0 -> c4
c4 :: c2 -> c4 -> c4
x :: s:0 -> s:0 -> s:0
0 :: s:0
plus :: s:0 -> s: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:

   x(v0, v1) -> null_x [0]
   plus(v0, v1) -> null_plus [0]
   PLUS(v0, v1) -> null_PLUS [0]
   X(v0, v1) -> null_X [0]

And the following fresh constants:    null_x, null_plus, null_PLUS, null_X

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

(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:

   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1)) [1]
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1)) [1]
   x(z0, 0) -> 0 [0]
   x(z0, s(z1)) -> plus(x(z0, z1), z0) [0]
   plus(z0, 0) -> z0 [0]
   plus(z0, s(z1)) -> s(plus(z0, z1)) [0]
   x(v0, v1) -> null_x [0]
   plus(v0, v1) -> null_plus [0]
   PLUS(v0, v1) -> null_PLUS [0]
   X(v0, v1) -> null_X [0]

The TRS has the following type information:
PLUS :: s:0:null_x:null_plus -> s:0:null_x:null_plus -> c2:null_PLUS
s :: s:0:null_x:null_plus -> s:0:null_x:null_plus
c2 :: c2:null_PLUS -> c2:null_PLUS
X :: s:0:null_x:null_plus -> s:0:null_x:null_plus -> c4:null_X
c4 :: c2:null_PLUS -> c4:null_X -> c4:null_X
x :: s:0:null_x:null_plus -> s:0:null_x:null_plus -> s:0:null_x:null_plus
0 :: s:0:null_x:null_plus
plus :: s:0:null_x:null_plus -> s:0:null_x:null_plus -> s:0:null_x:null_plus
null_x :: s:0:null_x:null_plus
null_plus :: s:0:null_x:null_plus
null_PLUS :: c2:null_PLUS
null_X :: c4:null_X

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_x => 0
   null_plus => 0
   null_PLUS => 0
   null_X => 0

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

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

   PLUS(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z0, z1) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   X(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   X(z, z') -{ 1 }-> 1 + PLUS(x(z0, z1), z0) + X(z0, z1) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   plus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 0 }-> 1 + plus(z0, z1) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   x(z, z') -{ 0 }-> plus(x(z0, z1), z0) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   x(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   x(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),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[fun1(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[x(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[plus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(fun(V1, V, Out),1,[fun(V3, V2, Ret1)],[Out = 1 + Ret1,V1 = V3,V2 >= 0,V3 >= 0,V = 1 + V2]).
eq(fun1(V1, V, Out),1,[x(V5, V4, Ret010),fun(Ret010, V5, Ret01),fun1(V5, V4, Ret11)],[Out = 1 + Ret01 + Ret11,V1 = V5,V4 >= 0,V5 >= 0,V = 1 + V4]).
eq(x(V1, V, Out),0,[],[Out = 0,V1 = V6,V6 >= 0,V = 0]).
eq(x(V1, V, Out),0,[x(V7, V8, Ret0),plus(Ret0, V7, Ret)],[Out = Ret,V1 = V7,V8 >= 0,V7 >= 0,V = 1 + V8]).
eq(plus(V1, V, Out),0,[],[Out = V9,V1 = V9,V9 >= 0,V = 0]).
eq(plus(V1, V, Out),0,[plus(V10, V11, Ret12)],[Out = 1 + Ret12,V1 = V10,V11 >= 0,V10 >= 0,V = 1 + V11]).
eq(x(V1, V, Out),0,[],[Out = 0,V13 >= 0,V12 >= 0,V1 = V13,V = V12]).
eq(plus(V1, V, Out),0,[],[Out = 0,V15 >= 0,V14 >= 0,V1 = V15,V = V14]).
eq(fun(V1, V, Out),0,[],[Out = 0,V17 >= 0,V16 >= 0,V1 = V17,V = V16]).
eq(fun1(V1, V, Out),0,[],[Out = 0,V18 >= 0,V19 >= 0,V1 = V18,V = V19]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,Out),[V1,V],[Out]).
input_output_vars(x(V1,V,Out),[V1,V],[Out]).
input_output_vars(plus(V1,V,Out),[V1,V],[Out]).


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

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

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

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

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


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

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

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


### Specialization of cost equations plus/3 
* CE 13 is refined into CE [16] 
* CE 11 is refined into CE [17] 
* CE 12 is refined into CE [18] 


### Cost equations --> "Loop" of plus/3 
* CEs [18] --> Loop 13 
* CEs [16] --> Loop 14 
* CEs [17] --> Loop 15 

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

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


### Specialization of cost equations x/3 
* CE 9 is refined into CE [19] 
* CE 10 is refined into CE [20,21,22,23] 


### Cost equations --> "Loop" of x/3 
* CEs [23] --> Loop 16 
* CEs [22] --> Loop 17 
* CEs [21] --> Loop 18 
* CEs [20] --> Loop 19 
* CEs [19] --> Loop 20 

### Ranking functions of CR x(V1,V,Out) 
* RF of phase [16,17,18,19]: [V]

#### Partial ranking functions of CR x(V1,V,Out) 
* Partial RF of phase [16,17,18,19]:
  - RF of loop [16:1,17:1,18:1,19:1]:
    V


### Specialization of cost equations fun1/3 
* CE 8 is refined into CE [24] 
* CE 7 is refined into CE [25,26,27,28] 


### Cost equations --> "Loop" of fun1/3 
* CEs [26,28] --> Loop 21 
* CEs [25,27] --> Loop 22 
* CEs [24] --> Loop 23 

### Ranking functions of CR fun1(V1,V,Out) 
* RF of phase [21,22]: [V]

#### Partial ranking functions of CR fun1(V1,V,Out) 
* Partial RF of phase [21,22]:
  - RF of loop [21:1,22:1]:
    V


### Specialization of cost equations start/2 
* CE 1 is refined into CE [29,30] 
* CE 2 is refined into CE [31,32] 
* CE 3 is refined into CE [33,34] 
* CE 4 is refined into CE [35,36,37,38] 


### Cost equations --> "Loop" of start/2 
* CEs [29,30,31,32,33,34,35,36,37,38] --> Loop 24 

### 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: [V1>=0,Out>=1,V>=Out] 

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


#### Cost of chains of plus(V1,V,Out):
* Chain [[13],15]: 0
  with precondition: [V+V1=Out,V1>=0,V>=1] 

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

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

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


#### Cost of chains of x(V1,V,Out):
* Chain [[16,17,18,19],20]: 0
  with precondition: [V1>=0,V>=1,Out>=0] 

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


#### Cost of chains of fun1(V1,V,Out):
* Chain [[21,22],23]: 2*it(21)+2*s(5)+0
  Such that:aux(4) =< V1
aux(7) =< V
it(21) =< aux(7)
s(6) =< it(21)*aux(4)
s(5) =< s(6)

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

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


#### Cost of chains of start(V1,V):
* Chain [24]: 3*s(7)+2*s(12)+0
  Such that:s(8) =< V1
aux(8) =< V
s(7) =< aux(8)
s(11) =< s(7)*s(8)
s(12) =< s(11)

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


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [24] with precondition: [V1>=0,V>=0] 
    - Upper bound: 2*V1*V+3*V 
    - Complexity: n^2 

### Maximum cost of start(V1,V): 2*V1*V+3*V 
Asymptotic class: n^2 
* Total analysis performed in 221 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:
   and(tt, z0) -> activate(z0)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   x(z0, 0) -> 0
   x(z0, s(z1)) -> plus(x(z0, z1), z0)
   activate(z0) -> z0
Tuples:
   AND(tt, z0) -> c(ACTIVATE(z0))
   PLUS(z0, 0) -> c1
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, 0) -> c3
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
   ACTIVATE(z0) -> c5
S tuples:
   AND(tt, z0) -> c(ACTIVATE(z0))
   PLUS(z0, 0) -> c1
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, 0) -> c3
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
   ACTIVATE(z0) -> c5
K tuples:none
Defined Rule Symbols:   and_2, plus_2, x_2, activate_1

Defined Pair Symbols:   AND_2, PLUS_2, X_2, ACTIVATE_1

Compound Symbols:   c_1, c1, c2_1, c3, c4_2, c5


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

(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^2, INF).


The TRS R consists of the following rules:

   AND(tt, z0) -> c(ACTIVATE(z0))
   PLUS(z0, 0) -> c1
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, 0) -> c3
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
   ACTIVATE(z0) -> c5

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

   and(tt, z0) -> activate(z0)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   x(z0, 0) -> 0
   x(z0, s(z1)) -> plus(x(z0, z1), z0)
   activate(z0) -> z0

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^2, INF).


The TRS R consists of the following rules:

   AND(tt, z0) -> c(ACTIVATE(z0))
   PLUS(z0, 0') -> c1
   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
   X(z0, 0') -> c3
   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
   ACTIVATE(z0) -> c5

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

   and(tt, z0) -> activate(z0)
   plus(z0, 0') -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   x(z0, 0') -> 0'
   x(z0, s(z1)) -> plus(x(z0, z1), z0)
   activate(z0) -> z0

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

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

(26)
Obligation:
Innermost TRS:
Rules:
AND(tt, z0) -> c(ACTIVATE(z0))
PLUS(z0, 0') -> c1
PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
X(z0, 0') -> c3
X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
ACTIVATE(z0) -> c5
and(tt, z0) -> activate(z0)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
x(z0, 0') -> 0'
x(z0, s(z1)) -> plus(x(z0, z1), z0)
activate(z0) -> z0

Types:
AND :: tt -> a -> c
tt :: tt
c :: c5 -> c
ACTIVATE :: a -> c5
PLUS :: 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c1:c2
s :: 0':s -> 0':s
c2 :: c1:c2 -> c1:c2
X :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c1:c2 -> c3:c4 -> c3:c4
x :: 0':s -> 0':s -> 0':s
c5 :: c5
and :: tt -> and:activate -> and:activate
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
hole_c1_6 :: c
hole_tt2_6 :: tt
hole_a3_6 :: a
hole_c54_6 :: c5
hole_c1:c25_6 :: c1:c2
hole_0':s6_6 :: 0':s
hole_c3:c47_6 :: c3:c4
hole_and:activate8_6 :: and:activate
gen_c1:c29_6 :: Nat -> c1:c2
gen_0':s10_6 :: Nat -> 0':s
gen_c3:c411_6 :: Nat -> c3:c4

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

(27) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
PLUS, X, x, plus

They will be analysed ascendingly in the following order:
PLUS < X
x < X
plus < x

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

(28)
Obligation:
Innermost TRS:
Rules:
AND(tt, z0) -> c(ACTIVATE(z0))
PLUS(z0, 0') -> c1
PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
X(z0, 0') -> c3
X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
ACTIVATE(z0) -> c5
and(tt, z0) -> activate(z0)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
x(z0, 0') -> 0'
x(z0, s(z1)) -> plus(x(z0, z1), z0)
activate(z0) -> z0

Types:
AND :: tt -> a -> c
tt :: tt
c :: c5 -> c
ACTIVATE :: a -> c5
PLUS :: 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c1:c2
s :: 0':s -> 0':s
c2 :: c1:c2 -> c1:c2
X :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c1:c2 -> c3:c4 -> c3:c4
x :: 0':s -> 0':s -> 0':s
c5 :: c5
and :: tt -> and:activate -> and:activate
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
hole_c1_6 :: c
hole_tt2_6 :: tt
hole_a3_6 :: a
hole_c54_6 :: c5
hole_c1:c25_6 :: c1:c2
hole_0':s6_6 :: 0':s
hole_c3:c47_6 :: c3:c4
hole_and:activate8_6 :: and:activate
gen_c1:c29_6 :: Nat -> c1:c2
gen_0':s10_6 :: Nat -> 0':s
gen_c3:c411_6 :: Nat -> c3:c4


Generator Equations:
gen_c1:c29_6(0) <=> c1
gen_c1:c29_6(+(x, 1)) <=> c2(gen_c1:c29_6(x))
gen_0':s10_6(0) <=> 0'
gen_0':s10_6(+(x, 1)) <=> s(gen_0':s10_6(x))
gen_c3:c411_6(0) <=> c3
gen_c3:c411_6(+(x, 1)) <=> c4(c1, gen_c3:c411_6(x))


The following defined symbols remain to be analysed:
PLUS, X, x, plus

They will be analysed ascendingly in the following order:
PLUS < X
x < X
plus < x

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
PLUS(gen_0':s10_6(a), gen_0':s10_6(n13_6)) -> gen_c1:c29_6(n13_6), rt in Omega(1 + n13_6)

Induction Base:
PLUS(gen_0':s10_6(a), gen_0':s10_6(0)) ->_R^Omega(1)
c1

Induction Step:
PLUS(gen_0':s10_6(a), gen_0':s10_6(+(n13_6, 1))) ->_R^Omega(1)
c2(PLUS(gen_0':s10_6(a), gen_0':s10_6(n13_6))) ->_IH
c2(gen_c1:c29_6(c14_6))

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:
AND(tt, z0) -> c(ACTIVATE(z0))
PLUS(z0, 0') -> c1
PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
X(z0, 0') -> c3
X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
ACTIVATE(z0) -> c5
and(tt, z0) -> activate(z0)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
x(z0, 0') -> 0'
x(z0, s(z1)) -> plus(x(z0, z1), z0)
activate(z0) -> z0

Types:
AND :: tt -> a -> c
tt :: tt
c :: c5 -> c
ACTIVATE :: a -> c5
PLUS :: 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c1:c2
s :: 0':s -> 0':s
c2 :: c1:c2 -> c1:c2
X :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c1:c2 -> c3:c4 -> c3:c4
x :: 0':s -> 0':s -> 0':s
c5 :: c5
and :: tt -> and:activate -> and:activate
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
hole_c1_6 :: c
hole_tt2_6 :: tt
hole_a3_6 :: a
hole_c54_6 :: c5
hole_c1:c25_6 :: c1:c2
hole_0':s6_6 :: 0':s
hole_c3:c47_6 :: c3:c4
hole_and:activate8_6 :: and:activate
gen_c1:c29_6 :: Nat -> c1:c2
gen_0':s10_6 :: Nat -> 0':s
gen_c3:c411_6 :: Nat -> c3:c4


Generator Equations:
gen_c1:c29_6(0) <=> c1
gen_c1:c29_6(+(x, 1)) <=> c2(gen_c1:c29_6(x))
gen_0':s10_6(0) <=> 0'
gen_0':s10_6(+(x, 1)) <=> s(gen_0':s10_6(x))
gen_c3:c411_6(0) <=> c3
gen_c3:c411_6(+(x, 1)) <=> c4(c1, gen_c3:c411_6(x))


The following defined symbols remain to be analysed:
PLUS, X, x, plus

They will be analysed ascendingly in the following order:
PLUS < X
x < X
plus < x

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

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

(33)
BOUNDS(n^1, INF)

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

(34)
Obligation:
Innermost TRS:
Rules:
AND(tt, z0) -> c(ACTIVATE(z0))
PLUS(z0, 0') -> c1
PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
X(z0, 0') -> c3
X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
ACTIVATE(z0) -> c5
and(tt, z0) -> activate(z0)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
x(z0, 0') -> 0'
x(z0, s(z1)) -> plus(x(z0, z1), z0)
activate(z0) -> z0

Types:
AND :: tt -> a -> c
tt :: tt
c :: c5 -> c
ACTIVATE :: a -> c5
PLUS :: 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c1:c2
s :: 0':s -> 0':s
c2 :: c1:c2 -> c1:c2
X :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c1:c2 -> c3:c4 -> c3:c4
x :: 0':s -> 0':s -> 0':s
c5 :: c5
and :: tt -> and:activate -> and:activate
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
hole_c1_6 :: c
hole_tt2_6 :: tt
hole_a3_6 :: a
hole_c54_6 :: c5
hole_c1:c25_6 :: c1:c2
hole_0':s6_6 :: 0':s
hole_c3:c47_6 :: c3:c4
hole_and:activate8_6 :: and:activate
gen_c1:c29_6 :: Nat -> c1:c2
gen_0':s10_6 :: Nat -> 0':s
gen_c3:c411_6 :: Nat -> c3:c4


Lemmas:
PLUS(gen_0':s10_6(a), gen_0':s10_6(n13_6)) -> gen_c1:c29_6(n13_6), rt in Omega(1 + n13_6)


Generator Equations:
gen_c1:c29_6(0) <=> c1
gen_c1:c29_6(+(x, 1)) <=> c2(gen_c1:c29_6(x))
gen_0':s10_6(0) <=> 0'
gen_0':s10_6(+(x, 1)) <=> s(gen_0':s10_6(x))
gen_c3:c411_6(0) <=> c3
gen_c3:c411_6(+(x, 1)) <=> c4(c1, gen_c3:c411_6(x))


The following defined symbols remain to be analysed:
plus, X, x

They will be analysed ascendingly in the following order:
x < X
plus < x

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

(35) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s10_6(a), gen_0':s10_6(n414_6)) -> gen_0':s10_6(+(n414_6, a)), rt in Omega(0)

Induction Base:
plus(gen_0':s10_6(a), gen_0':s10_6(0)) ->_R^Omega(0)
gen_0':s10_6(a)

Induction Step:
plus(gen_0':s10_6(a), gen_0':s10_6(+(n414_6, 1))) ->_R^Omega(0)
s(plus(gen_0':s10_6(a), gen_0':s10_6(n414_6))) ->_IH
s(gen_0':s10_6(+(a, c415_6)))

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:
AND(tt, z0) -> c(ACTIVATE(z0))
PLUS(z0, 0') -> c1
PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
X(z0, 0') -> c3
X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
ACTIVATE(z0) -> c5
and(tt, z0) -> activate(z0)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
x(z0, 0') -> 0'
x(z0, s(z1)) -> plus(x(z0, z1), z0)
activate(z0) -> z0

Types:
AND :: tt -> a -> c
tt :: tt
c :: c5 -> c
ACTIVATE :: a -> c5
PLUS :: 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c1:c2
s :: 0':s -> 0':s
c2 :: c1:c2 -> c1:c2
X :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c1:c2 -> c3:c4 -> c3:c4
x :: 0':s -> 0':s -> 0':s
c5 :: c5
and :: tt -> and:activate -> and:activate
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
hole_c1_6 :: c
hole_tt2_6 :: tt
hole_a3_6 :: a
hole_c54_6 :: c5
hole_c1:c25_6 :: c1:c2
hole_0':s6_6 :: 0':s
hole_c3:c47_6 :: c3:c4
hole_and:activate8_6 :: and:activate
gen_c1:c29_6 :: Nat -> c1:c2
gen_0':s10_6 :: Nat -> 0':s
gen_c3:c411_6 :: Nat -> c3:c4


Lemmas:
PLUS(gen_0':s10_6(a), gen_0':s10_6(n13_6)) -> gen_c1:c29_6(n13_6), rt in Omega(1 + n13_6)
plus(gen_0':s10_6(a), gen_0':s10_6(n414_6)) -> gen_0':s10_6(+(n414_6, a)), rt in Omega(0)


Generator Equations:
gen_c1:c29_6(0) <=> c1
gen_c1:c29_6(+(x, 1)) <=> c2(gen_c1:c29_6(x))
gen_0':s10_6(0) <=> 0'
gen_0':s10_6(+(x, 1)) <=> s(gen_0':s10_6(x))
gen_c3:c411_6(0) <=> c3
gen_c3:c411_6(+(x, 1)) <=> c4(c1, gen_c3:c411_6(x))


The following defined symbols remain to be analysed:
x, X

They will be analysed ascendingly in the following order:
x < X

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

(37) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
x(gen_0':s10_6(a), gen_0':s10_6(n1311_6)) -> gen_0':s10_6(*(n1311_6, a)), rt in Omega(0)

Induction Base:
x(gen_0':s10_6(a), gen_0':s10_6(0)) ->_R^Omega(0)
0'

Induction Step:
x(gen_0':s10_6(a), gen_0':s10_6(+(n1311_6, 1))) ->_R^Omega(0)
plus(x(gen_0':s10_6(a), gen_0':s10_6(n1311_6)), gen_0':s10_6(a)) ->_IH
plus(gen_0':s10_6(*(c1312_6, a)), gen_0':s10_6(a)) ->_L^Omega(0)
gen_0':s10_6(+(a, *(n1311_6, a)))

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

(38)
Obligation:
Innermost TRS:
Rules:
AND(tt, z0) -> c(ACTIVATE(z0))
PLUS(z0, 0') -> c1
PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
X(z0, 0') -> c3
X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
ACTIVATE(z0) -> c5
and(tt, z0) -> activate(z0)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
x(z0, 0') -> 0'
x(z0, s(z1)) -> plus(x(z0, z1), z0)
activate(z0) -> z0

Types:
AND :: tt -> a -> c
tt :: tt
c :: c5 -> c
ACTIVATE :: a -> c5
PLUS :: 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c1:c2
s :: 0':s -> 0':s
c2 :: c1:c2 -> c1:c2
X :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c1:c2 -> c3:c4 -> c3:c4
x :: 0':s -> 0':s -> 0':s
c5 :: c5
and :: tt -> and:activate -> and:activate
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
hole_c1_6 :: c
hole_tt2_6 :: tt
hole_a3_6 :: a
hole_c54_6 :: c5
hole_c1:c25_6 :: c1:c2
hole_0':s6_6 :: 0':s
hole_c3:c47_6 :: c3:c4
hole_and:activate8_6 :: and:activate
gen_c1:c29_6 :: Nat -> c1:c2
gen_0':s10_6 :: Nat -> 0':s
gen_c3:c411_6 :: Nat -> c3:c4


Lemmas:
PLUS(gen_0':s10_6(a), gen_0':s10_6(n13_6)) -> gen_c1:c29_6(n13_6), rt in Omega(1 + n13_6)
plus(gen_0':s10_6(a), gen_0':s10_6(n414_6)) -> gen_0':s10_6(+(n414_6, a)), rt in Omega(0)
x(gen_0':s10_6(a), gen_0':s10_6(n1311_6)) -> gen_0':s10_6(*(n1311_6, a)), rt in Omega(0)


Generator Equations:
gen_c1:c29_6(0) <=> c1
gen_c1:c29_6(+(x, 1)) <=> c2(gen_c1:c29_6(x))
gen_0':s10_6(0) <=> 0'
gen_0':s10_6(+(x, 1)) <=> s(gen_0':s10_6(x))
gen_c3:c411_6(0) <=> c3
gen_c3:c411_6(+(x, 1)) <=> c4(c1, gen_c3:c411_6(x))


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

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
X(gen_0':s10_6(a), gen_0':s10_6(n2333_6)) -> *12_6, rt in Omega(a*n2333_6 + n2333_6)

Induction Base:
X(gen_0':s10_6(a), gen_0':s10_6(0))

Induction Step:
X(gen_0':s10_6(a), gen_0':s10_6(+(n2333_6, 1))) ->_R^Omega(1)
c4(PLUS(x(gen_0':s10_6(a), gen_0':s10_6(n2333_6)), gen_0':s10_6(a)), X(gen_0':s10_6(a), gen_0':s10_6(n2333_6))) ->_L^Omega(0)
c4(PLUS(gen_0':s10_6(*(n2333_6, a)), gen_0':s10_6(a)), X(gen_0':s10_6(a), gen_0':s10_6(n2333_6))) ->_L^Omega(1 + a)
c4(gen_c1:c29_6(a), X(gen_0':s10_6(*(n2333_6, a)), gen_0':s10_6(n2333_6))) ->_IH
c4(gen_c1:c29_6(*(n2333_6, a)), *12_6)

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(40)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
AND(tt, z0) -> c(ACTIVATE(z0))
PLUS(z0, 0') -> c1
PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
X(z0, 0') -> c3
X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
ACTIVATE(z0) -> c5
and(tt, z0) -> activate(z0)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
x(z0, 0') -> 0'
x(z0, s(z1)) -> plus(x(z0, z1), z0)
activate(z0) -> z0

Types:
AND :: tt -> a -> c
tt :: tt
c :: c5 -> c
ACTIVATE :: a -> c5
PLUS :: 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c1:c2
s :: 0':s -> 0':s
c2 :: c1:c2 -> c1:c2
X :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c1:c2 -> c3:c4 -> c3:c4
x :: 0':s -> 0':s -> 0':s
c5 :: c5
and :: tt -> and:activate -> and:activate
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
hole_c1_6 :: c
hole_tt2_6 :: tt
hole_a3_6 :: a
hole_c54_6 :: c5
hole_c1:c25_6 :: c1:c2
hole_0':s6_6 :: 0':s
hole_c3:c47_6 :: c3:c4
hole_and:activate8_6 :: and:activate
gen_c1:c29_6 :: Nat -> c1:c2
gen_0':s10_6 :: Nat -> 0':s
gen_c3:c411_6 :: Nat -> c3:c4


Lemmas:
PLUS(gen_0':s10_6(a), gen_0':s10_6(n13_6)) -> gen_c1:c29_6(n13_6), rt in Omega(1 + n13_6)
plus(gen_0':s10_6(a), gen_0':s10_6(n414_6)) -> gen_0':s10_6(+(n414_6, a)), rt in Omega(0)
x(gen_0':s10_6(a), gen_0':s10_6(n1311_6)) -> gen_0':s10_6(*(n1311_6, a)), rt in Omega(0)


Generator Equations:
gen_c1:c29_6(0) <=> c1
gen_c1:c29_6(+(x, 1)) <=> c2(gen_c1:c29_6(x))
gen_0':s10_6(0) <=> 0'
gen_0':s10_6(+(x, 1)) <=> s(gen_0':s10_6(x))
gen_c3:c411_6(0) <=> c3
gen_c3:c411_6(+(x, 1)) <=> c4(c1, gen_c3:c411_6(x))


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

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

(42)
BOUNDS(n^2, INF)