/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 139 ms]
(4) CpxRelTRS
(5) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
    (7) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxWeightedTrs
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedTrs
    (11) CompletionProof [UPPER BOUND(ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (14) CpxRNTS
    (15) CompleteCoflocoProof [FINISHED, 614 ms]
    (16) BOUNDS(1, n^2)
(17) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) CpxRelTRS
    (19) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) typed CpxTrs
    (21) OrderProof [LOWER BOUND(ID), 0 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 307 ms]
    (24) BEST
        (25) proven lower bound
            (26) LowerBoundPropagationProof [FINISHED, 0 ms]
            (27) BOUNDS(n^1, INF)
        (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 22 ms]
            (30) BOUNDS(1, INF)


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

(0)
Obligation:
The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   rev(ls) -> r1(ls, empty)
   r1(empty, a) -> a
   r1(cons(x, k), a) -> r1(k, cons(x, a))

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(empty) -> empty
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_rev(x_1)) -> rev(encArg(x_1))
   encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2))
   encode_rev(x_1) -> rev(encArg(x_1))
   encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2))
   encode_empty -> empty
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))


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

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


The TRS R consists of the following rules:

   rev(ls) -> r1(ls, empty)
   r1(empty, a) -> a
   r1(cons(x, k), a) -> r1(k, cons(x, a))

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

   encArg(empty) -> empty
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_rev(x_1)) -> rev(encArg(x_1))
   encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2))
   encode_rev(x_1) -> rev(encArg(x_1))
   encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2))
   encode_empty -> empty
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))

Rewrite Strategy: FULL
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   rev(ls) -> r1(ls, empty)
   r1(empty, a) -> a
   r1(cons(x, k), a) -> r1(k, cons(x, a))

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

   encArg(empty) -> empty
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_rev(x_1)) -> rev(encArg(x_1))
   encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2))
   encode_rev(x_1) -> rev(encArg(x_1))
   encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2))
   encode_empty -> empty
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))

Rewrite Strategy: FULL
----------------------------------------

(5) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.
----------------------------------------

(6)
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:

   rev(ls) -> r1(ls, empty)
   r1(empty, a) -> a
   r1(cons(x, k), a) -> r1(k, cons(x, a))

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

   encArg(empty) -> empty
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_rev(x_1)) -> rev(encArg(x_1))
   encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2))
   encode_rev(x_1) -> rev(encArg(x_1))
   encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2))
   encode_empty -> empty
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))

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

(7) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(8)
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:

   rev(ls) -> r1(ls, empty) [1]
   r1(empty, a) -> a [1]
   r1(cons(x, k), a) -> r1(k, cons(x, a)) [1]
   encArg(empty) -> empty [0]
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_rev(x_1)) -> rev(encArg(x_1)) [0]
   encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2)) [0]
   encode_rev(x_1) -> rev(encArg(x_1)) [0]
   encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2)) [0]
   encode_empty -> empty [0]
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) [0]

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

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

(10)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   rev(ls) -> r1(ls, empty) [1]
   r1(empty, a) -> a [1]
   r1(cons(x, k), a) -> r1(k, cons(x, a)) [1]
   encArg(empty) -> empty [0]
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_rev(x_1)) -> rev(encArg(x_1)) [0]
   encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2)) [0]
   encode_rev(x_1) -> rev(encArg(x_1)) [0]
   encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2)) [0]
   encode_empty -> empty [0]
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) [0]

The TRS has the following type information:
rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
empty :: empty:cons:cons_rev:cons_r1
cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encArg :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_empty :: empty:cons:cons_rev:cons_r1
encode_cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1

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

(11) 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:

   encArg(v0) -> null_encArg [0]
   encode_rev(v0) -> null_encode_rev [0]
   encode_r1(v0, v1) -> null_encode_r1 [0]
   encode_empty -> null_encode_empty [0]
   encode_cons(v0, v1) -> null_encode_cons [0]
   r1(v0, v1) -> null_r1 [0]

And the following fresh constants:    null_encArg, null_encode_rev, null_encode_r1, null_encode_empty, null_encode_cons, null_r1

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

(12)
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:

   rev(ls) -> r1(ls, empty) [1]
   r1(empty, a) -> a [1]
   r1(cons(x, k), a) -> r1(k, cons(x, a)) [1]
   encArg(empty) -> empty [0]
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_rev(x_1)) -> rev(encArg(x_1)) [0]
   encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2)) [0]
   encode_rev(x_1) -> rev(encArg(x_1)) [0]
   encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2)) [0]
   encode_empty -> empty [0]
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) [0]
   encArg(v0) -> null_encArg [0]
   encode_rev(v0) -> null_encode_rev [0]
   encode_r1(v0, v1) -> null_encode_r1 [0]
   encode_empty -> null_encode_empty [0]
   encode_cons(v0, v1) -> null_encode_cons [0]
   r1(v0, v1) -> null_r1 [0]

The TRS has the following type information:
rev :: empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1 -> empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1
r1 :: empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1 -> empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1 -> empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1
empty :: empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1
cons :: empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1 -> empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1 -> empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1
encArg :: empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1 -> empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1
cons_rev :: empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1 -> empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1
cons_r1 :: empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1 -> empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1 -> empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1
encode_rev :: empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1 -> empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1
encode_r1 :: empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1 -> empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1 -> empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1
encode_empty :: empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1
encode_cons :: empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1 -> empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1 -> empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1
null_encArg :: empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1
null_encode_rev :: empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1
null_encode_r1 :: empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1
null_encode_empty :: empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1
null_encode_cons :: empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1
null_r1 :: empty:cons:cons_rev:cons_r1:null_encArg:null_encode_rev:null_encode_r1:null_encode_empty:null_encode_cons:null_r1

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

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

   empty => 0
   null_encArg => 0
   null_encode_rev => 0
   null_encode_r1 => 0
   null_encode_empty => 0
   null_encode_cons => 0
   null_r1 => 0

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

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> rev(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> r1(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_cons(z, z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 0 }-> r1(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_r1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_rev(z) -{ 0 }-> rev(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_rev(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   r1(z, z') -{ 1 }-> a :|: z' = a, a >= 0, z = 0
   r1(z, z') -{ 1 }-> r1(k, 1 + x + a) :|: z' = a, a >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   rev(z) -{ 1 }-> r1(ls, 0) :|: ls >= 0, z = ls

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V, V2),0,[rev(V, Out)],[V >= 0]).
eq(start(V, V2),0,[r1(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(start(V, V2),0,[encArg(V, Out)],[V >= 0]).
eq(start(V, V2),0,[fun(V, Out)],[V >= 0]).
eq(start(V, V2),0,[fun1(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(start(V, V2),0,[fun2(Out)],[]).
eq(start(V, V2),0,[fun3(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(rev(V, Out),1,[r1(V1, 0, Ret)],[Out = Ret,V1 >= 0,V = V1]).
eq(r1(V, V2, Out),1,[],[Out = V3,V2 = V3,V3 >= 0,V = 0]).
eq(r1(V, V2, Out),1,[r1(V6, 1 + V5 + V4, Ret1)],[Out = Ret1,V2 = V4,V4 >= 0,V5 >= 0,V6 >= 0,V = 1 + V5 + V6]).
eq(encArg(V, Out),0,[],[Out = 0,V = 0]).
eq(encArg(V, Out),0,[encArg(V8, Ret01),encArg(V7, Ret11)],[Out = 1 + Ret01 + Ret11,V8 >= 0,V = 1 + V7 + V8,V7 >= 0]).
eq(encArg(V, Out),0,[encArg(V9, Ret0),rev(Ret0, Ret2)],[Out = Ret2,V = 1 + V9,V9 >= 0]).
eq(encArg(V, Out),0,[encArg(V10, Ret02),encArg(V11, Ret12),r1(Ret02, Ret12, Ret3)],[Out = Ret3,V10 >= 0,V = 1 + V10 + V11,V11 >= 0]).
eq(fun(V, Out),0,[encArg(V12, Ret03),rev(Ret03, Ret4)],[Out = Ret4,V12 >= 0,V = V12]).
eq(fun1(V, V2, Out),0,[encArg(V14, Ret04),encArg(V13, Ret13),r1(Ret04, Ret13, Ret5)],[Out = Ret5,V14 >= 0,V13 >= 0,V = V14,V2 = V13]).
eq(fun2(Out),0,[],[Out = 0]).
eq(fun3(V, V2, Out),0,[encArg(V16, Ret011),encArg(V15, Ret14)],[Out = 1 + Ret011 + Ret14,V16 >= 0,V15 >= 0,V = V16,V2 = V15]).
eq(encArg(V, Out),0,[],[Out = 0,V17 >= 0,V = V17]).
eq(fun(V, Out),0,[],[Out = 0,V18 >= 0,V = V18]).
eq(fun1(V, V2, Out),0,[],[Out = 0,V20 >= 0,V19 >= 0,V = V20,V2 = V19]).
eq(fun3(V, V2, Out),0,[],[Out = 0,V21 >= 0,V22 >= 0,V = V21,V2 = V22]).
eq(r1(V, V2, Out),0,[],[Out = 0,V23 >= 0,V24 >= 0,V = V23,V2 = V24]).
input_output_vars(rev(V,Out),[V],[Out]).
input_output_vars(r1(V,V2,Out),[V,V2],[Out]).
input_output_vars(encArg(V,Out),[V],[Out]).
input_output_vars(fun(V,Out),[V],[Out]).
input_output_vars(fun1(V,V2,Out),[V,V2],[Out]).
input_output_vars(fun2(Out),[],[Out]).
input_output_vars(fun3(V,V2,Out),[V,V2],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [r1/3]
1. non_recursive  : [rev/2]
2. recursive [non_tail,multiple] : [encArg/2]
3. non_recursive  : [fun/2]
4. non_recursive  : [fun1/3]
5. non_recursive  : [fun2/1]
6. non_recursive  : [fun3/3]
7. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into r1/3
1. SCC is completely evaluated into other SCCs
2. SCC is partially evaluated into encArg/2
3. SCC is partially evaluated into fun/2
4. SCC is partially evaluated into fun1/3
5. SCC is completely evaluated into other SCCs
6. SCC is partially evaluated into fun3/3
7. SCC is partially evaluated into start/2

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

### Specialization of cost equations r1/3 
* CE 10 is refined into CE [21] 
* CE 8 is refined into CE [22] 
* CE 9 is refined into CE [23] 


### Cost equations --> "Loop" of r1/3 
* CEs [23] --> Loop 11 
* CEs [21] --> Loop 12 
* CEs [22] --> Loop 13 

### Ranking functions of CR r1(V,V2,Out) 
* RF of phase [11]: [V]

#### Partial ranking functions of CR r1(V,V2,Out) 
* Partial RF of phase [11]:
  - RF of loop [11:1]:
    V


### Specialization of cost equations encArg/2 
* CE 11 is refined into CE [24] 
* CE 12 is refined into CE [25] 
* CE 14 is refined into CE [26,27,28] 
* CE 13 is refined into CE [29,30,31] 


### Cost equations --> "Loop" of encArg/2 
* CEs [31] --> Loop 14 
* CEs [29,30] --> Loop 15 
* CEs [25] --> Loop 16 
* CEs [28] --> Loop 17 
* CEs [26] --> Loop 18 
* CEs [27] --> Loop 19 
* CEs [24] --> Loop 20 

### Ranking functions of CR encArg(V,Out) 
* RF of phase [14,15,16,17,18,19]: [V]

#### Partial ranking functions of CR encArg(V,Out) 
* Partial RF of phase [14,15,16,17,18,19]:
  - RF of loop [14:1,15:1,16:1,16:2,17:1,17:2,18:1,18:2,19:1,19:2]:
    V


### Specialization of cost equations fun/2 
* CE 15 is refined into CE [32,33,34,35,36] 
* CE 16 is refined into CE [37] 


### Cost equations --> "Loop" of fun/2 
* CEs [36] --> Loop 21 
* CEs [32,33,34,35,37] --> Loop 22 

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

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


### Specialization of cost equations fun1/3 
* CE 17 is refined into CE [38,39,40,41,42,43,44,45,46,47] 
* CE 18 is refined into CE [48] 


### Cost equations --> "Loop" of fun1/3 
* CEs [44] --> Loop 23 
* CEs [47] --> Loop 24 
* CEs [40,45] --> Loop 25 
* CEs [38,39,41,42,43,46,48] --> Loop 26 

### Ranking functions of CR fun1(V,V2,Out) 

#### Partial ranking functions of CR fun1(V,V2,Out) 


### Specialization of cost equations fun3/3 
* CE 19 is refined into CE [49,50,51,52] 
* CE 20 is refined into CE [53] 


### Cost equations --> "Loop" of fun3/3 
* CEs [52] --> Loop 27 
* CEs [51] --> Loop 28 
* CEs [50] --> Loop 29 
* CEs [49] --> Loop 30 
* CEs [53] --> Loop 31 

### Ranking functions of CR fun3(V,V2,Out) 

#### Partial ranking functions of CR fun3(V,V2,Out) 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [54,55,56] 
* CE 2 is refined into CE [57,58,59] 
* CE 3 is refined into CE [60,61] 
* CE 4 is refined into CE [62,63] 
* CE 5 is refined into CE [64,65,66,67] 
* CE 6 is refined into CE [68] 
* CE 7 is refined into CE [69,70,71,72,73] 


### Cost equations --> "Loop" of start/2 
* CEs [54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73] --> Loop 32 

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

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


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

#### Cost of chains of r1(V,V2,Out):
* Chain [[11],13]: 1*it(11)+1
  Such that:it(11) =< -V2+Out

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

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

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

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

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


#### Cost of chains of encArg(V,Out):
* Chain [20]: 0
  with precondition: [Out=0,V>=0] 

* Chain [multiple([14,15,16,17,18,19],[[20]])]: 5*it(14)+1*it(17)+1*s(10)+1*s(11)+1*s(12)+1*s(13)+0
  Such that:aux(9) =< V
aux(10) =< 2/3*V
it(14) =< aux(9)
it(16) =< aux(9)
it(17) =< aux(9)
it(17) =< aux(10)
aux(4) =< aux(9)
aux(2) =< aux(9)+1
s(10) =< it(14)*aux(9)
s(11) =< it(14)*aux(2)
s(13) =< it(16)*aux(4)
s(12) =< it(17)*aux(4)

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


#### Cost of chains of fun(V,Out):
* Chain [22]: 11*s(17)+2*s(19)+2*s(22)+2*s(23)+2*s(24)+2*s(25)+2
  Such that:aux(12) =< V
aux(13) =< 2/3*V
s(17) =< aux(12)
s(19) =< aux(12)
s(19) =< aux(13)
s(20) =< aux(12)
s(21) =< aux(12)+1
s(22) =< s(17)*aux(12)
s(23) =< s(17)*s(21)
s(24) =< aux(12)*s(20)
s(25) =< s(19)*s(20)

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

* Chain [21]: 6*s(40)+1*s(42)+1*s(45)+1*s(46)+1*s(47)+1*s(48)+2
  Such that:s(39) =< 2/3*V
aux(14) =< V
s(40) =< aux(14)
s(42) =< aux(14)
s(42) =< s(39)
s(43) =< aux(14)
s(44) =< aux(14)+1
s(45) =< s(40)*aux(14)
s(46) =< s(40)*s(44)
s(47) =< aux(14)*s(43)
s(48) =< s(42)*s(43)

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


#### Cost of chains of fun1(V,V2,Out):
* Chain [26]: 10*s(53)+2*s(55)+2*s(58)+2*s(59)+2*s(60)+2*s(61)+17*s(65)+3*s(67)+3*s(70)+3*s(71)+3*s(72)+3*s(73)+1
  Such that:aux(17) =< V
aux(18) =< 2/3*V
aux(19) =< V2
aux(20) =< 2/3*V2
s(65) =< aux(17)
s(67) =< aux(17)
s(67) =< aux(18)
s(68) =< aux(17)
s(69) =< aux(17)+1
s(70) =< s(65)*aux(17)
s(71) =< s(65)*s(69)
s(72) =< aux(17)*s(68)
s(73) =< s(67)*s(68)
s(53) =< aux(19)
s(55) =< aux(19)
s(55) =< aux(20)
s(56) =< aux(19)
s(57) =< aux(19)+1
s(58) =< s(53)*aux(19)
s(59) =< s(53)*s(57)
s(60) =< aux(19)*s(56)
s(61) =< s(55)*s(56)

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

* Chain [25]: 10*s(111)+2*s(113)+2*s(116)+2*s(117)+2*s(118)+2*s(119)+5*s(122)+1*s(124)+1*s(127)+1*s(128)+1*s(129)+1*s(130)+1
  Such that:s(120) =< V
s(121) =< 2/3*V
aux(21) =< V2
aux(22) =< 2/3*V2
s(111) =< aux(21)
s(113) =< aux(21)
s(113) =< aux(22)
s(114) =< aux(21)
s(115) =< aux(21)+1
s(116) =< s(111)*aux(21)
s(117) =< s(111)*s(115)
s(118) =< aux(21)*s(114)
s(119) =< s(113)*s(114)
s(122) =< s(120)
s(124) =< s(120)
s(124) =< s(121)
s(125) =< s(120)
s(126) =< s(120)+1
s(127) =< s(122)*s(120)
s(128) =< s(122)*s(126)
s(129) =< s(120)*s(125)
s(130) =< s(124)*s(125)

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

* Chain [24]: 6*s(144)+1*s(146)+1*s(149)+1*s(150)+1*s(151)+1*s(152)+5*s(155)+1*s(157)+1*s(160)+1*s(161)+1*s(162)+1*s(163)+1
  Such that:s(143) =< 2/3*V
s(153) =< V2
s(154) =< 2/3*V2
aux(23) =< V
s(144) =< aux(23)
s(155) =< s(153)
s(157) =< s(153)
s(157) =< s(154)
s(158) =< s(153)
s(159) =< s(153)+1
s(160) =< s(155)*s(153)
s(161) =< s(155)*s(159)
s(162) =< s(153)*s(158)
s(163) =< s(157)*s(158)
s(146) =< aux(23)
s(146) =< s(143)
s(147) =< aux(23)
s(148) =< aux(23)+1
s(149) =< s(144)*aux(23)
s(150) =< s(144)*s(148)
s(151) =< aux(23)*s(147)
s(152) =< s(146)*s(147)

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

* Chain [23]: 6*s(167)+1*s(169)+1*s(172)+1*s(173)+1*s(174)+1*s(175)+1
  Such that:s(166) =< 2/3*V
aux(24) =< V
s(167) =< aux(24)
s(169) =< aux(24)
s(169) =< s(166)
s(170) =< aux(24)
s(171) =< aux(24)+1
s(172) =< s(167)*aux(24)
s(173) =< s(167)*s(171)
s(174) =< aux(24)*s(170)
s(175) =< s(169)*s(170)

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


#### Cost of chains of fun3(V,V2,Out):
* Chain [31]: 0
  with precondition: [Out=0,V>=0,V2>=0] 

* Chain [30]: 0
  with precondition: [Out=1,V>=0,V2>=0] 

* Chain [29]: 5*s(179)+1*s(181)+1*s(184)+1*s(185)+1*s(186)+1*s(187)+0
  Such that:s(177) =< V2
s(178) =< 2/3*V2
s(179) =< s(177)
s(181) =< s(177)
s(181) =< s(178)
s(182) =< s(177)
s(183) =< s(177)+1
s(184) =< s(179)*s(177)
s(185) =< s(179)*s(183)
s(186) =< s(177)*s(182)
s(187) =< s(181)*s(182)

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

* Chain [28]: 5*s(190)+1*s(192)+1*s(195)+1*s(196)+1*s(197)+1*s(198)+0
  Such that:s(188) =< V
s(189) =< 2/3*V
s(190) =< s(188)
s(192) =< s(188)
s(192) =< s(189)
s(193) =< s(188)
s(194) =< s(188)+1
s(195) =< s(190)*s(188)
s(196) =< s(190)*s(194)
s(197) =< s(188)*s(193)
s(198) =< s(192)*s(193)

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

* Chain [27]: 5*s(201)+1*s(203)+1*s(206)+1*s(207)+1*s(208)+1*s(209)+5*s(212)+1*s(214)+1*s(217)+1*s(218)+1*s(219)+1*s(220)+0
  Such that:s(199) =< V
s(200) =< 2/3*V
s(210) =< V2
s(211) =< 2/3*V2
s(212) =< s(210)
s(214) =< s(210)
s(214) =< s(211)
s(215) =< s(210)
s(216) =< s(210)+1
s(217) =< s(212)*s(210)
s(218) =< s(212)*s(216)
s(219) =< s(210)*s(215)
s(220) =< s(214)*s(215)
s(201) =< s(199)
s(203) =< s(199)
s(203) =< s(200)
s(204) =< s(199)
s(205) =< s(199)+1
s(206) =< s(201)*s(199)
s(207) =< s(201)*s(205)
s(208) =< s(199)*s(204)
s(209) =< s(203)*s(204)

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


#### Cost of chains of start(V,V2):
* Chain [32]: 70*s(221)+12*s(229)+12*s(232)+12*s(233)+12*s(234)+12*s(235)+35*s(268)+7*s(269)+7*s(272)+7*s(273)+7*s(274)+7*s(275)+2
  Such that:aux(25) =< V
aux(26) =< 2/3*V
aux(27) =< V2
aux(28) =< 2/3*V2
s(221) =< aux(25)
s(229) =< aux(25)
s(229) =< aux(26)
s(230) =< aux(25)
s(231) =< aux(25)+1
s(232) =< s(221)*aux(25)
s(233) =< s(221)*s(231)
s(234) =< aux(25)*s(230)
s(235) =< s(229)*s(230)
s(268) =< aux(27)
s(269) =< aux(27)
s(269) =< aux(28)
s(270) =< aux(27)
s(271) =< aux(27)+1
s(272) =< s(268)*aux(27)
s(273) =< s(268)*s(271)
s(274) =< aux(27)*s(270)
s(275) =< s(269)*s(270)

  with precondition: [] 


Closed-form bounds of start(V,V2): 
-------------------------------------
* Chain [32] with precondition: [] 
    - Upper bound: nat(V)*94+2+nat(V)*48*nat(V)+nat(V2)*49+nat(V2)*28*nat(V2) 
    - Complexity: n^2 

### Maximum cost of start(V,V2): nat(V)*94+2+nat(V)*48*nat(V)+nat(V2)*49+nat(V2)*28*nat(V2) 
Asymptotic class: n^2 
* Total analysis performed in 506 ms.


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

(16)
BOUNDS(1, n^2)

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

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

(18)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   rev(ls) -> r1(ls, empty)
   r1(empty, a) -> a
   r1(cons(x, k), a) -> r1(k, cons(x, a))

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

   encArg(empty) -> empty
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_rev(x_1)) -> rev(encArg(x_1))
   encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2))
   encode_rev(x_1) -> rev(encArg(x_1))
   encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2))
   encode_empty -> empty
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))

Rewrite Strategy: FULL
----------------------------------------

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

(20)
Obligation:
TRS:
Rules:
rev(ls) -> r1(ls, empty)
r1(empty, a) -> a
r1(cons(x, k), a) -> r1(k, cons(x, a))
encArg(empty) -> empty
encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
encArg(cons_rev(x_1)) -> rev(encArg(x_1))
encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2))
encode_rev(x_1) -> rev(encArg(x_1))
encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2))
encode_empty -> empty
encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))

Types:
rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
empty :: empty:cons:cons_rev:cons_r1
cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encArg :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_empty :: empty:cons:cons_rev:cons_r1
encode_cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
hole_empty:cons:cons_rev:cons_r11_0 :: empty:cons:cons_rev:cons_r1
gen_empty:cons:cons_rev:cons_r12_0 :: Nat -> empty:cons:cons_rev:cons_r1

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

(21) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
r1, encArg

They will be analysed ascendingly in the following order:
r1 < encArg

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

(22)
Obligation:
TRS:
Rules:
rev(ls) -> r1(ls, empty)
r1(empty, a) -> a
r1(cons(x, k), a) -> r1(k, cons(x, a))
encArg(empty) -> empty
encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
encArg(cons_rev(x_1)) -> rev(encArg(x_1))
encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2))
encode_rev(x_1) -> rev(encArg(x_1))
encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2))
encode_empty -> empty
encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))

Types:
rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
empty :: empty:cons:cons_rev:cons_r1
cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encArg :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_empty :: empty:cons:cons_rev:cons_r1
encode_cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
hole_empty:cons:cons_rev:cons_r11_0 :: empty:cons:cons_rev:cons_r1
gen_empty:cons:cons_rev:cons_r12_0 :: Nat -> empty:cons:cons_rev:cons_r1


Generator Equations:
gen_empty:cons:cons_rev:cons_r12_0(0) <=> empty
gen_empty:cons:cons_rev:cons_r12_0(+(x, 1)) <=> cons(empty, gen_empty:cons:cons_rev:cons_r12_0(x))


The following defined symbols remain to be analysed:
r1, encArg

They will be analysed ascendingly in the following order:
r1 < encArg

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
r1(gen_empty:cons:cons_rev:cons_r12_0(n4_0), gen_empty:cons:cons_rev:cons_r12_0(b)) -> gen_empty:cons:cons_rev:cons_r12_0(+(n4_0, b)), rt in Omega(1 + n4_0)

Induction Base:
r1(gen_empty:cons:cons_rev:cons_r12_0(0), gen_empty:cons:cons_rev:cons_r12_0(b)) ->_R^Omega(1)
gen_empty:cons:cons_rev:cons_r12_0(b)

Induction Step:
r1(gen_empty:cons:cons_rev:cons_r12_0(+(n4_0, 1)), gen_empty:cons:cons_rev:cons_r12_0(b)) ->_R^Omega(1)
r1(gen_empty:cons:cons_rev:cons_r12_0(n4_0), cons(empty, gen_empty:cons:cons_rev:cons_r12_0(b))) ->_IH
gen_empty:cons:cons_rev:cons_r12_0(+(+(b, 1), c5_0))

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

(24)
Complex Obligation (BEST)

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

(25)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
rev(ls) -> r1(ls, empty)
r1(empty, a) -> a
r1(cons(x, k), a) -> r1(k, cons(x, a))
encArg(empty) -> empty
encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
encArg(cons_rev(x_1)) -> rev(encArg(x_1))
encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2))
encode_rev(x_1) -> rev(encArg(x_1))
encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2))
encode_empty -> empty
encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))

Types:
rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
empty :: empty:cons:cons_rev:cons_r1
cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encArg :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_empty :: empty:cons:cons_rev:cons_r1
encode_cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
hole_empty:cons:cons_rev:cons_r11_0 :: empty:cons:cons_rev:cons_r1
gen_empty:cons:cons_rev:cons_r12_0 :: Nat -> empty:cons:cons_rev:cons_r1


Generator Equations:
gen_empty:cons:cons_rev:cons_r12_0(0) <=> empty
gen_empty:cons:cons_rev:cons_r12_0(+(x, 1)) <=> cons(empty, gen_empty:cons:cons_rev:cons_r12_0(x))


The following defined symbols remain to be analysed:
r1, encArg

They will be analysed ascendingly in the following order:
r1 < encArg

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

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

(27)
BOUNDS(n^1, INF)

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

(28)
Obligation:
TRS:
Rules:
rev(ls) -> r1(ls, empty)
r1(empty, a) -> a
r1(cons(x, k), a) -> r1(k, cons(x, a))
encArg(empty) -> empty
encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
encArg(cons_rev(x_1)) -> rev(encArg(x_1))
encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2))
encode_rev(x_1) -> rev(encArg(x_1))
encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2))
encode_empty -> empty
encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))

Types:
rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
empty :: empty:cons:cons_rev:cons_r1
cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encArg :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_empty :: empty:cons:cons_rev:cons_r1
encode_cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
hole_empty:cons:cons_rev:cons_r11_0 :: empty:cons:cons_rev:cons_r1
gen_empty:cons:cons_rev:cons_r12_0 :: Nat -> empty:cons:cons_rev:cons_r1


Lemmas:
r1(gen_empty:cons:cons_rev:cons_r12_0(n4_0), gen_empty:cons:cons_rev:cons_r12_0(b)) -> gen_empty:cons:cons_rev:cons_r12_0(+(n4_0, b)), rt in Omega(1 + n4_0)


Generator Equations:
gen_empty:cons:cons_rev:cons_r12_0(0) <=> empty
gen_empty:cons:cons_rev:cons_r12_0(+(x, 1)) <=> cons(empty, gen_empty:cons:cons_rev:cons_r12_0(x))


The following defined symbols remain to be analysed:
encArg
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(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_empty:cons:cons_rev:cons_r12_0(n558_0)) -> gen_empty:cons:cons_rev:cons_r12_0(n558_0), rt in Omega(0)

Induction Base:
encArg(gen_empty:cons:cons_rev:cons_r12_0(0)) ->_R^Omega(0)
empty

Induction Step:
encArg(gen_empty:cons:cons_rev:cons_r12_0(+(n558_0, 1))) ->_R^Omega(0)
cons(encArg(empty), encArg(gen_empty:cons:cons_rev:cons_r12_0(n558_0))) ->_R^Omega(0)
cons(empty, encArg(gen_empty:cons:cons_rev:cons_r12_0(n558_0))) ->_IH
cons(empty, gen_empty:cons:cons_rev:cons_r12_0(c559_0))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(30)
BOUNDS(1, INF)