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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 169 ms]
(4) CpxRelTRS
(5) NonCtorToCtorProof [UPPER BOUND(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, 579 ms]
    (16) BOUNDS(1, n^1)
(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), 246 ms]
    (24) proven lower bound
    (25) LowerBoundPropagationProof [FINISHED, 0 ms]
    (26) BOUNDS(n^1, INF)


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

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


The TRS R consists of the following rules:

   .(1, x) -> x
   .(x, 1) -> x
   .(i(x), x) -> 1
   .(x, i(x)) -> 1
   i(1) -> 1
   i(i(x)) -> x
   .(i(y), .(y, z)) -> z
   .(y, .(i(y), z)) -> z

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

(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(1) -> 1
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_i(x_1) -> i(encArg(x_1))


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

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


The TRS R consists of the following rules:

   .(1, x) -> x
   .(x, 1) -> x
   .(i(x), x) -> 1
   .(x, i(x)) -> 1
   i(1) -> 1
   i(i(x)) -> x
   .(i(y), .(y, z)) -> z
   .(y, .(i(y), z)) -> z

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

   encArg(1) -> 1
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_i(x_1) -> i(encArg(x_1))

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

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

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


The TRS R consists of the following rules:

   .(1, x) -> x
   .(x, 1) -> x
   .(i(x), x) -> 1
   .(x, i(x)) -> 1
   i(1) -> 1
   i(i(x)) -> x
   .(i(y), .(y, z)) -> z
   .(y, .(i(y), z)) -> z

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

   encArg(1) -> 1
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_i(x_1) -> i(encArg(x_1))

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

(5) NonCtorToCtorProof (UPPER BOUND(ID))
transformed non-ctor to ctor-system
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   .(1, x) -> x
   .(x, 1) -> x
   i(1) -> 1
   .(c_i(x), x) -> 1
   .(x, c_i(x)) -> 1
   i(c_i(x)) -> x
   .(y, c_.(c_i(y), z)) -> z
   .(c_i(y), c_.(y, z)) -> z

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

   encArg(1) -> 1
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_i(x_1) -> i(encArg(x_1))
   i(x0) -> c_i(x0)
   .(x0, x1) -> c_.(x0, x1)

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^1).


The TRS R consists of the following rules:

   .(1, x) -> x [1]
   .(x, 1) -> x [1]
   i(1) -> 1 [1]
   .(c_i(x), x) -> 1 [1]
   .(x, c_i(x)) -> 1 [1]
   i(c_i(x)) -> x [1]
   .(y, c_.(c_i(y), z)) -> z [1]
   .(c_i(y), c_.(y, z)) -> z [1]
   encArg(1) -> 1 [0]
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_i(x_1)) -> i(encArg(x_1)) [0]
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2)) [0]
   encode_1 -> 1 [0]
   encode_i(x_1) -> i(encArg(x_1)) [0]
   i(x0) -> c_i(x0) [0]
   .(x0, x1) -> c_.(x0, x1) [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:

   .(1, x) -> x [1]
   .(x, 1) -> x [1]
   i(1) -> 1 [1]
   .(c_i(x), x) -> 1 [1]
   .(x, c_i(x)) -> 1 [1]
   i(c_i(x)) -> x [1]
   .(y, c_.(c_i(y), z)) -> z [1]
   .(c_i(y), c_.(y, z)) -> z [1]
   encArg(1) -> 1 [0]
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_i(x_1)) -> i(encArg(x_1)) [0]
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2)) [0]
   encode_1 -> 1 [0]
   encode_i(x_1) -> i(encArg(x_1)) [0]
   i(x0) -> c_i(x0) [0]
   .(x0, x1) -> c_.(x0, x1) [0]

The TRS has the following type information:
. :: 1:c_i:c_.:cons_.:cons_i -> 1:c_i:c_.:cons_.:cons_i -> 1:c_i:c_.:cons_.:cons_i
1 :: 1:c_i:c_.:cons_.:cons_i
i :: 1:c_i:c_.:cons_.:cons_i -> 1:c_i:c_.:cons_.:cons_i
c_i :: 1:c_i:c_.:cons_.:cons_i -> 1:c_i:c_.:cons_.:cons_i
c_. :: 1:c_i:c_.:cons_.:cons_i -> 1:c_i:c_.:cons_.:cons_i -> 1:c_i:c_.:cons_.:cons_i
encArg :: 1:c_i:c_.:cons_.:cons_i -> 1:c_i:c_.:cons_.:cons_i
cons_. :: 1:c_i:c_.:cons_.:cons_i -> 1:c_i:c_.:cons_.:cons_i -> 1:c_i:c_.:cons_.:cons_i
cons_i :: 1:c_i:c_.:cons_.:cons_i -> 1:c_i:c_.:cons_.:cons_i
encode_. :: 1:c_i:c_.:cons_.:cons_i -> 1:c_i:c_.:cons_.:cons_i -> 1:c_i:c_.:cons_.:cons_i
encode_1 :: 1:c_i:c_.:cons_.:cons_i
encode_i :: 1:c_i:c_.:cons_.:cons_i -> 1:c_i:c_.:cons_.:cons_i

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_.(v0, v1) -> null_encode_. [0]
   encode_1 -> null_encode_1 [0]
   encode_i(v0) -> null_encode_i [0]
   i(v0) -> null_i [0]
   .(v0, v1) -> null_. [0]

And the following fresh constants:    null_encArg, null_encode_., null_encode_1, null_encode_i, null_i, null_.

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

(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:

   .(1, x) -> x [1]
   .(x, 1) -> x [1]
   i(1) -> 1 [1]
   .(c_i(x), x) -> 1 [1]
   .(x, c_i(x)) -> 1 [1]
   i(c_i(x)) -> x [1]
   .(y, c_.(c_i(y), z)) -> z [1]
   .(c_i(y), c_.(y, z)) -> z [1]
   encArg(1) -> 1 [0]
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_i(x_1)) -> i(encArg(x_1)) [0]
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2)) [0]
   encode_1 -> 1 [0]
   encode_i(x_1) -> i(encArg(x_1)) [0]
   i(x0) -> c_i(x0) [0]
   .(x0, x1) -> c_.(x0, x1) [0]
   encArg(v0) -> null_encArg [0]
   encode_.(v0, v1) -> null_encode_. [0]
   encode_1 -> null_encode_1 [0]
   encode_i(v0) -> null_encode_i [0]
   i(v0) -> null_i [0]
   .(v0, v1) -> null_. [0]

The TRS has the following type information:
. :: 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_. -> 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_. -> 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_.
1 :: 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_.
i :: 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_. -> 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_.
c_i :: 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_. -> 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_.
c_. :: 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_. -> 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_. -> 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_.
encArg :: 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_. -> 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_.
cons_. :: 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_. -> 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_. -> 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_.
cons_i :: 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_. -> 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_.
encode_. :: 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_. -> 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_. -> 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_.
encode_1 :: 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_.
encode_i :: 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_. -> 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_.
null_encArg :: 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_.
null_encode_. :: 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_.
null_encode_1 :: 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_.
null_encode_i :: 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_.
null_i :: 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_.
null_. :: 1:c_i:c_.:cons_.:cons_i:null_encArg:null_encode_.:null_encode_1:null_encode_i:null_i:null_.

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:

   1 => 0
   null_encArg => 0
   null_encode_. => 0
   null_encode_1 => 0
   null_encode_i => 0
   null_i => 0
   null_. => 0

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

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

   .(z', z'') -{ 1 }-> x :|: x >= 0, z'' = x, z' = 0
   .(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0
   .(z', z'') -{ 1 }-> z :|: z >= 0, z'' = 1 + (1 + y) + z, y >= 0, z' = y
   .(z', z'') -{ 1 }-> z :|: z' = 1 + y, z >= 0, y >= 0, z'' = 1 + y + z
   .(z', z'') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z'' = x
   .(z', z'') -{ 1 }-> 0 :|: z' = x, x >= 0, z'' = 1 + x
   .(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   .(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z'' = x1, x0 >= 0, x1 >= 0, z' = x0
   encArg(z') -{ 0 }-> i(encArg(x_1)) :|: x_1 >= 0, z' = 1 + x_1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   encArg(z') -{ 0 }-> .(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_.(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_.(z', z'') -{ 0 }-> .(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_1 -{ 0 }-> 0 :|: 
   encode_i(z') -{ 0 }-> i(encArg(x_1)) :|: x_1 >= 0, z' = x_1
   encode_i(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   i(z') -{ 1 }-> x :|: z' = 1 + x, x >= 0
   i(z') -{ 1 }-> 0 :|: z' = 0
   i(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   i(z') -{ 0 }-> 1 + x0 :|: x0 >= 0, z' = x0

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,[fun(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(start(V, V2),0,[i(V, Out)],[V >= 0]).
eq(start(V, V2),0,[encArg(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, Out)],[V >= 0]).
eq(fun(V, V2, Out),1,[],[Out = V1,V1 >= 0,V2 = V1,V = 0]).
eq(fun(V, V2, Out),1,[],[Out = V3,V2 = 0,V = V3,V3 >= 0]).
eq(i(V, Out),1,[],[Out = 0,V = 0]).
eq(fun(V, V2, Out),1,[],[Out = 0,V = 1 + V4,V4 >= 0,V2 = V4]).
eq(fun(V, V2, Out),1,[],[Out = 0,V = V5,V5 >= 0,V2 = 1 + V5]).
eq(i(V, Out),1,[],[Out = V6,V = 1 + V6,V6 >= 0]).
eq(fun(V, V2, Out),1,[],[Out = V8,V8 >= 0,V2 = 2 + V7 + V8,V7 >= 0,V = V7]).
eq(fun(V, V2, Out),1,[],[Out = V9,V = 1 + V10,V9 >= 0,V10 >= 0,V2 = 1 + V10 + V9]).
eq(encArg(V, Out),0,[],[Out = 0,V = 0]).
eq(encArg(V, Out),0,[encArg(V12, Ret0),encArg(V11, Ret1),fun(Ret0, Ret1, Ret)],[Out = Ret,V12 >= 0,V11 >= 0,V = 1 + V11 + V12]).
eq(encArg(V, Out),0,[encArg(V13, Ret01),i(Ret01, Ret2)],[Out = Ret2,V13 >= 0,V = 1 + V13]).
eq(fun1(V, V2, Out),0,[encArg(V14, Ret02),encArg(V15, Ret11),fun(Ret02, Ret11, Ret3)],[Out = Ret3,V14 >= 0,V = V14,V15 >= 0,V2 = V15]).
eq(fun2(Out),0,[],[Out = 0]).
eq(fun3(V, Out),0,[encArg(V16, Ret03),i(Ret03, Ret4)],[Out = Ret4,V16 >= 0,V = V16]).
eq(i(V, Out),0,[],[Out = 1 + V17,V17 >= 0,V = V17]).
eq(fun(V, V2, Out),0,[],[Out = 1 + V18 + V19,V2 = V19,V18 >= 0,V19 >= 0,V = V18]).
eq(encArg(V, Out),0,[],[Out = 0,V20 >= 0,V = V20]).
eq(fun1(V, V2, Out),0,[],[Out = 0,V22 >= 0,V21 >= 0,V2 = V21,V = V22]).
eq(fun3(V, Out),0,[],[Out = 0,V23 >= 0,V = V23]).
eq(i(V, Out),0,[],[Out = 0,V24 >= 0,V = V24]).
eq(fun(V, V2, Out),0,[],[Out = 0,V25 >= 0,V26 >= 0,V2 = V26,V = V25]).
input_output_vars(fun(V,V2,Out),[V,V2],[Out]).
input_output_vars(i(V,Out),[V],[Out]).
input_output_vars(encArg(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,Out),[V],[Out]).


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

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

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

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

### Specialization of cost equations fun/3 
* CE 13 is refined into CE [26] 
* CE 12 is refined into CE [27] 
* CE 11 is refined into CE [28] 
* CE 10 is refined into CE [29] 
* CE 9 is refined into CE [30] 
* CE 14 is refined into CE [31] 
* CE 8 is refined into CE [32] 
* CE 7 is refined into CE [33] 


### Cost equations --> "Loop" of fun/3 
* CEs [26] --> Loop 17 
* CEs [27] --> Loop 18 
* CEs [28] --> Loop 19 
* CEs [29] --> Loop 20 
* CEs [30,31] --> Loop 21 
* CEs [32] --> Loop 22 
* CEs [33] --> Loop 23 

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

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


### Specialization of cost equations i/2 
* CE 17 is refined into CE [34] 
* CE 16 is refined into CE [35] 
* CE 15 is refined into CE [36] 
* CE 18 is refined into CE [37] 


### Cost equations --> "Loop" of i/2 
* CEs [34] --> Loop 24 
* CEs [35] --> Loop 25 
* CEs [36,37] --> Loop 26 

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

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


### Specialization of cost equations encArg/2 
* CE 19 is refined into CE [38] 
* CE 21 is refined into CE [39,40,41] 
* CE 20 is refined into CE [42,43,44,45,46,47] 


### Cost equations --> "Loop" of encArg/2 
* CEs [45] --> Loop 27 
* CEs [46] --> Loop 28 
* CEs [47] --> Loop 29 
* CEs [43] --> Loop 30 
* CEs [42] --> Loop 31 
* CEs [44] --> Loop 32 
* CEs [41] --> Loop 33 
* CEs [40] --> Loop 34 
* CEs [39] --> Loop 35 
* CEs [38] --> Loop 36 

### Ranking functions of CR encArg(V,Out) 
* RF of phase [27,28,29,30,31,32,33,34,35]: [V]

#### Partial ranking functions of CR encArg(V,Out) 
* Partial RF of phase [27,28,29,30,31,32,33,34,35]:
  - RF of loop [27:1,27:2,28:1,28:2,29:1,29:2,30:1,30:2,31:1,31:2,32:1,32:2,33:1,34:1,35:1]:
    V


### Specialization of cost equations fun1/3 
* CE 22 is refined into CE [48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66] 
* CE 23 is refined into CE [67] 


### Cost equations --> "Loop" of fun1/3 
* CEs [66] --> Loop 37 
* CEs [60] --> Loop 38 
* CEs [58,62] --> Loop 39 
* CEs [56] --> Loop 40 
* CEs [52,55,61,64,65] --> Loop 41 
* CEs [51] --> Loop 42 
* CEs [48,49,50,53,54,57,59,63,67] --> Loop 43 

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

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


### Specialization of cost equations fun3/2 
* CE 24 is refined into CE [68,69,70,71,72] 
* CE 25 is refined into CE [73] 


### Cost equations --> "Loop" of fun3/2 
* CEs [72] --> Loop 44 
* CEs [71] --> Loop 45 
* CEs [69] --> Loop 46 
* CEs [68,70,73] --> Loop 47 

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

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


### Specialization of cost equations start/2 
* CE 1 is refined into CE [74,75,76,77,78,79] 
* CE 2 is refined into CE [80,81,82] 
* CE 3 is refined into CE [83,84] 
* CE 4 is refined into CE [85,86,87,88,89,90,91] 
* CE 5 is refined into CE [92] 
* CE 6 is refined into CE [93,94,95,96] 


### Cost equations --> "Loop" of start/2 
* CEs [74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96] --> Loop 48 

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

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


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

#### Cost of chains of fun(V,V2,Out):
* Chain [23]: 1
  with precondition: [V=0,V2=Out,V2>=0] 

* Chain [22]: 1
  with precondition: [V2=0,V=Out,V>=0] 

* Chain [21]: 1
  with precondition: [Out=0,V>=0,V2>=0] 

* Chain [20]: 1
  with precondition: [Out=0,V+1=V2,V>=0] 

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

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

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


#### Cost of chains of i(V,Out):
* Chain [26]: 1
  with precondition: [Out=0,V>=0] 

* Chain [25]: 1
  with precondition: [V=Out+1,V>=1] 

* Chain [24]: 0
  with precondition: [V+1=Out,V>=0] 


#### Cost of chains of encArg(V,Out):
* Chain [36]: 0
  with precondition: [Out=0,V>=0] 

* Chain [multiple([27,28,29,30,31,32,33,34,35],[[36]])]: 2*it(27)+3*it(30)+2*it(33)+0
  Such that:it([36]) =< V+1
aux(3) =< V
aux(4) =< V/2
it(30) =< aux(3)
it(33) =< aux(3)
it(27) =< aux(4)
it(27) =< it([36])*(1/2)+aux(4)
it(30) =< it([36])*(1/2)+aux(4)

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


#### Cost of chains of fun1(V,V2,Out):
* Chain [43]: 9*s(4)+6*s(5)+6*s(6)+9*s(16)+6*s(17)+6*s(18)+1
  Such that:aux(5) =< V
aux(6) =< V+1
aux(7) =< V/2
aux(8) =< V2
aux(9) =< V2+1
aux(10) =< V2/2
s(16) =< aux(5)
s(17) =< aux(5)
s(18) =< aux(7)
s(18) =< aux(6)*(1/2)+aux(7)
s(16) =< aux(6)*(1/2)+aux(7)
s(4) =< aux(8)
s(5) =< aux(8)
s(6) =< aux(10)
s(6) =< aux(9)*(1/2)+aux(10)
s(4) =< aux(9)*(1/2)+aux(10)

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

* Chain [42]: 0
  with precondition: [Out=1,V>=0,V2>=0] 

* Chain [41]: 15*s(40)+10*s(41)+10*s(42)+9*s(52)+6*s(53)+6*s(54)+1
  Such that:aux(11) =< V
aux(12) =< V+1
aux(13) =< V/2
aux(14) =< V2
aux(15) =< V2+1
aux(16) =< V2/2
s(40) =< aux(14)
s(41) =< aux(14)
s(42) =< aux(16)
s(42) =< aux(15)*(1/2)+aux(16)
s(40) =< aux(15)*(1/2)+aux(16)
s(52) =< aux(11)
s(53) =< aux(11)
s(54) =< aux(13)
s(54) =< aux(12)*(1/2)+aux(13)
s(52) =< aux(12)*(1/2)+aux(13)

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

* Chain [40]: 3*s(88)+2*s(89)+2*s(90)+0
  Such that:s(86) =< V2
s(85) =< V2+1
s(87) =< V2/2
s(88) =< s(86)
s(89) =< s(86)
s(90) =< s(87)
s(90) =< s(85)*(1/2)+s(87)
s(88) =< s(85)*(1/2)+s(87)

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

* Chain [39]: 6*s(94)+4*s(95)+4*s(96)+3*s(106)+2*s(107)+2*s(108)+1
  Such that:s(104) =< V2
s(103) =< V2+1
s(105) =< V2/2
aux(17) =< V
aux(18) =< V+1
aux(19) =< V/2
s(94) =< aux(17)
s(95) =< aux(17)
s(96) =< aux(19)
s(96) =< aux(18)*(1/2)+aux(19)
s(94) =< aux(18)*(1/2)+aux(19)
s(106) =< s(104)
s(107) =< s(104)
s(108) =< s(105)
s(108) =< s(103)*(1/2)+s(105)
s(106) =< s(103)*(1/2)+s(105)

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

* Chain [38]: 3*s(112)+2*s(113)+2*s(114)+0
  Such that:s(110) =< V
s(109) =< V+1
s(111) =< V/2
s(112) =< s(110)
s(113) =< s(110)
s(114) =< s(111)
s(114) =< s(109)*(1/2)+s(111)
s(112) =< s(109)*(1/2)+s(111)

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

* Chain [37]: 3*s(118)+2*s(119)+2*s(120)+3*s(124)+2*s(125)+2*s(126)+0
  Such that:s(116) =< V
s(115) =< V+1
s(117) =< V/2
s(122) =< V2
s(121) =< V2+1
s(123) =< V2/2
s(124) =< s(122)
s(125) =< s(122)
s(126) =< s(123)
s(126) =< s(121)*(1/2)+s(123)
s(124) =< s(121)*(1/2)+s(123)
s(118) =< s(116)
s(119) =< s(116)
s(120) =< s(117)
s(120) =< s(115)*(1/2)+s(117)
s(118) =< s(115)*(1/2)+s(117)

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


#### Cost of chains of fun3(V,Out):
* Chain [47]: 3*s(130)+2*s(131)+2*s(132)+1
  Such that:s(128) =< V
s(127) =< V+1
s(129) =< V/2
s(130) =< s(128)
s(131) =< s(128)
s(132) =< s(129)
s(132) =< s(127)*(1/2)+s(129)
s(130) =< s(127)*(1/2)+s(129)

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

* Chain [46]: 0
  with precondition: [Out=1,V>=0] 

* Chain [45]: 3*s(136)+2*s(137)+2*s(138)+0
  Such that:s(134) =< V
s(133) =< V+1
s(135) =< V/2
s(136) =< s(134)
s(137) =< s(134)
s(138) =< s(135)
s(138) =< s(133)*(1/2)+s(135)
s(136) =< s(133)*(1/2)+s(135)

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

* Chain [44]: 3*s(142)+2*s(143)+2*s(144)+1
  Such that:s(140) =< V
s(139) =< V+1
s(141) =< V/2
s(142) =< s(140)
s(143) =< s(140)
s(144) =< s(141)
s(144) =< s(139)*(1/2)+s(141)
s(142) =< s(139)*(1/2)+s(141)

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


#### Cost of chains of start(V,V2):
* Chain [48]: 42*s(148)+28*s(149)+28*s(150)+33*s(160)+22*s(161)+22*s(162)+1
  Such that:aux(20) =< V
aux(21) =< V+1
aux(22) =< V/2
aux(23) =< V2
aux(24) =< V2+1
aux(25) =< V2/2
s(148) =< aux(20)
s(149) =< aux(20)
s(150) =< aux(22)
s(150) =< aux(21)*(1/2)+aux(22)
s(148) =< aux(21)*(1/2)+aux(22)
s(160) =< aux(23)
s(161) =< aux(23)
s(162) =< aux(25)
s(162) =< aux(24)*(1/2)+aux(25)
s(160) =< aux(24)*(1/2)+aux(25)

  with precondition: [] 


Closed-form bounds of start(V,V2): 
-------------------------------------
* Chain [48] with precondition: [] 
    - Upper bound: nat(V)*70+1+nat(V2)*55+nat(V/2)*28+nat(V2/2)*22 
    - Complexity: n 

### Maximum cost of start(V,V2): nat(V)*70+1+nat(V2)*55+nat(V/2)*28+nat(V2/2)*22 
Asymptotic class: n 
* Total analysis performed in 503 ms.


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

(16)
BOUNDS(1, n^1)

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

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

(18)
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:

   .(1', x) -> x
   .(x, 1') -> x
   .(i(x), x) -> 1'
   .(x, i(x)) -> 1'
   i(1') -> 1'
   i(i(x)) -> x
   .(i(y), .(y, z)) -> z
   .(y, .(i(y), z)) -> z

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

   encArg(1') -> 1'
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_1 -> 1'
   encode_i(x_1) -> i(encArg(x_1))

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

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

(20)
Obligation:
Innermost TRS:
Rules:
.(1', x) -> x
.(x, 1') -> x
.(i(x), x) -> 1'
.(x, i(x)) -> 1'
i(1') -> 1'
i(i(x)) -> x
.(i(y), .(y, z)) -> z
.(y, .(i(y), z)) -> z
encArg(1') -> 1'
encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
encArg(cons_i(x_1)) -> i(encArg(x_1))
encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
encode_1 -> 1'
encode_i(x_1) -> i(encArg(x_1))

Types:
. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
1' :: 1':cons_.:cons_i
i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
encArg :: 1':cons_.:cons_i -> 1':cons_.:cons_i
cons_. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
cons_i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
encode_. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
encode_1 :: 1':cons_.:cons_i
encode_i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
hole_1':cons_.:cons_i1_0 :: 1':cons_.:cons_i
gen_1':cons_.:cons_i2_0 :: Nat -> 1':cons_.:cons_i

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

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

(22)
Obligation:
Innermost TRS:
Rules:
.(1', x) -> x
.(x, 1') -> x
.(i(x), x) -> 1'
.(x, i(x)) -> 1'
i(1') -> 1'
i(i(x)) -> x
.(i(y), .(y, z)) -> z
.(y, .(i(y), z)) -> z
encArg(1') -> 1'
encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
encArg(cons_i(x_1)) -> i(encArg(x_1))
encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
encode_1 -> 1'
encode_i(x_1) -> i(encArg(x_1))

Types:
. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
1' :: 1':cons_.:cons_i
i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
encArg :: 1':cons_.:cons_i -> 1':cons_.:cons_i
cons_. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
cons_i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
encode_. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
encode_1 :: 1':cons_.:cons_i
encode_i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
hole_1':cons_.:cons_i1_0 :: 1':cons_.:cons_i
gen_1':cons_.:cons_i2_0 :: Nat -> 1':cons_.:cons_i


Generator Equations:
gen_1':cons_.:cons_i2_0(0) <=> 1'
gen_1':cons_.:cons_i2_0(+(x, 1)) <=> cons_.(1', gen_1':cons_.:cons_i2_0(x))


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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_1':cons_.:cons_i2_0(n4_0)) -> gen_1':cons_.:cons_i2_0(0), rt in Omega(n4_0)

Induction Base:
encArg(gen_1':cons_.:cons_i2_0(0)) ->_R^Omega(0)
1'

Induction Step:
encArg(gen_1':cons_.:cons_i2_0(+(n4_0, 1))) ->_R^Omega(0)
.(encArg(1'), encArg(gen_1':cons_.:cons_i2_0(n4_0))) ->_R^Omega(0)
.(1', encArg(gen_1':cons_.:cons_i2_0(n4_0))) ->_IH
.(1', gen_1':cons_.:cons_i2_0(0)) ->_R^Omega(1)
gen_1':cons_.:cons_i2_0(0)

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

(24)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
.(1', x) -> x
.(x, 1') -> x
.(i(x), x) -> 1'
.(x, i(x)) -> 1'
i(1') -> 1'
i(i(x)) -> x
.(i(y), .(y, z)) -> z
.(y, .(i(y), z)) -> z
encArg(1') -> 1'
encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
encArg(cons_i(x_1)) -> i(encArg(x_1))
encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
encode_1 -> 1'
encode_i(x_1) -> i(encArg(x_1))

Types:
. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
1' :: 1':cons_.:cons_i
i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
encArg :: 1':cons_.:cons_i -> 1':cons_.:cons_i
cons_. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
cons_i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
encode_. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
encode_1 :: 1':cons_.:cons_i
encode_i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
hole_1':cons_.:cons_i1_0 :: 1':cons_.:cons_i
gen_1':cons_.:cons_i2_0 :: Nat -> 1':cons_.:cons_i


Generator Equations:
gen_1':cons_.:cons_i2_0(0) <=> 1'
gen_1':cons_.:cons_i2_0(+(x, 1)) <=> cons_.(1', gen_1':cons_.:cons_i2_0(x))


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

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

(26)
BOUNDS(n^1, INF)