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


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WORST_CASE(Omega(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 175 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 352 ms]
(12) proven lower bound
(13) LowerBoundPropagationProof [FINISHED, 0 ms]
(14) BOUNDS(n^1, INF)


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

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


The TRS R consists of the following rules:

   app(id, x) -> x
   app(add, 0) -> id
   app(app(add, app(s, x)), y) -> app(s, app(app(add, x), y))
   app(app(map, f), nil) -> nil
   app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))

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(id) -> id
   encArg(add) -> add
   encArg(0) -> 0
   encArg(s) -> s
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_id -> id
   encode_add -> add
   encode_0 -> 0
   encode_s -> s
   encode_map -> map
   encode_nil -> nil
   encode_cons -> cons


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

(2)
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:

   app(id, x) -> x
   app(add, 0) -> id
   app(app(add, app(s, x)), y) -> app(s, app(app(add, x), y))
   app(app(map, f), nil) -> nil
   app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))

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

   encArg(id) -> id
   encArg(add) -> add
   encArg(0) -> 0
   encArg(s) -> s
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_id -> id
   encode_add -> add
   encode_0 -> 0
   encode_s -> s
   encode_map -> map
   encode_nil -> nil
   encode_cons -> cons

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, INF).


The TRS R consists of the following rules:

   app(id, x) -> x
   app(add, 0) -> id
   app(app(add, app(s, x)), y) -> app(s, app(app(add, x), y))
   app(app(map, f), nil) -> nil
   app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))

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

   encArg(id) -> id
   encArg(add) -> add
   encArg(0) -> 0
   encArg(s) -> s
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_id -> id
   encode_add -> add
   encode_0 -> 0
   encode_s -> s
   encode_map -> map
   encode_nil -> nil
   encode_cons -> cons

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

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

(6)
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:

   app(id, x) -> x
   app(add, 0') -> id
   app(app(add, app(s, x)), y) -> app(s, app(app(add, x), y))
   app(app(map, f), nil) -> nil
   app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))

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

   encArg(id) -> id
   encArg(add) -> add
   encArg(0') -> 0'
   encArg(s) -> s
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_id -> id
   encode_add -> add
   encode_0 -> 0'
   encode_s -> s
   encode_map -> map
   encode_nil -> nil
   encode_cons -> cons

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

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

(8)
Obligation:
Innermost TRS:
Rules:
app(id, x) -> x
app(add, 0') -> id
app(app(add, app(s, x)), y) -> app(s, app(app(add, x), y))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
encArg(id) -> id
encArg(add) -> add
encArg(0') -> 0'
encArg(s) -> s
encArg(map) -> map
encArg(nil) -> nil
encArg(cons) -> cons
encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
encode_id -> id
encode_add -> add
encode_0 -> 0'
encode_s -> s
encode_map -> map
encode_nil -> nil
encode_cons -> cons

Types:
app :: id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app
id :: id:add:0':s:map:nil:cons:cons_app
add :: id:add:0':s:map:nil:cons:cons_app
0' :: id:add:0':s:map:nil:cons:cons_app
s :: id:add:0':s:map:nil:cons:cons_app
map :: id:add:0':s:map:nil:cons:cons_app
nil :: id:add:0':s:map:nil:cons:cons_app
cons :: id:add:0':s:map:nil:cons:cons_app
encArg :: id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app
cons_app :: id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app
encode_app :: id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app
encode_id :: id:add:0':s:map:nil:cons:cons_app
encode_add :: id:add:0':s:map:nil:cons:cons_app
encode_0 :: id:add:0':s:map:nil:cons:cons_app
encode_s :: id:add:0':s:map:nil:cons:cons_app
encode_map :: id:add:0':s:map:nil:cons:cons_app
encode_nil :: id:add:0':s:map:nil:cons:cons_app
encode_cons :: id:add:0':s:map:nil:cons:cons_app
hole_id:add:0':s:map:nil:cons:cons_app1_3 :: id:add:0':s:map:nil:cons:cons_app
gen_id:add:0':s:map:nil:cons:cons_app2_3 :: Nat -> id:add:0':s:map:nil:cons:cons_app

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

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

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

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

(10)
Obligation:
Innermost TRS:
Rules:
app(id, x) -> x
app(add, 0') -> id
app(app(add, app(s, x)), y) -> app(s, app(app(add, x), y))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
encArg(id) -> id
encArg(add) -> add
encArg(0') -> 0'
encArg(s) -> s
encArg(map) -> map
encArg(nil) -> nil
encArg(cons) -> cons
encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
encode_id -> id
encode_add -> add
encode_0 -> 0'
encode_s -> s
encode_map -> map
encode_nil -> nil
encode_cons -> cons

Types:
app :: id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app
id :: id:add:0':s:map:nil:cons:cons_app
add :: id:add:0':s:map:nil:cons:cons_app
0' :: id:add:0':s:map:nil:cons:cons_app
s :: id:add:0':s:map:nil:cons:cons_app
map :: id:add:0':s:map:nil:cons:cons_app
nil :: id:add:0':s:map:nil:cons:cons_app
cons :: id:add:0':s:map:nil:cons:cons_app
encArg :: id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app
cons_app :: id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app
encode_app :: id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app
encode_id :: id:add:0':s:map:nil:cons:cons_app
encode_add :: id:add:0':s:map:nil:cons:cons_app
encode_0 :: id:add:0':s:map:nil:cons:cons_app
encode_s :: id:add:0':s:map:nil:cons:cons_app
encode_map :: id:add:0':s:map:nil:cons:cons_app
encode_nil :: id:add:0':s:map:nil:cons:cons_app
encode_cons :: id:add:0':s:map:nil:cons:cons_app
hole_id:add:0':s:map:nil:cons:cons_app1_3 :: id:add:0':s:map:nil:cons:cons_app
gen_id:add:0':s:map:nil:cons:cons_app2_3 :: Nat -> id:add:0':s:map:nil:cons:cons_app


Generator Equations:
gen_id:add:0':s:map:nil:cons:cons_app2_3(0) <=> id
gen_id:add:0':s:map:nil:cons:cons_app2_3(+(x, 1)) <=> cons_app(id, gen_id:add:0':s:map:nil:cons:cons_app2_3(x))


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

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_id:add:0':s:map:nil:cons:cons_app2_3(n134_3)) -> gen_id:add:0':s:map:nil:cons:cons_app2_3(0), rt in Omega(n134_3)

Induction Base:
encArg(gen_id:add:0':s:map:nil:cons:cons_app2_3(0)) ->_R^Omega(0)
id

Induction Step:
encArg(gen_id:add:0':s:map:nil:cons:cons_app2_3(+(n134_3, 1))) ->_R^Omega(0)
app(encArg(id), encArg(gen_id:add:0':s:map:nil:cons:cons_app2_3(n134_3))) ->_R^Omega(0)
app(id, encArg(gen_id:add:0':s:map:nil:cons:cons_app2_3(n134_3))) ->_IH
app(id, gen_id:add:0':s:map:nil:cons:cons_app2_3(0)) ->_R^Omega(1)
gen_id:add:0':s:map:nil:cons:cons_app2_3(0)

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

(12)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
app(id, x) -> x
app(add, 0') -> id
app(app(add, app(s, x)), y) -> app(s, app(app(add, x), y))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
encArg(id) -> id
encArg(add) -> add
encArg(0') -> 0'
encArg(s) -> s
encArg(map) -> map
encArg(nil) -> nil
encArg(cons) -> cons
encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
encode_id -> id
encode_add -> add
encode_0 -> 0'
encode_s -> s
encode_map -> map
encode_nil -> nil
encode_cons -> cons

Types:
app :: id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app
id :: id:add:0':s:map:nil:cons:cons_app
add :: id:add:0':s:map:nil:cons:cons_app
0' :: id:add:0':s:map:nil:cons:cons_app
s :: id:add:0':s:map:nil:cons:cons_app
map :: id:add:0':s:map:nil:cons:cons_app
nil :: id:add:0':s:map:nil:cons:cons_app
cons :: id:add:0':s:map:nil:cons:cons_app
encArg :: id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app
cons_app :: id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app
encode_app :: id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app -> id:add:0':s:map:nil:cons:cons_app
encode_id :: id:add:0':s:map:nil:cons:cons_app
encode_add :: id:add:0':s:map:nil:cons:cons_app
encode_0 :: id:add:0':s:map:nil:cons:cons_app
encode_s :: id:add:0':s:map:nil:cons:cons_app
encode_map :: id:add:0':s:map:nil:cons:cons_app
encode_nil :: id:add:0':s:map:nil:cons:cons_app
encode_cons :: id:add:0':s:map:nil:cons:cons_app
hole_id:add:0':s:map:nil:cons:cons_app1_3 :: id:add:0':s:map:nil:cons:cons_app
gen_id:add:0':s:map:nil:cons:cons_app2_3 :: Nat -> id:add:0':s:map:nil:cons:cons_app


Generator Equations:
gen_id:add:0':s:map:nil:cons:cons_app2_3(0) <=> id
gen_id:add:0':s:map:nil:cons:cons_app2_3(+(x, 1)) <=> cons_app(id, gen_id:add:0':s:map:nil:cons:cons_app2_3(x))


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

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

(14)
BOUNDS(n^1, INF)