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


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 1264 ms]
    (10) BOUNDS(1, n^2)
(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTRS
    (13) SlicingProof [LOWER BOUND(ID), 0 ms]
    (14) CpxTRS
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) typed CpxTrs
    (17) OrderProof [LOWER BOUND(ID), 0 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 333 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 27 ms]
            (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 28 ms]
            (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 49 ms]
            (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
            (32) typed CpxTrs
            (33) RewriteLemmaProof [LOWER BOUND(ID), 2763 ms]
            (34) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   fstsplit(0, x) -> nil
   fstsplit(s(n), nil) -> nil
   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
   sndsplit(0, x) -> x
   sndsplit(s(n), nil) -> nil
   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
   empty(nil) -> true
   empty(cons(h, t)) -> false
   leq(0, m) -> true
   leq(s(n), 0) -> false
   leq(s(n), s(m)) -> leq(n, m)
   length(nil) -> 0
   length(cons(h, t)) -> s(length(t))
   app(nil, x) -> x
   app(cons(h, t), x) -> cons(h, app(t, x))
   map_f(pid, nil) -> nil
   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
   process(store, m) -> if1(store, m, leq(m, length(store)))
   if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store)))
   if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
   if2(store, m, false) -> process(app(map_f(self, nil), sndsplit(m, store)), m)
   if3(store, m, false) -> process(sndsplit(m, app(map_f(self, nil), store)), m)

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

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

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


The TRS R consists of the following rules:

   fstsplit(0, x) -> nil [1]
   fstsplit(s(n), nil) -> nil [1]
   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t)) [1]
   sndsplit(0, x) -> x [1]
   sndsplit(s(n), nil) -> nil [1]
   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t) [1]
   empty(nil) -> true [1]
   empty(cons(h, t)) -> false [1]
   leq(0, m) -> true [1]
   leq(s(n), 0) -> false [1]
   leq(s(n), s(m)) -> leq(n, m) [1]
   length(nil) -> 0 [1]
   length(cons(h, t)) -> s(length(t)) [1]
   app(nil, x) -> x [1]
   app(cons(h, t), x) -> cons(h, app(t, x)) [1]
   map_f(pid, nil) -> nil [1]
   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t)) [1]
   process(store, m) -> if1(store, m, leq(m, length(store))) [1]
   if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store))) [1]
   if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store)))) [1]
   if2(store, m, false) -> process(app(map_f(self, nil), sndsplit(m, store)), m) [1]
   if3(store, m, false) -> process(sndsplit(m, app(map_f(self, nil), store)), m) [1]

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

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

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   fstsplit(0, x) -> nil [1]
   fstsplit(s(n), nil) -> nil [1]
   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t)) [1]
   sndsplit(0, x) -> x [1]
   sndsplit(s(n), nil) -> nil [1]
   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t) [1]
   empty(nil) -> true [1]
   empty(cons(h, t)) -> false [1]
   leq(0, m) -> true [1]
   leq(s(n), 0) -> false [1]
   leq(s(n), s(m)) -> leq(n, m) [1]
   length(nil) -> 0 [1]
   length(cons(h, t)) -> s(length(t)) [1]
   app(nil, x) -> x [1]
   app(cons(h, t), x) -> cons(h, app(t, x)) [1]
   map_f(pid, nil) -> nil [1]
   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t)) [1]
   process(store, m) -> if1(store, m, leq(m, length(store))) [1]
   if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store))) [1]
   if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store)))) [1]
   if2(store, m, false) -> process(app(map_f(self, nil), sndsplit(m, store)), m) [1]
   if3(store, m, false) -> process(sndsplit(m, app(map_f(self, nil), store)), m) [1]

The TRS has the following type information:
fstsplit :: 0:s -> nil:cons:f -> nil:cons:f
0 :: 0:s
nil :: nil:cons:f
s :: 0:s -> 0:s
cons :: a -> nil:cons:f -> nil:cons:f
sndsplit :: 0:s -> nil:cons:f -> nil:cons:f
empty :: nil:cons:f -> true:false
true :: true:false
false :: true:false
leq :: 0:s -> 0:s -> true:false
length :: nil:cons:f -> 0:s
app :: nil:cons:f -> nil:cons:f -> nil:cons:f
map_f :: self -> nil:cons:f -> nil:cons:f
f :: self -> a -> nil:cons:f
process :: nil:cons:f -> 0:s -> process:if1:if2:if3
if1 :: nil:cons:f -> 0:s -> true:false -> process:if1:if2:if3
if2 :: nil:cons:f -> 0:s -> true:false -> process:if1:if2:if3
if3 :: nil:cons:f -> 0:s -> true:false -> process:if1:if2:if3
self :: self

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

(5) 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:

   fstsplit(v0, v1) -> null_fstsplit [0]
   sndsplit(v0, v1) -> null_sndsplit [0]
   empty(v0) -> null_empty [0]
   length(v0) -> null_length [0]
   app(v0, v1) -> null_app [0]
   map_f(v0, v1) -> null_map_f [0]
   if2(v0, v1, v2) -> null_if2 [0]
   if3(v0, v1, v2) -> null_if3 [0]
   leq(v0, v1) -> null_leq [0]
   if1(v0, v1, v2) -> null_if1 [0]

And the following fresh constants:    null_fstsplit, null_sndsplit, null_empty, null_length, null_app, null_map_f, null_if2, null_if3, null_leq, null_if1, const

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

(6)
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:

   fstsplit(0, x) -> nil [1]
   fstsplit(s(n), nil) -> nil [1]
   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t)) [1]
   sndsplit(0, x) -> x [1]
   sndsplit(s(n), nil) -> nil [1]
   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t) [1]
   empty(nil) -> true [1]
   empty(cons(h, t)) -> false [1]
   leq(0, m) -> true [1]
   leq(s(n), 0) -> false [1]
   leq(s(n), s(m)) -> leq(n, m) [1]
   length(nil) -> 0 [1]
   length(cons(h, t)) -> s(length(t)) [1]
   app(nil, x) -> x [1]
   app(cons(h, t), x) -> cons(h, app(t, x)) [1]
   map_f(pid, nil) -> nil [1]
   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t)) [1]
   process(store, m) -> if1(store, m, leq(m, length(store))) [1]
   if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store))) [1]
   if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store)))) [1]
   if2(store, m, false) -> process(app(map_f(self, nil), sndsplit(m, store)), m) [1]
   if3(store, m, false) -> process(sndsplit(m, app(map_f(self, nil), store)), m) [1]
   fstsplit(v0, v1) -> null_fstsplit [0]
   sndsplit(v0, v1) -> null_sndsplit [0]
   empty(v0) -> null_empty [0]
   length(v0) -> null_length [0]
   app(v0, v1) -> null_app [0]
   map_f(v0, v1) -> null_map_f [0]
   if2(v0, v1, v2) -> null_if2 [0]
   if3(v0, v1, v2) -> null_if3 [0]
   leq(v0, v1) -> null_leq [0]
   if1(v0, v1, v2) -> null_if1 [0]

The TRS has the following type information:
fstsplit :: 0:s:null_length -> nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f -> nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f
0 :: 0:s:null_length
nil :: nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f
s :: 0:s:null_length -> 0:s:null_length
cons :: a -> nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f -> nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f
sndsplit :: 0:s:null_length -> nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f -> nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f
empty :: nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f -> true:false:null_empty:null_leq
true :: true:false:null_empty:null_leq
false :: true:false:null_empty:null_leq
leq :: 0:s:null_length -> 0:s:null_length -> true:false:null_empty:null_leq
length :: nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f -> 0:s:null_length
app :: nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f -> nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f -> nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f
map_f :: self -> nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f -> nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f
f :: self -> a -> nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f
process :: nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f -> 0:s:null_length -> null_if2:null_if3:null_if1
if1 :: nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f -> 0:s:null_length -> true:false:null_empty:null_leq -> null_if2:null_if3:null_if1
if2 :: nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f -> 0:s:null_length -> true:false:null_empty:null_leq -> null_if2:null_if3:null_if1
if3 :: nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f -> 0:s:null_length -> true:false:null_empty:null_leq -> null_if2:null_if3:null_if1
self :: self
null_fstsplit :: nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f
null_sndsplit :: nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f
null_empty :: true:false:null_empty:null_leq
null_length :: 0:s:null_length
null_app :: nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f
null_map_f :: nil:cons:f:null_fstsplit:null_sndsplit:null_app:null_map_f
null_if2 :: null_if2:null_if3:null_if1
null_if3 :: null_if2:null_if3:null_if1
null_leq :: true:false:null_empty:null_leq
null_if1 :: null_if2:null_if3:null_if1
const :: a

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

(7) 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
   nil => 0
   true => 2
   false => 1
   self => 0
   null_fstsplit => 0
   null_sndsplit => 0
   null_empty => 0
   null_length => 0
   null_app => 0
   null_map_f => 0
   null_if2 => 0
   null_if3 => 0
   null_leq => 0
   null_if1 => 0
   const => 0

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

(8)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   app(z, z') -{ 1 }-> 1 + h + app(t, x) :|: z = 1 + h + t, z' = x, x >= 0, h >= 0, t >= 0
   empty(z) -{ 1 }-> 2 :|: z = 0
   empty(z) -{ 1 }-> 1 :|: z = 1 + h + t, h >= 0, t >= 0
   empty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   fstsplit(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
   fstsplit(z, z') -{ 1 }-> 0 :|: n >= 0, z = 1 + n, z' = 0
   fstsplit(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   fstsplit(z, z') -{ 1 }-> 1 + h + fstsplit(n, t) :|: z' = 1 + h + t, n >= 0, h >= 0, t >= 0, z = 1 + n
   if1(z, z', z'') -{ 1 }-> if3(store, m, empty(fstsplit(m, app(map_f(0, 0), store)))) :|: z = store, z' = m, store >= 0, z'' = 1, m >= 0
   if1(z, z', z'') -{ 1 }-> if2(store, m, empty(fstsplit(m, store))) :|: z = store, z' = m, z'' = 2, store >= 0, m >= 0
   if1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if2(z, z', z'') -{ 1 }-> process(app(map_f(0, 0), sndsplit(m, store)), m) :|: z = store, z' = m, store >= 0, z'' = 1, m >= 0
   if2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if3(z, z', z'') -{ 1 }-> process(sndsplit(m, app(map_f(0, 0), store)), m) :|: z = store, z' = m, store >= 0, z'' = 1, m >= 0
   if3(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   length(z) -{ 1 }-> 1 + length(t) :|: z = 1 + h + t, h >= 0, t >= 0
   leq(z, z') -{ 1 }-> leq(n, m) :|: n >= 0, z' = 1 + m, z = 1 + n, m >= 0
   leq(z, z') -{ 1 }-> 2 :|: z' = m, z = 0, m >= 0
   leq(z, z') -{ 1 }-> 1 :|: n >= 0, z = 1 + n, z' = 0
   leq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   map_f(z, z') -{ 1 }-> app(1 + pid + h, map_f(pid, t)) :|: z' = 1 + h + t, z = pid, h >= 0, t >= 0, pid >= 0
   map_f(z, z') -{ 1 }-> 0 :|: z = pid, pid >= 0, z' = 0
   map_f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   process(z, z') -{ 1 }-> if1(store, m, leq(m, length(store))) :|: z = store, z' = m, store >= 0, m >= 0
   sndsplit(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0
   sndsplit(z, z') -{ 1 }-> sndsplit(n, t) :|: z' = 1 + h + t, n >= 0, h >= 0, t >= 0, z = 1 + n
   sndsplit(z, z') -{ 1 }-> 0 :|: n >= 0, z = 1 + n, z' = 0
   sndsplit(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V, V31),0,[fstsplit(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V31),0,[sndsplit(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V31),0,[empty(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V31),0,[leq(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V31),0,[length(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V31),0,[app(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V31),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V31),0,[process(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V31),0,[if1(V1, V, V31, Out)],[V1 >= 0,V >= 0,V31 >= 0]).
eq(start(V1, V, V31),0,[if2(V1, V, V31, Out)],[V1 >= 0,V >= 0,V31 >= 0]).
eq(start(V1, V, V31),0,[if3(V1, V, V31, Out)],[V1 >= 0,V >= 0,V31 >= 0]).
eq(fstsplit(V1, V, Out),1,[],[Out = 0,V = V2,V2 >= 0,V1 = 0]).
eq(fstsplit(V1, V, Out),1,[],[Out = 0,V3 >= 0,V1 = 1 + V3,V = 0]).
eq(fstsplit(V1, V, Out),1,[fstsplit(V5, V4, Ret1)],[Out = 1 + Ret1 + V6,V = 1 + V4 + V6,V5 >= 0,V6 >= 0,V4 >= 0,V1 = 1 + V5]).
eq(sndsplit(V1, V, Out),1,[],[Out = V7,V = V7,V7 >= 0,V1 = 0]).
eq(sndsplit(V1, V, Out),1,[],[Out = 0,V8 >= 0,V1 = 1 + V8,V = 0]).
eq(sndsplit(V1, V, Out),1,[sndsplit(V11, V10, Ret)],[Out = Ret,V = 1 + V10 + V9,V11 >= 0,V9 >= 0,V10 >= 0,V1 = 1 + V11]).
eq(empty(V1, Out),1,[],[Out = 2,V1 = 0]).
eq(empty(V1, Out),1,[],[Out = 1,V1 = 1 + V12 + V13,V12 >= 0,V13 >= 0]).
eq(leq(V1, V, Out),1,[],[Out = 2,V = V14,V1 = 0,V14 >= 0]).
eq(leq(V1, V, Out),1,[],[Out = 1,V15 >= 0,V1 = 1 + V15,V = 0]).
eq(leq(V1, V, Out),1,[leq(V17, V16, Ret2)],[Out = Ret2,V17 >= 0,V = 1 + V16,V1 = 1 + V17,V16 >= 0]).
eq(length(V1, Out),1,[],[Out = 0,V1 = 0]).
eq(length(V1, Out),1,[length(V19, Ret11)],[Out = 1 + Ret11,V1 = 1 + V18 + V19,V18 >= 0,V19 >= 0]).
eq(app(V1, V, Out),1,[],[Out = V20,V = V20,V20 >= 0,V1 = 0]).
eq(app(V1, V, Out),1,[app(V23, V21, Ret12)],[Out = 1 + Ret12 + V22,V1 = 1 + V22 + V23,V = V21,V21 >= 0,V22 >= 0,V23 >= 0]).
eq(fun(V1, V, Out),1,[],[Out = 0,V1 = V24,V24 >= 0,V = 0]).
eq(fun(V1, V, Out),1,[fun(V27, V26, Ret13),app(1 + V27 + V25, Ret13, Ret3)],[Out = Ret3,V = 1 + V25 + V26,V1 = V27,V25 >= 0,V26 >= 0,V27 >= 0]).
eq(process(V1, V, Out),1,[length(V28, Ret21),leq(V29, Ret21, Ret22),if1(V28, V29, Ret22, Ret4)],[Out = Ret4,V1 = V28,V = V29,V28 >= 0,V29 >= 0]).
eq(if1(V1, V, V31, Out),1,[fstsplit(V32, V30, Ret20),empty(Ret20, Ret23),if2(V30, V32, Ret23, Ret5)],[Out = Ret5,V1 = V30,V = V32,V31 = 2,V30 >= 0,V32 >= 0]).
eq(if1(V1, V, V31, Out),1,[fun(0, 0, Ret2010),app(Ret2010, V34, Ret201),fstsplit(V33, Ret201, Ret202),empty(Ret202, Ret24),if3(V34, V33, Ret24, Ret6)],[Out = Ret6,V1 = V34,V = V33,V34 >= 0,V31 = 1,V33 >= 0]).
eq(if2(V1, V, V31, Out),1,[fun(0, 0, Ret00),sndsplit(V35, V36, Ret01),app(Ret00, Ret01, Ret0),process(Ret0, V35, Ret7)],[Out = Ret7,V1 = V36,V = V35,V36 >= 0,V31 = 1,V35 >= 0]).
eq(if3(V1, V, V31, Out),1,[fun(0, 0, Ret010),app(Ret010, V38, Ret011),sndsplit(V37, Ret011, Ret02),process(Ret02, V37, Ret8)],[Out = Ret8,V1 = V38,V = V37,V38 >= 0,V31 = 1,V37 >= 0]).
eq(fstsplit(V1, V, Out),0,[],[Out = 0,V40 >= 0,V39 >= 0,V1 = V40,V = V39]).
eq(sndsplit(V1, V, Out),0,[],[Out = 0,V42 >= 0,V41 >= 0,V1 = V42,V = V41]).
eq(empty(V1, Out),0,[],[Out = 0,V43 >= 0,V1 = V43]).
eq(length(V1, Out),0,[],[Out = 0,V44 >= 0,V1 = V44]).
eq(app(V1, V, Out),0,[],[Out = 0,V45 >= 0,V46 >= 0,V1 = V45,V = V46]).
eq(fun(V1, V, Out),0,[],[Out = 0,V47 >= 0,V48 >= 0,V1 = V47,V = V48]).
eq(if2(V1, V, V31, Out),0,[],[Out = 0,V50 >= 0,V31 = V51,V49 >= 0,V1 = V50,V = V49,V51 >= 0]).
eq(if3(V1, V, V31, Out),0,[],[Out = 0,V52 >= 0,V31 = V54,V53 >= 0,V1 = V52,V = V53,V54 >= 0]).
eq(leq(V1, V, Out),0,[],[Out = 0,V55 >= 0,V56 >= 0,V1 = V55,V = V56]).
eq(if1(V1, V, V31, Out),0,[],[Out = 0,V57 >= 0,V31 = V59,V58 >= 0,V1 = V57,V = V58,V59 >= 0]).
input_output_vars(fstsplit(V1,V,Out),[V1,V],[Out]).
input_output_vars(sndsplit(V1,V,Out),[V1,V],[Out]).
input_output_vars(empty(V1,Out),[V1],[Out]).
input_output_vars(leq(V1,V,Out),[V1,V],[Out]).
input_output_vars(length(V1,Out),[V1],[Out]).
input_output_vars(app(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(process(V1,V,Out),[V1,V],[Out]).
input_output_vars(if1(V1,V,V31,Out),[V1,V,V31],[Out]).
input_output_vars(if2(V1,V,V31,Out),[V1,V,V31],[Out]).
input_output_vars(if3(V1,V,V31,Out),[V1,V,V31],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [app/3]
1. non_recursive  : [empty/2]
2. recursive  : [fstsplit/3]
3. recursive [non_tail] : [fun/3]
4. recursive  : [length/2]
5. recursive  : [leq/3]
6. recursive  : [sndsplit/3]
7. recursive  : [if1/4,if2/4,if3/4,process/3]
8. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into app/3
1. SCC is partially evaluated into empty/2
2. SCC is partially evaluated into fstsplit/3
3. SCC is partially evaluated into fun/3
4. SCC is partially evaluated into length/2
5. SCC is partially evaluated into leq/3
6. SCC is partially evaluated into sndsplit/3
7. SCC is partially evaluated into process/3
8. SCC is partially evaluated into start/3

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

### Specialization of cost equations app/3 
* CE 21 is refined into CE [45] 
* CE 19 is refined into CE [46] 
* CE 20 is refined into CE [47] 


### Cost equations --> "Loop" of app/3 
* CEs [47] --> Loop 25 
* CEs [45] --> Loop 26 
* CEs [46] --> Loop 27 

### Ranking functions of CR app(V1,V,Out) 
* RF of phase [25]: [V1]

#### Partial ranking functions of CR app(V1,V,Out) 
* Partial RF of phase [25]:
  - RF of loop [25:1]:
    V1


### Specialization of cost equations empty/2 
* CE 27 is refined into CE [48] 
* CE 28 is refined into CE [49] 
* CE 26 is refined into CE [50] 


### Cost equations --> "Loop" of empty/2 
* CEs [48] --> Loop 28 
* CEs [49] --> Loop 29 
* CEs [50] --> Loop 30 

### Ranking functions of CR empty(V1,Out) 

#### Partial ranking functions of CR empty(V1,Out) 


### Specialization of cost equations fstsplit/3 
* CE 23 is refined into CE [51] 
* CE 22 is refined into CE [52] 
* CE 25 is refined into CE [53] 
* CE 24 is refined into CE [54] 


### Cost equations --> "Loop" of fstsplit/3 
* CEs [54] --> Loop 31 
* CEs [51] --> Loop 32 
* CEs [52,53] --> Loop 33 

### Ranking functions of CR fstsplit(V1,V,Out) 
* RF of phase [31]: [V,V1]

#### Partial ranking functions of CR fstsplit(V1,V,Out) 
* Partial RF of phase [31]:
  - RF of loop [31:1]:
    V
    V1


### Specialization of cost equations fun/3 
* CE 16 is refined into CE [55] 
* CE 18 is refined into CE [56] 
* CE 17 is refined into CE [57,58,59] 


### Cost equations --> "Loop" of fun/3 
* CEs [58] --> Loop 34 
* CEs [59] --> Loop 35 
* CEs [57] --> Loop 36 
* CEs [55,56] --> Loop 37 

### Ranking functions of CR fun(V1,V,Out) 
* RF of phase [34,35,36]: [V]

#### Partial ranking functions of CR fun(V1,V,Out) 
* Partial RF of phase [34,35,36]:
  - RF of loop [34:1,35:1,36:1]:
    V


### Specialization of cost equations length/2 
* CE 42 is refined into CE [60] 
* CE 44 is refined into CE [61] 
* CE 43 is refined into CE [62] 


### Cost equations --> "Loop" of length/2 
* CEs [62] --> Loop 38 
* CEs [60,61] --> Loop 39 

### Ranking functions of CR length(V1,Out) 
* RF of phase [38]: [V1]

#### Partial ranking functions of CR length(V1,Out) 
* Partial RF of phase [38]:
  - RF of loop [38:1]:
    V1


### Specialization of cost equations leq/3 
* CE 41 is refined into CE [63] 
* CE 39 is refined into CE [64] 
* CE 38 is refined into CE [65] 
* CE 40 is refined into CE [66] 


### Cost equations --> "Loop" of leq/3 
* CEs [66] --> Loop 40 
* CEs [63] --> Loop 41 
* CEs [64] --> Loop 42 
* CEs [65] --> Loop 43 

### Ranking functions of CR leq(V1,V,Out) 
* RF of phase [40]: [V,V1]

#### Partial ranking functions of CR leq(V1,V,Out) 
* Partial RF of phase [40]:
  - RF of loop [40:1]:
    V
    V1


### Specialization of cost equations sndsplit/3 
* CE 30 is refined into CE [67] 
* CE 32 is refined into CE [68] 
* CE 29 is refined into CE [69] 
* CE 31 is refined into CE [70] 


### Cost equations --> "Loop" of sndsplit/3 
* CEs [70] --> Loop 44 
* CEs [67,68] --> Loop 45 
* CEs [69] --> Loop 46 

### Ranking functions of CR sndsplit(V1,V,Out) 
* RF of phase [44]: [V,V1]

#### Partial ranking functions of CR sndsplit(V1,V,Out) 
* Partial RF of phase [44]:
  - RF of loop [44:1]:
    V
    V1


### Specialization of cost equations process/3 
* CE 33 is refined into CE [71,72,73,74,75,76,77] 
* CE 35 is refined into CE [78,79,80,81,82,83,84,85,86,87,88,89] 
* CE 37 is refined into CE [90,91,92,93,94,95,96,97] 
* CE 34 is refined into CE [98,99,100,101,102,103] 
* CE 36 is refined into CE [104,105,106,107] 


### Cost equations --> "Loop" of process/3 
* CEs [99,102,106] --> Loop 47 
* CEs [98,100,101,103,104,105,107] --> Loop 48 
* CEs [71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97] --> Loop 49 

### Ranking functions of CR process(V1,V,Out) 
* RF of phase [47]: [V1,V1-V+1]

#### Partial ranking functions of CR process(V1,V,Out) 
* Partial RF of phase [47]:
  - RF of loop [47:1]:
    V1
    V1-V+1


### Specialization of cost equations start/3 
* CE 4 is refined into CE [108,109,110,111] 
* CE 5 is refined into CE [112,113,114,115] 
* CE 1 is refined into CE [116] 
* CE 2 is refined into CE [117,118,119] 
* CE 3 is refined into CE [120,121,122,123,124,125] 
* CE 6 is refined into CE [126,127,128,129,130,131] 
* CE 7 is refined into CE [132,133,134,135,136] 
* CE 8 is refined into CE [137,138] 
* CE 9 is refined into CE [139,140,141] 
* CE 10 is refined into CE [142,143,144] 
* CE 11 is refined into CE [145,146,147,148,149] 
* CE 12 is refined into CE [150,151] 
* CE 13 is refined into CE [152,153,154,155] 
* CE 14 is refined into CE [156,157] 
* CE 15 is refined into CE [158] 


### Cost equations --> "Loop" of start/3 
* CEs [108,109,110,111,112,113,114,115] --> Loop 50 
* CEs [146] --> Loop 51 
* CEs [117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136] --> Loop 52 
* CEs [116,137,138,139,140,141,142,143,144,145,147,148,149,150,151,152,153,154,155,156,157,158] --> Loop 53 

### Ranking functions of CR start(V1,V,V31) 

#### Partial ranking functions of CR start(V1,V,V31) 


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

#### Cost of chains of app(V1,V,Out):
* Chain [[25],27]: 1*it(25)+1
  Such that:it(25) =< -V+Out

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

* Chain [[25],26]: 1*it(25)+0
  Such that:it(25) =< Out

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

* Chain [27]: 1
  with precondition: [V1=0,V=Out,V>=0] 

* Chain [26]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of empty(V1,Out):
* Chain [30]: 1
  with precondition: [V1=0,Out=2] 

* Chain [29]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [28]: 1
  with precondition: [Out=1,V1>=1] 


#### Cost of chains of fstsplit(V1,V,Out):
* Chain [[31],33]: 1*it(31)+1
  Such that:it(31) =< V1

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

* Chain [[31],32]: 1*it(31)+1
  Such that:it(31) =< V1

  with precondition: [V=Out,V1>=2,V>=1] 

* Chain [33]: 1
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [32]: 1
  with precondition: [V=0,Out=0,V1>=1] 


#### Cost of chains of fun(V1,V,Out):
* Chain [[34,35,36],37]: 4*it(34)+1*s(7)+1*s(8)+1
  Such that:aux(2) =< V1+V
aux(6) =< V
it(34) =< aux(6)
aux(3) =< aux(2)
s(7) =< it(34)*aux(2)
s(8) =< it(34)*aux(3)

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

* Chain [37]: 1
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of length(V1,Out):
* Chain [[38],39]: 1*it(38)+1
  Such that:it(38) =< V1

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

* Chain [39]: 1
  with precondition: [Out=0,V1>=0] 


#### Cost of chains of leq(V1,V,Out):
* Chain [[40],43]: 1*it(40)+1
  Such that:it(40) =< V1

  with precondition: [Out=2,V1>=1,V>=V1] 

* Chain [[40],42]: 1*it(40)+1
  Such that:it(40) =< V

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

* Chain [[40],41]: 1*it(40)+0
  Such that:it(40) =< V

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

* Chain [43]: 1
  with precondition: [V1=0,Out=2,V>=0] 

* Chain [42]: 1
  with precondition: [V=0,Out=1,V1>=1] 

* Chain [41]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of sndsplit(V1,V,Out):
* Chain [[44],46]: 1*it(44)+1
  Such that:it(44) =< V1

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

* Chain [[44],45]: 1*it(44)+1
  Such that:it(44) =< V

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

* Chain [46]: 1
  with precondition: [V1=0,V=Out,V>=0] 

* Chain [45]: 1
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of process(V1,V,Out):
* Chain [[47],49]: 45*it(47)+19*s(18)+2*s(71)+8
  Such that:aux(18) =< V
aux(27) =< V1
it(47) =< aux(27)
s(18) =< aux(18)
s(73) =< it(47)*aux(27)
s(71) =< s(73)

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

* Chain [[47],48,49]: 34*it(47)+37*s(18)+2*s(71)+20
  Such that:aux(37) =< V
aux(38) =< V1
it(47) =< aux(38)
s(18) =< aux(37)
s(73) =< it(47)*aux(38)
s(71) =< s(73)

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

* Chain [49]: 22*s(12)+19*s(18)+8
  Such that:aux(17) =< V1
aux(18) =< V
s(12) =< aux(17)
s(18) =< aux(18)

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

* Chain [48,49]: 37*s(18)+11*s(76)+20
  Such that:aux(35) =< V1
aux(37) =< V
s(18) =< aux(37)
s(76) =< aux(35)

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


#### Cost of chains of start(V1,V,V31):
* Chain [53]: 119*s(126)+119*s(127)+1*s(139)+1*s(140)+4*s(146)+20
  Such that:s(135) =< V1+V
aux(41) =< V1
aux(42) =< V
s(126) =< aux(41)
s(127) =< aux(42)
s(145) =< s(126)*aux(41)
s(146) =< s(145)
s(138) =< s(135)
s(139) =< s(127)*s(135)
s(140) =< s(127)*s(138)

  with precondition: [V1>=0] 

* Chain [52]: 1134*s(148)+228*s(149)+336*s(162)+12*s(164)+8*s(183)+29
  Such that:aux(53) =< V1
aux(54) =< V1-V
aux(55) =< V
s(149) =< aux(53)
s(148) =< aux(55)
s(182) =< s(149)*aux(53)
s(183) =< s(182)
s(162) =< aux(54)
s(163) =< s(162)*aux(54)
s(164) =< s(163)

  with precondition: [V31=1,V1>=0,V>=0] 

* Chain [51]: 1
  with precondition: [V=0,V1>=1] 

* Chain [50]: 462*s(252)+2*s(253)+112*s(275)+4*s(277)+27
  Such that:s(272) =< V1-V
aux(60) =< V1
aux(61) =< V
s(253) =< aux(60)
s(252) =< aux(61)
s(275) =< s(272)
s(276) =< s(275)*s(272)
s(277) =< s(276)

  with precondition: [V31=2,V1>=0,V>=0] 


Closed-form bounds of start(V1,V,V31): 
-------------------------------------
* Chain [53] with precondition: [V1>=0] 
    - Upper bound: 119*V1+20+4*V1*V1+nat(V)*119+nat(V)*2*nat(V1+V) 
    - Complexity: n^2 
* Chain [52] with precondition: [V31=1,V1>=0,V>=0] 
    - Upper bound: 228*V1+29+8*V1*V1+1134*V+nat(V1-V)*336+nat(V1-V)*12*nat(V1-V) 
    - Complexity: n^2 
* Chain [51] with precondition: [V=0,V1>=1] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [50] with precondition: [V31=2,V1>=0,V>=0] 
    - Upper bound: 2*V1+462*V+27+nat(V1-V)*112+nat(V1-V)*4*nat(V1-V) 
    - Complexity: n^2 

### Maximum cost of start(V1,V,V31): 2*V1+19+nat(V)*119+max([4*V1*V1+117*V1+max([nat(V)*2*nat(V1+V),109*V1+9+4*V1*V1+nat(V)*1015+nat(V1-V)*336+nat(V1-V)*12*nat(V1-V)]),nat(V)*343+7+nat(V1-V)*112+nat(V1-V)*4*nat(V1-V)])+1 
Asymptotic class: n^2 
* Total analysis performed in 1093 ms.


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

(10)
BOUNDS(1, n^2)

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

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

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   fstsplit(0', x) -> nil
   fstsplit(s(n), nil) -> nil
   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
   sndsplit(0', x) -> x
   sndsplit(s(n), nil) -> nil
   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
   empty(nil) -> true
   empty(cons(h, t)) -> false
   leq(0', m) -> true
   leq(s(n), 0') -> false
   leq(s(n), s(m)) -> leq(n, m)
   length(nil) -> 0'
   length(cons(h, t)) -> s(length(t))
   app(nil, x) -> x
   app(cons(h, t), x) -> cons(h, app(t, x))
   map_f(pid, nil) -> nil
   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
   process(store, m) -> if1(store, m, leq(m, length(store)))
   if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store)))
   if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
   if2(store, m, false) -> process(app(map_f(self, nil), sndsplit(m, store)), m)
   if3(store, m, false) -> process(sndsplit(m, app(map_f(self, nil), store)), m)

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

(13) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
cons/0
map_f/0
f/0
f/1

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

(14)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   fstsplit(0', x) -> nil
   fstsplit(s(n), nil) -> nil
   fstsplit(s(n), cons(t)) -> cons(fstsplit(n, t))
   sndsplit(0', x) -> x
   sndsplit(s(n), nil) -> nil
   sndsplit(s(n), cons(t)) -> sndsplit(n, t)
   empty(nil) -> true
   empty(cons(t)) -> false
   leq(0', m) -> true
   leq(s(n), 0') -> false
   leq(s(n), s(m)) -> leq(n, m)
   length(nil) -> 0'
   length(cons(t)) -> s(length(t))
   app(nil, x) -> x
   app(cons(t), x) -> cons(app(t, x))
   map_f(nil) -> nil
   map_f(cons(t)) -> app(f, map_f(t))
   process(store, m) -> if1(store, m, leq(m, length(store)))
   if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store)))
   if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(nil), store))))
   if2(store, m, false) -> process(app(map_f(nil), sndsplit(m, store)), m)
   if3(store, m, false) -> process(sndsplit(m, app(map_f(nil), store)), m)

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

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

(16)
Obligation:
Innermost TRS:
Rules:
fstsplit(0', x) -> nil
fstsplit(s(n), nil) -> nil
fstsplit(s(n), cons(t)) -> cons(fstsplit(n, t))
sndsplit(0', x) -> x
sndsplit(s(n), nil) -> nil
sndsplit(s(n), cons(t)) -> sndsplit(n, t)
empty(nil) -> true
empty(cons(t)) -> false
leq(0', m) -> true
leq(s(n), 0') -> false
leq(s(n), s(m)) -> leq(n, m)
length(nil) -> 0'
length(cons(t)) -> s(length(t))
app(nil, x) -> x
app(cons(t), x) -> cons(app(t, x))
map_f(nil) -> nil
map_f(cons(t)) -> app(f, map_f(t))
process(store, m) -> if1(store, m, leq(m, length(store)))
if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(nil), store))))
if2(store, m, false) -> process(app(map_f(nil), sndsplit(m, store)), m)
if3(store, m, false) -> process(sndsplit(m, app(map_f(nil), store)), m)

Types:
fstsplit :: 0':s -> nil:cons:f -> nil:cons:f
0' :: 0':s
nil :: nil:cons:f
s :: 0':s -> 0':s
cons :: nil:cons:f -> nil:cons:f
sndsplit :: 0':s -> nil:cons:f -> nil:cons:f
empty :: nil:cons:f -> true:false
true :: true:false
false :: true:false
leq :: 0':s -> 0':s -> true:false
length :: nil:cons:f -> 0':s
app :: nil:cons:f -> nil:cons:f -> nil:cons:f
map_f :: nil:cons:f -> nil:cons:f
f :: nil:cons:f
process :: nil:cons:f -> 0':s -> process:if1:if2:if3
if1 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
if2 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
if3 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
hole_nil:cons:f1_0 :: nil:cons:f
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_process:if1:if2:if34_0 :: process:if1:if2:if3
gen_nil:cons:f5_0 :: Nat -> nil:cons:f
gen_0':s6_0 :: Nat -> 0':s

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

(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
fstsplit, sndsplit, leq, length, app, map_f, process

They will be analysed ascendingly in the following order:
fstsplit < process
sndsplit < process
leq < process
length < process
app < map_f
app < process
map_f < process

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

(18)
Obligation:
Innermost TRS:
Rules:
fstsplit(0', x) -> nil
fstsplit(s(n), nil) -> nil
fstsplit(s(n), cons(t)) -> cons(fstsplit(n, t))
sndsplit(0', x) -> x
sndsplit(s(n), nil) -> nil
sndsplit(s(n), cons(t)) -> sndsplit(n, t)
empty(nil) -> true
empty(cons(t)) -> false
leq(0', m) -> true
leq(s(n), 0') -> false
leq(s(n), s(m)) -> leq(n, m)
length(nil) -> 0'
length(cons(t)) -> s(length(t))
app(nil, x) -> x
app(cons(t), x) -> cons(app(t, x))
map_f(nil) -> nil
map_f(cons(t)) -> app(f, map_f(t))
process(store, m) -> if1(store, m, leq(m, length(store)))
if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(nil), store))))
if2(store, m, false) -> process(app(map_f(nil), sndsplit(m, store)), m)
if3(store, m, false) -> process(sndsplit(m, app(map_f(nil), store)), m)

Types:
fstsplit :: 0':s -> nil:cons:f -> nil:cons:f
0' :: 0':s
nil :: nil:cons:f
s :: 0':s -> 0':s
cons :: nil:cons:f -> nil:cons:f
sndsplit :: 0':s -> nil:cons:f -> nil:cons:f
empty :: nil:cons:f -> true:false
true :: true:false
false :: true:false
leq :: 0':s -> 0':s -> true:false
length :: nil:cons:f -> 0':s
app :: nil:cons:f -> nil:cons:f -> nil:cons:f
map_f :: nil:cons:f -> nil:cons:f
f :: nil:cons:f
process :: nil:cons:f -> 0':s -> process:if1:if2:if3
if1 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
if2 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
if3 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
hole_nil:cons:f1_0 :: nil:cons:f
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_process:if1:if2:if34_0 :: process:if1:if2:if3
gen_nil:cons:f5_0 :: Nat -> nil:cons:f
gen_0':s6_0 :: Nat -> 0':s


Generator Equations:
gen_nil:cons:f5_0(0) <=> nil
gen_nil:cons:f5_0(+(x, 1)) <=> cons(gen_nil:cons:f5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
fstsplit, sndsplit, leq, length, app, map_f, process

They will be analysed ascendingly in the following order:
fstsplit < process
sndsplit < process
leq < process
length < process
app < map_f
app < process
map_f < process

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
fstsplit(gen_0':s6_0(n8_0), gen_nil:cons:f5_0(n8_0)) -> gen_nil:cons:f5_0(n8_0), rt in Omega(1 + n8_0)

Induction Base:
fstsplit(gen_0':s6_0(0), gen_nil:cons:f5_0(0)) ->_R^Omega(1)
nil

Induction Step:
fstsplit(gen_0':s6_0(+(n8_0, 1)), gen_nil:cons:f5_0(+(n8_0, 1))) ->_R^Omega(1)
cons(fstsplit(gen_0':s6_0(n8_0), gen_nil:cons:f5_0(n8_0))) ->_IH
cons(gen_nil:cons:f5_0(c9_0))

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

(20)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
fstsplit(0', x) -> nil
fstsplit(s(n), nil) -> nil
fstsplit(s(n), cons(t)) -> cons(fstsplit(n, t))
sndsplit(0', x) -> x
sndsplit(s(n), nil) -> nil
sndsplit(s(n), cons(t)) -> sndsplit(n, t)
empty(nil) -> true
empty(cons(t)) -> false
leq(0', m) -> true
leq(s(n), 0') -> false
leq(s(n), s(m)) -> leq(n, m)
length(nil) -> 0'
length(cons(t)) -> s(length(t))
app(nil, x) -> x
app(cons(t), x) -> cons(app(t, x))
map_f(nil) -> nil
map_f(cons(t)) -> app(f, map_f(t))
process(store, m) -> if1(store, m, leq(m, length(store)))
if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(nil), store))))
if2(store, m, false) -> process(app(map_f(nil), sndsplit(m, store)), m)
if3(store, m, false) -> process(sndsplit(m, app(map_f(nil), store)), m)

Types:
fstsplit :: 0':s -> nil:cons:f -> nil:cons:f
0' :: 0':s
nil :: nil:cons:f
s :: 0':s -> 0':s
cons :: nil:cons:f -> nil:cons:f
sndsplit :: 0':s -> nil:cons:f -> nil:cons:f
empty :: nil:cons:f -> true:false
true :: true:false
false :: true:false
leq :: 0':s -> 0':s -> true:false
length :: nil:cons:f -> 0':s
app :: nil:cons:f -> nil:cons:f -> nil:cons:f
map_f :: nil:cons:f -> nil:cons:f
f :: nil:cons:f
process :: nil:cons:f -> 0':s -> process:if1:if2:if3
if1 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
if2 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
if3 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
hole_nil:cons:f1_0 :: nil:cons:f
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_process:if1:if2:if34_0 :: process:if1:if2:if3
gen_nil:cons:f5_0 :: Nat -> nil:cons:f
gen_0':s6_0 :: Nat -> 0':s


Generator Equations:
gen_nil:cons:f5_0(0) <=> nil
gen_nil:cons:f5_0(+(x, 1)) <=> cons(gen_nil:cons:f5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
fstsplit, sndsplit, leq, length, app, map_f, process

They will be analysed ascendingly in the following order:
fstsplit < process
sndsplit < process
leq < process
length < process
app < map_f
app < process
map_f < process

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
Innermost TRS:
Rules:
fstsplit(0', x) -> nil
fstsplit(s(n), nil) -> nil
fstsplit(s(n), cons(t)) -> cons(fstsplit(n, t))
sndsplit(0', x) -> x
sndsplit(s(n), nil) -> nil
sndsplit(s(n), cons(t)) -> sndsplit(n, t)
empty(nil) -> true
empty(cons(t)) -> false
leq(0', m) -> true
leq(s(n), 0') -> false
leq(s(n), s(m)) -> leq(n, m)
length(nil) -> 0'
length(cons(t)) -> s(length(t))
app(nil, x) -> x
app(cons(t), x) -> cons(app(t, x))
map_f(nil) -> nil
map_f(cons(t)) -> app(f, map_f(t))
process(store, m) -> if1(store, m, leq(m, length(store)))
if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(nil), store))))
if2(store, m, false) -> process(app(map_f(nil), sndsplit(m, store)), m)
if3(store, m, false) -> process(sndsplit(m, app(map_f(nil), store)), m)

Types:
fstsplit :: 0':s -> nil:cons:f -> nil:cons:f
0' :: 0':s
nil :: nil:cons:f
s :: 0':s -> 0':s
cons :: nil:cons:f -> nil:cons:f
sndsplit :: 0':s -> nil:cons:f -> nil:cons:f
empty :: nil:cons:f -> true:false
true :: true:false
false :: true:false
leq :: 0':s -> 0':s -> true:false
length :: nil:cons:f -> 0':s
app :: nil:cons:f -> nil:cons:f -> nil:cons:f
map_f :: nil:cons:f -> nil:cons:f
f :: nil:cons:f
process :: nil:cons:f -> 0':s -> process:if1:if2:if3
if1 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
if2 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
if3 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
hole_nil:cons:f1_0 :: nil:cons:f
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_process:if1:if2:if34_0 :: process:if1:if2:if3
gen_nil:cons:f5_0 :: Nat -> nil:cons:f
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
fstsplit(gen_0':s6_0(n8_0), gen_nil:cons:f5_0(n8_0)) -> gen_nil:cons:f5_0(n8_0), rt in Omega(1 + n8_0)


Generator Equations:
gen_nil:cons:f5_0(0) <=> nil
gen_nil:cons:f5_0(+(x, 1)) <=> cons(gen_nil:cons:f5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
sndsplit, leq, length, app, map_f, process

They will be analysed ascendingly in the following order:
sndsplit < process
leq < process
length < process
app < map_f
app < process
map_f < process

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sndsplit(gen_0':s6_0(n548_0), gen_nil:cons:f5_0(n548_0)) -> gen_nil:cons:f5_0(0), rt in Omega(1 + n548_0)

Induction Base:
sndsplit(gen_0':s6_0(0), gen_nil:cons:f5_0(0)) ->_R^Omega(1)
gen_nil:cons:f5_0(0)

Induction Step:
sndsplit(gen_0':s6_0(+(n548_0, 1)), gen_nil:cons:f5_0(+(n548_0, 1))) ->_R^Omega(1)
sndsplit(gen_0':s6_0(n548_0), gen_nil:cons:f5_0(n548_0)) ->_IH
gen_nil:cons:f5_0(0)

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

(26)
Obligation:
Innermost TRS:
Rules:
fstsplit(0', x) -> nil
fstsplit(s(n), nil) -> nil
fstsplit(s(n), cons(t)) -> cons(fstsplit(n, t))
sndsplit(0', x) -> x
sndsplit(s(n), nil) -> nil
sndsplit(s(n), cons(t)) -> sndsplit(n, t)
empty(nil) -> true
empty(cons(t)) -> false
leq(0', m) -> true
leq(s(n), 0') -> false
leq(s(n), s(m)) -> leq(n, m)
length(nil) -> 0'
length(cons(t)) -> s(length(t))
app(nil, x) -> x
app(cons(t), x) -> cons(app(t, x))
map_f(nil) -> nil
map_f(cons(t)) -> app(f, map_f(t))
process(store, m) -> if1(store, m, leq(m, length(store)))
if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(nil), store))))
if2(store, m, false) -> process(app(map_f(nil), sndsplit(m, store)), m)
if3(store, m, false) -> process(sndsplit(m, app(map_f(nil), store)), m)

Types:
fstsplit :: 0':s -> nil:cons:f -> nil:cons:f
0' :: 0':s
nil :: nil:cons:f
s :: 0':s -> 0':s
cons :: nil:cons:f -> nil:cons:f
sndsplit :: 0':s -> nil:cons:f -> nil:cons:f
empty :: nil:cons:f -> true:false
true :: true:false
false :: true:false
leq :: 0':s -> 0':s -> true:false
length :: nil:cons:f -> 0':s
app :: nil:cons:f -> nil:cons:f -> nil:cons:f
map_f :: nil:cons:f -> nil:cons:f
f :: nil:cons:f
process :: nil:cons:f -> 0':s -> process:if1:if2:if3
if1 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
if2 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
if3 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
hole_nil:cons:f1_0 :: nil:cons:f
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_process:if1:if2:if34_0 :: process:if1:if2:if3
gen_nil:cons:f5_0 :: Nat -> nil:cons:f
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
fstsplit(gen_0':s6_0(n8_0), gen_nil:cons:f5_0(n8_0)) -> gen_nil:cons:f5_0(n8_0), rt in Omega(1 + n8_0)
sndsplit(gen_0':s6_0(n548_0), gen_nil:cons:f5_0(n548_0)) -> gen_nil:cons:f5_0(0), rt in Omega(1 + n548_0)


Generator Equations:
gen_nil:cons:f5_0(0) <=> nil
gen_nil:cons:f5_0(+(x, 1)) <=> cons(gen_nil:cons:f5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
leq, length, app, map_f, process

They will be analysed ascendingly in the following order:
leq < process
length < process
app < map_f
app < process
map_f < process

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
leq(gen_0':s6_0(n1156_0), gen_0':s6_0(n1156_0)) -> true, rt in Omega(1 + n1156_0)

Induction Base:
leq(gen_0':s6_0(0), gen_0':s6_0(0)) ->_R^Omega(1)
true

Induction Step:
leq(gen_0':s6_0(+(n1156_0, 1)), gen_0':s6_0(+(n1156_0, 1))) ->_R^Omega(1)
leq(gen_0':s6_0(n1156_0), gen_0':s6_0(n1156_0)) ->_IH
true

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

(28)
Obligation:
Innermost TRS:
Rules:
fstsplit(0', x) -> nil
fstsplit(s(n), nil) -> nil
fstsplit(s(n), cons(t)) -> cons(fstsplit(n, t))
sndsplit(0', x) -> x
sndsplit(s(n), nil) -> nil
sndsplit(s(n), cons(t)) -> sndsplit(n, t)
empty(nil) -> true
empty(cons(t)) -> false
leq(0', m) -> true
leq(s(n), 0') -> false
leq(s(n), s(m)) -> leq(n, m)
length(nil) -> 0'
length(cons(t)) -> s(length(t))
app(nil, x) -> x
app(cons(t), x) -> cons(app(t, x))
map_f(nil) -> nil
map_f(cons(t)) -> app(f, map_f(t))
process(store, m) -> if1(store, m, leq(m, length(store)))
if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(nil), store))))
if2(store, m, false) -> process(app(map_f(nil), sndsplit(m, store)), m)
if3(store, m, false) -> process(sndsplit(m, app(map_f(nil), store)), m)

Types:
fstsplit :: 0':s -> nil:cons:f -> nil:cons:f
0' :: 0':s
nil :: nil:cons:f
s :: 0':s -> 0':s
cons :: nil:cons:f -> nil:cons:f
sndsplit :: 0':s -> nil:cons:f -> nil:cons:f
empty :: nil:cons:f -> true:false
true :: true:false
false :: true:false
leq :: 0':s -> 0':s -> true:false
length :: nil:cons:f -> 0':s
app :: nil:cons:f -> nil:cons:f -> nil:cons:f
map_f :: nil:cons:f -> nil:cons:f
f :: nil:cons:f
process :: nil:cons:f -> 0':s -> process:if1:if2:if3
if1 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
if2 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
if3 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
hole_nil:cons:f1_0 :: nil:cons:f
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_process:if1:if2:if34_0 :: process:if1:if2:if3
gen_nil:cons:f5_0 :: Nat -> nil:cons:f
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
fstsplit(gen_0':s6_0(n8_0), gen_nil:cons:f5_0(n8_0)) -> gen_nil:cons:f5_0(n8_0), rt in Omega(1 + n8_0)
sndsplit(gen_0':s6_0(n548_0), gen_nil:cons:f5_0(n548_0)) -> gen_nil:cons:f5_0(0), rt in Omega(1 + n548_0)
leq(gen_0':s6_0(n1156_0), gen_0':s6_0(n1156_0)) -> true, rt in Omega(1 + n1156_0)


Generator Equations:
gen_nil:cons:f5_0(0) <=> nil
gen_nil:cons:f5_0(+(x, 1)) <=> cons(gen_nil:cons:f5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
length, app, map_f, process

They will be analysed ascendingly in the following order:
length < process
app < map_f
app < process
map_f < process

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_nil:cons:f5_0(n1503_0)) -> gen_0':s6_0(n1503_0), rt in Omega(1 + n1503_0)

Induction Base:
length(gen_nil:cons:f5_0(0)) ->_R^Omega(1)
0'

Induction Step:
length(gen_nil:cons:f5_0(+(n1503_0, 1))) ->_R^Omega(1)
s(length(gen_nil:cons:f5_0(n1503_0))) ->_IH
s(gen_0':s6_0(c1504_0))

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

(30)
Obligation:
Innermost TRS:
Rules:
fstsplit(0', x) -> nil
fstsplit(s(n), nil) -> nil
fstsplit(s(n), cons(t)) -> cons(fstsplit(n, t))
sndsplit(0', x) -> x
sndsplit(s(n), nil) -> nil
sndsplit(s(n), cons(t)) -> sndsplit(n, t)
empty(nil) -> true
empty(cons(t)) -> false
leq(0', m) -> true
leq(s(n), 0') -> false
leq(s(n), s(m)) -> leq(n, m)
length(nil) -> 0'
length(cons(t)) -> s(length(t))
app(nil, x) -> x
app(cons(t), x) -> cons(app(t, x))
map_f(nil) -> nil
map_f(cons(t)) -> app(f, map_f(t))
process(store, m) -> if1(store, m, leq(m, length(store)))
if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(nil), store))))
if2(store, m, false) -> process(app(map_f(nil), sndsplit(m, store)), m)
if3(store, m, false) -> process(sndsplit(m, app(map_f(nil), store)), m)

Types:
fstsplit :: 0':s -> nil:cons:f -> nil:cons:f
0' :: 0':s
nil :: nil:cons:f
s :: 0':s -> 0':s
cons :: nil:cons:f -> nil:cons:f
sndsplit :: 0':s -> nil:cons:f -> nil:cons:f
empty :: nil:cons:f -> true:false
true :: true:false
false :: true:false
leq :: 0':s -> 0':s -> true:false
length :: nil:cons:f -> 0':s
app :: nil:cons:f -> nil:cons:f -> nil:cons:f
map_f :: nil:cons:f -> nil:cons:f
f :: nil:cons:f
process :: nil:cons:f -> 0':s -> process:if1:if2:if3
if1 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
if2 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
if3 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
hole_nil:cons:f1_0 :: nil:cons:f
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_process:if1:if2:if34_0 :: process:if1:if2:if3
gen_nil:cons:f5_0 :: Nat -> nil:cons:f
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
fstsplit(gen_0':s6_0(n8_0), gen_nil:cons:f5_0(n8_0)) -> gen_nil:cons:f5_0(n8_0), rt in Omega(1 + n8_0)
sndsplit(gen_0':s6_0(n548_0), gen_nil:cons:f5_0(n548_0)) -> gen_nil:cons:f5_0(0), rt in Omega(1 + n548_0)
leq(gen_0':s6_0(n1156_0), gen_0':s6_0(n1156_0)) -> true, rt in Omega(1 + n1156_0)
length(gen_nil:cons:f5_0(n1503_0)) -> gen_0':s6_0(n1503_0), rt in Omega(1 + n1503_0)


Generator Equations:
gen_nil:cons:f5_0(0) <=> nil
gen_nil:cons:f5_0(+(x, 1)) <=> cons(gen_nil:cons:f5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
app, map_f, process

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

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
app(gen_nil:cons:f5_0(n1823_0), gen_nil:cons:f5_0(b)) -> gen_nil:cons:f5_0(+(n1823_0, b)), rt in Omega(1 + n1823_0)

Induction Base:
app(gen_nil:cons:f5_0(0), gen_nil:cons:f5_0(b)) ->_R^Omega(1)
gen_nil:cons:f5_0(b)

Induction Step:
app(gen_nil:cons:f5_0(+(n1823_0, 1)), gen_nil:cons:f5_0(b)) ->_R^Omega(1)
cons(app(gen_nil:cons:f5_0(n1823_0), gen_nil:cons:f5_0(b))) ->_IH
cons(gen_nil:cons:f5_0(+(b, c1824_0)))

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

(32)
Obligation:
Innermost TRS:
Rules:
fstsplit(0', x) -> nil
fstsplit(s(n), nil) -> nil
fstsplit(s(n), cons(t)) -> cons(fstsplit(n, t))
sndsplit(0', x) -> x
sndsplit(s(n), nil) -> nil
sndsplit(s(n), cons(t)) -> sndsplit(n, t)
empty(nil) -> true
empty(cons(t)) -> false
leq(0', m) -> true
leq(s(n), 0') -> false
leq(s(n), s(m)) -> leq(n, m)
length(nil) -> 0'
length(cons(t)) -> s(length(t))
app(nil, x) -> x
app(cons(t), x) -> cons(app(t, x))
map_f(nil) -> nil
map_f(cons(t)) -> app(f, map_f(t))
process(store, m) -> if1(store, m, leq(m, length(store)))
if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(nil), store))))
if2(store, m, false) -> process(app(map_f(nil), sndsplit(m, store)), m)
if3(store, m, false) -> process(sndsplit(m, app(map_f(nil), store)), m)

Types:
fstsplit :: 0':s -> nil:cons:f -> nil:cons:f
0' :: 0':s
nil :: nil:cons:f
s :: 0':s -> 0':s
cons :: nil:cons:f -> nil:cons:f
sndsplit :: 0':s -> nil:cons:f -> nil:cons:f
empty :: nil:cons:f -> true:false
true :: true:false
false :: true:false
leq :: 0':s -> 0':s -> true:false
length :: nil:cons:f -> 0':s
app :: nil:cons:f -> nil:cons:f -> nil:cons:f
map_f :: nil:cons:f -> nil:cons:f
f :: nil:cons:f
process :: nil:cons:f -> 0':s -> process:if1:if2:if3
if1 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
if2 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
if3 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
hole_nil:cons:f1_0 :: nil:cons:f
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_process:if1:if2:if34_0 :: process:if1:if2:if3
gen_nil:cons:f5_0 :: Nat -> nil:cons:f
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
fstsplit(gen_0':s6_0(n8_0), gen_nil:cons:f5_0(n8_0)) -> gen_nil:cons:f5_0(n8_0), rt in Omega(1 + n8_0)
sndsplit(gen_0':s6_0(n548_0), gen_nil:cons:f5_0(n548_0)) -> gen_nil:cons:f5_0(0), rt in Omega(1 + n548_0)
leq(gen_0':s6_0(n1156_0), gen_0':s6_0(n1156_0)) -> true, rt in Omega(1 + n1156_0)
length(gen_nil:cons:f5_0(n1503_0)) -> gen_0':s6_0(n1503_0), rt in Omega(1 + n1503_0)
app(gen_nil:cons:f5_0(n1823_0), gen_nil:cons:f5_0(b)) -> gen_nil:cons:f5_0(+(n1823_0, b)), rt in Omega(1 + n1823_0)


Generator Equations:
gen_nil:cons:f5_0(0) <=> nil
gen_nil:cons:f5_0(+(x, 1)) <=> cons(gen_nil:cons:f5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
map_f, process

They will be analysed ascendingly in the following order:
map_f < process

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

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
map_f(gen_nil:cons:f5_0(+(1, n2800_0))) -> *7_0, rt in Omega(n2800_0)

Induction Base:
map_f(gen_nil:cons:f5_0(+(1, 0)))

Induction Step:
map_f(gen_nil:cons:f5_0(+(1, +(n2800_0, 1)))) ->_R^Omega(1)
app(f, map_f(gen_nil:cons:f5_0(+(1, n2800_0)))) ->_IH
app(f, *7_0)

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

(34)
Obligation:
Innermost TRS:
Rules:
fstsplit(0', x) -> nil
fstsplit(s(n), nil) -> nil
fstsplit(s(n), cons(t)) -> cons(fstsplit(n, t))
sndsplit(0', x) -> x
sndsplit(s(n), nil) -> nil
sndsplit(s(n), cons(t)) -> sndsplit(n, t)
empty(nil) -> true
empty(cons(t)) -> false
leq(0', m) -> true
leq(s(n), 0') -> false
leq(s(n), s(m)) -> leq(n, m)
length(nil) -> 0'
length(cons(t)) -> s(length(t))
app(nil, x) -> x
app(cons(t), x) -> cons(app(t, x))
map_f(nil) -> nil
map_f(cons(t)) -> app(f, map_f(t))
process(store, m) -> if1(store, m, leq(m, length(store)))
if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(nil), store))))
if2(store, m, false) -> process(app(map_f(nil), sndsplit(m, store)), m)
if3(store, m, false) -> process(sndsplit(m, app(map_f(nil), store)), m)

Types:
fstsplit :: 0':s -> nil:cons:f -> nil:cons:f
0' :: 0':s
nil :: nil:cons:f
s :: 0':s -> 0':s
cons :: nil:cons:f -> nil:cons:f
sndsplit :: 0':s -> nil:cons:f -> nil:cons:f
empty :: nil:cons:f -> true:false
true :: true:false
false :: true:false
leq :: 0':s -> 0':s -> true:false
length :: nil:cons:f -> 0':s
app :: nil:cons:f -> nil:cons:f -> nil:cons:f
map_f :: nil:cons:f -> nil:cons:f
f :: nil:cons:f
process :: nil:cons:f -> 0':s -> process:if1:if2:if3
if1 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
if2 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
if3 :: nil:cons:f -> 0':s -> true:false -> process:if1:if2:if3
hole_nil:cons:f1_0 :: nil:cons:f
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_process:if1:if2:if34_0 :: process:if1:if2:if3
gen_nil:cons:f5_0 :: Nat -> nil:cons:f
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
fstsplit(gen_0':s6_0(n8_0), gen_nil:cons:f5_0(n8_0)) -> gen_nil:cons:f5_0(n8_0), rt in Omega(1 + n8_0)
sndsplit(gen_0':s6_0(n548_0), gen_nil:cons:f5_0(n548_0)) -> gen_nil:cons:f5_0(0), rt in Omega(1 + n548_0)
leq(gen_0':s6_0(n1156_0), gen_0':s6_0(n1156_0)) -> true, rt in Omega(1 + n1156_0)
length(gen_nil:cons:f5_0(n1503_0)) -> gen_0':s6_0(n1503_0), rt in Omega(1 + n1503_0)
app(gen_nil:cons:f5_0(n1823_0), gen_nil:cons:f5_0(b)) -> gen_nil:cons:f5_0(+(n1823_0, b)), rt in Omega(1 + n1823_0)
map_f(gen_nil:cons:f5_0(+(1, n2800_0))) -> *7_0, rt in Omega(n2800_0)


Generator Equations:
gen_nil:cons:f5_0(0) <=> nil
gen_nil:cons:f5_0(+(x, 1)) <=> cons(gen_nil:cons:f5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
process