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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTRS
    (5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 282 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 66 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 349 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 85 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 393 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 176 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 978 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 629 ms]
    (42) CpxRNTS
    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 2054 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 930 ms]
    (48) CpxRNTS
    (49) FinalProof [FINISHED, 0 ms]
    (50) BOUNDS(1, n^2)
(51) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(52) CpxTRS
    (53) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) typed CpxTrs
    (55) OrderProof [LOWER BOUND(ID), 0 ms]
    (56) typed CpxTrs
    (57) RewriteLemmaProof [LOWER BOUND(ID), 273 ms]
    (58) BEST
        (59) proven lower bound
            (60) LowerBoundPropagationProof [FINISHED, 0 ms]
            (61) BOUNDS(n^1, INF)
        (62) typed CpxTrs
            (63) RewriteLemmaProof [LOWER BOUND(ID), 63 ms]
            (64) typed CpxTrs


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

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


The TRS R consists of the following rules:

   if(true, x, y) -> x
   if(false, x, y) -> y
   eq(0, 0) -> true
   eq(0, s(x)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   app(nil, l) -> l
   app(cons(x, l1), l2) -> cons(x, app(l1, l2))
   app(app(l1, l2), l3) -> app(l1, app(l2, l3))
   mem(x, nil) -> false
   mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l)
   ifmem(true, x, l) -> true
   ifmem(false, x, l) -> mem(x, l)
   inter(x, nil) -> nil
   inter(nil, x) -> nil
   inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
   inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
   inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
   inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
   ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
   ifinter(false, x, l1, l2) -> inter(l1, l2)

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of ifinter: ifmem, eq, mem
The following defined symbols can occur below the 0th argument of ifmem: eq

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
app(app(l1, l2), l3) -> app(l1, app(l2, l3))
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))

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

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


The TRS R consists of the following rules:

   if(true, x, y) -> x
   if(false, x, y) -> y
   eq(0, 0) -> true
   eq(0, s(x)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   app(nil, l) -> l
   app(cons(x, l1), l2) -> cons(x, app(l1, l2))
   mem(x, nil) -> false
   mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l)
   ifmem(true, x, l) -> true
   ifmem(false, x, l) -> mem(x, l)
   inter(x, nil) -> nil
   inter(nil, x) -> nil
   inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
   inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
   ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
   ifinter(false, x, l1, l2) -> inter(l1, l2)

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

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

The duplicating contexts are:
mem([], cons(y, l))
inter(cons([], l1), l2)
inter(cons(x, l1), [])
inter(l1, cons([], l2))
inter([], cons(x, l2))


The defined contexts are:
ifinter([], x1, x2, x3)
ifmem([], x1, x2)


[] just represents basic- or constructor-terms in the following defined contexts:
ifmem([], x1, x2)


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
----------------------------------------

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


The TRS R consists of the following rules:

   if(true, x, y) -> x
   if(false, x, y) -> y
   eq(0, 0) -> true
   eq(0, s(x)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   app(nil, l) -> l
   app(cons(x, l1), l2) -> cons(x, app(l1, l2))
   mem(x, nil) -> false
   mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l)
   ifmem(true, x, l) -> true
   ifmem(false, x, l) -> mem(x, l)
   inter(x, nil) -> nil
   inter(nil, x) -> nil
   inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
   inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
   ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
   ifinter(false, x, l1, l2) -> inter(l1, l2)

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

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

(6)
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:

   if(true, x, y) -> x [1]
   if(false, x, y) -> y [1]
   eq(0, 0) -> true [1]
   eq(0, s(x)) -> false [1]
   eq(s(x), 0) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   app(nil, l) -> l [1]
   app(cons(x, l1), l2) -> cons(x, app(l1, l2)) [1]
   mem(x, nil) -> false [1]
   mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l) [1]
   ifmem(true, x, l) -> true [1]
   ifmem(false, x, l) -> mem(x, l) [1]
   inter(x, nil) -> nil [1]
   inter(nil, x) -> nil [1]
   inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2) [1]
   inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1) [1]
   ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2)) [1]
   ifinter(false, x, l1, l2) -> inter(l1, l2) [1]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   if(true, x, y) -> x [1]
   if(false, x, y) -> y [1]
   eq(0, 0) -> true [1]
   eq(0, s(x)) -> false [1]
   eq(s(x), 0) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   app(nil, l) -> l [1]
   app(cons(x, l1), l2) -> cons(x, app(l1, l2)) [1]
   mem(x, nil) -> false [1]
   mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l) [1]
   ifmem(true, x, l) -> true [1]
   ifmem(false, x, l) -> mem(x, l) [1]
   inter(x, nil) -> nil [1]
   inter(nil, x) -> nil [1]
   inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2) [1]
   inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1) [1]
   ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2)) [1]
   ifinter(false, x, l1, l2) -> inter(l1, l2) [1]

The TRS has the following type information:
if :: true:false -> if -> if -> if
true :: true:false
false :: true:false
eq :: 0:s -> 0:s -> true:false
0 :: 0:s
s :: 0:s -> 0:s
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s -> nil:cons -> nil:cons
mem :: 0:s -> nil:cons -> true:false
ifmem :: true:false -> 0:s -> nil:cons -> true:false
inter :: nil:cons -> nil:cons -> nil:cons
ifinter :: true:false -> 0:s -> nil:cons -> nil:cons -> nil:cons

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

(9) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   if_3
   app_2
   inter_2
   ifinter_4

(c) The following functions are completely defined:

   mem_2
   eq_2
   ifmem_3

Due to the following rules being added:
none

And the following fresh constants:    const

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

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   if(true, x, y) -> x [1]
   if(false, x, y) -> y [1]
   eq(0, 0) -> true [1]
   eq(0, s(x)) -> false [1]
   eq(s(x), 0) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   app(nil, l) -> l [1]
   app(cons(x, l1), l2) -> cons(x, app(l1, l2)) [1]
   mem(x, nil) -> false [1]
   mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l) [1]
   ifmem(true, x, l) -> true [1]
   ifmem(false, x, l) -> mem(x, l) [1]
   inter(x, nil) -> nil [1]
   inter(nil, x) -> nil [1]
   inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2) [1]
   inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1) [1]
   ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2)) [1]
   ifinter(false, x, l1, l2) -> inter(l1, l2) [1]

The TRS has the following type information:
if :: true:false -> if -> if -> if
true :: true:false
false :: true:false
eq :: 0:s -> 0:s -> true:false
0 :: 0:s
s :: 0:s -> 0:s
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s -> nil:cons -> nil:cons
mem :: 0:s -> nil:cons -> true:false
ifmem :: true:false -> 0:s -> nil:cons -> true:false
inter :: nil:cons -> nil:cons -> nil:cons
ifinter :: true:false -> 0:s -> nil:cons -> nil:cons -> nil:cons
const :: if

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

(11) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(12)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   if(true, x, y) -> x [1]
   if(false, x, y) -> y [1]
   eq(0, 0) -> true [1]
   eq(0, s(x)) -> false [1]
   eq(s(x), 0) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   app(nil, l) -> l [1]
   app(cons(x, l1), l2) -> cons(x, app(l1, l2)) [1]
   mem(x, nil) -> false [1]
   mem(0, cons(0, l)) -> ifmem(true, 0, l) [2]
   mem(0, cons(s(x'), l)) -> ifmem(false, 0, l) [2]
   mem(s(x''), cons(0, l)) -> ifmem(false, s(x''), l) [2]
   mem(s(x1), cons(s(y'), l)) -> ifmem(eq(x1, y'), s(x1), l) [2]
   ifmem(true, x, l) -> true [1]
   ifmem(false, x, l) -> mem(x, l) [1]
   inter(x, nil) -> nil [1]
   inter(nil, x) -> nil [1]
   inter(cons(x, l1), nil) -> ifinter(false, x, l1, nil) [2]
   inter(cons(x, l1), cons(y'', l')) -> ifinter(ifmem(eq(x, y''), x, l'), x, l1, cons(y'', l')) [2]
   inter(nil, cons(x, l2)) -> ifinter(false, x, l2, nil) [2]
   inter(cons(y1, l''), cons(x, l2)) -> ifinter(ifmem(eq(x, y1), x, l''), x, l2, cons(y1, l'')) [2]
   ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2)) [1]
   ifinter(false, x, l1, l2) -> inter(l1, l2) [1]

The TRS has the following type information:
if :: true:false -> if -> if -> if
true :: true:false
false :: true:false
eq :: 0:s -> 0:s -> true:false
0 :: 0:s
s :: 0:s -> 0:s
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s -> nil:cons -> nil:cons
mem :: 0:s -> nil:cons -> true:false
ifmem :: true:false -> 0:s -> nil:cons -> true:false
inter :: nil:cons -> nil:cons -> nil:cons
ifinter :: true:false -> 0:s -> nil:cons -> nil:cons -> nil:cons
const :: if

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:

   true => 1
   false => 0
   0 => 0
   nil => 0
   const => 0

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

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

   app(z, z') -{ 1 }-> l :|: z' = l, l >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(l1, l2) :|: x >= 0, z = 1 + x + l1, z' = l2, l1 >= 0, l2 >= 0
   eq(z, z') -{ 1 }-> eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   if(z, z', z'') -{ 1 }-> x :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   ifinter(z, z', z'', z1) -{ 1 }-> inter(l1, l2) :|: z' = x, x >= 0, z1 = l2, z'' = l1, l1 >= 0, z = 0, l2 >= 0
   ifinter(z, z', z'', z1) -{ 1 }-> 1 + x + inter(l1, l2) :|: z' = x, z = 1, x >= 0, z1 = l2, z'' = l1, l1 >= 0, l2 >= 0
   ifmem(z, z', z'') -{ 1 }-> mem(x, l) :|: z' = x, x >= 0, l >= 0, z = 0, z'' = l
   ifmem(z, z', z'') -{ 1 }-> 1 :|: z' = x, z = 1, x >= 0, l >= 0, z'' = l
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y''), x, l'), x, l1, 1 + y'' + l') :|: x >= 0, z = 1 + x + l1, l' >= 0, y'' >= 0, z' = 1 + y'' + l', l1 >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y1), x, l''), x, l2, 1 + y1 + l'') :|: y1 >= 0, l'' >= 0, x >= 0, z' = 1 + x + l2, l2 >= 0, z = 1 + y1 + l''
   inter(z, z') -{ 2 }-> ifinter(0, x, l1, 0) :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' = 0
   inter(z, z') -{ 2 }-> ifinter(0, x, l2, 0) :|: x >= 0, z' = 1 + x + l2, z = 0, l2 >= 0
   inter(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   inter(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(eq(x1, y'), 1 + x1, l) :|: z' = 1 + (1 + y') + l, x1 >= 0, z = 1 + x1, y' >= 0, l >= 0
   mem(z, z') -{ 2 }-> ifmem(1, 0, l) :|: l >= 0, z = 0, z' = 1 + 0 + l
   mem(z, z') -{ 2 }-> ifmem(0, 0, l) :|: x' >= 0, l >= 0, z' = 1 + (1 + x') + l, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 1 + x'', l) :|: z = 1 + x'', l >= 0, x'' >= 0, z' = 1 + 0 + l
   mem(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0


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

(15) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(l1, z') :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' >= 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   ifinter(z, z', z'', z1) -{ 1 }-> inter(z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   ifinter(z, z', z'', z1) -{ 1 }-> 1 + z' + inter(z'', z1) :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0
   ifmem(z, z', z'') -{ 1 }-> mem(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   ifmem(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y''), x, l'), x, l1, 1 + y'' + l') :|: x >= 0, z = 1 + x + l1, l' >= 0, y'' >= 0, z' = 1 + y'' + l', l1 >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y1), x, l''), x, l2, 1 + y1 + l'') :|: y1 >= 0, l'' >= 0, x >= 0, z' = 1 + x + l2, l2 >= 0, z = 1 + y1 + l''
   inter(z, z') -{ 2 }-> ifinter(0, x, l1, 0) :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' = 0
   inter(z, z') -{ 2 }-> ifinter(0, x, l2, 0) :|: x >= 0, z' = 1 + x + l2, z = 0, l2 >= 0
   inter(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   inter(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(eq(z - 1, y'), 1 + (z - 1), l) :|: z' = 1 + (1 + y') + l, z - 1 >= 0, y' >= 0, l >= 0
   mem(z, z') -{ 2 }-> ifmem(1, 0, z' - 1) :|: z' - 1 >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 0, l) :|: x' >= 0, l >= 0, z' = 1 + (1 + x') + l, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 1 + (z - 1), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   mem(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0


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

(17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { if }
   { app }
   { eq }
   { ifmem, mem }
   { inter, ifinter }

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

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(l1, z') :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' >= 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   ifinter(z, z', z'', z1) -{ 1 }-> inter(z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   ifinter(z, z', z'', z1) -{ 1 }-> 1 + z' + inter(z'', z1) :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0
   ifmem(z, z', z'') -{ 1 }-> mem(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   ifmem(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y''), x, l'), x, l1, 1 + y'' + l') :|: x >= 0, z = 1 + x + l1, l' >= 0, y'' >= 0, z' = 1 + y'' + l', l1 >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y1), x, l''), x, l2, 1 + y1 + l'') :|: y1 >= 0, l'' >= 0, x >= 0, z' = 1 + x + l2, l2 >= 0, z = 1 + y1 + l''
   inter(z, z') -{ 2 }-> ifinter(0, x, l1, 0) :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' = 0
   inter(z, z') -{ 2 }-> ifinter(0, x, l2, 0) :|: x >= 0, z' = 1 + x + l2, z = 0, l2 >= 0
   inter(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   inter(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(eq(z - 1, y'), 1 + (z - 1), l) :|: z' = 1 + (1 + y') + l, z - 1 >= 0, y' >= 0, l >= 0
   mem(z, z') -{ 2 }-> ifmem(1, 0, z' - 1) :|: z' - 1 >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 0, l) :|: x' >= 0, l >= 0, z' = 1 + (1 + x') + l, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 1 + (z - 1), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   mem(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {if}, {app}, {eq}, {ifmem,mem}, {inter,ifinter}

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

(19) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(l1, z') :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' >= 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   ifinter(z, z', z'', z1) -{ 1 }-> inter(z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   ifinter(z, z', z'', z1) -{ 1 }-> 1 + z' + inter(z'', z1) :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0
   ifmem(z, z', z'') -{ 1 }-> mem(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   ifmem(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y''), x, l'), x, l1, 1 + y'' + l') :|: x >= 0, z = 1 + x + l1, l' >= 0, y'' >= 0, z' = 1 + y'' + l', l1 >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y1), x, l''), x, l2, 1 + y1 + l'') :|: y1 >= 0, l'' >= 0, x >= 0, z' = 1 + x + l2, l2 >= 0, z = 1 + y1 + l''
   inter(z, z') -{ 2 }-> ifinter(0, x, l1, 0) :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' = 0
   inter(z, z') -{ 2 }-> ifinter(0, x, l2, 0) :|: x >= 0, z' = 1 + x + l2, z = 0, l2 >= 0
   inter(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   inter(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(eq(z - 1, y'), 1 + (z - 1), l) :|: z' = 1 + (1 + y') + l, z - 1 >= 0, y' >= 0, l >= 0
   mem(z, z') -{ 2 }-> ifmem(1, 0, z' - 1) :|: z' - 1 >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 0, l) :|: x' >= 0, l >= 0, z' = 1 + (1 + x') + l, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 1 + (z - 1), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   mem(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {if}, {app}, {eq}, {ifmem,mem}, {inter,ifinter}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z' + z''

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

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(l1, z') :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' >= 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   ifinter(z, z', z'', z1) -{ 1 }-> inter(z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   ifinter(z, z', z'', z1) -{ 1 }-> 1 + z' + inter(z'', z1) :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0
   ifmem(z, z', z'') -{ 1 }-> mem(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   ifmem(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y''), x, l'), x, l1, 1 + y'' + l') :|: x >= 0, z = 1 + x + l1, l' >= 0, y'' >= 0, z' = 1 + y'' + l', l1 >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y1), x, l''), x, l2, 1 + y1 + l'') :|: y1 >= 0, l'' >= 0, x >= 0, z' = 1 + x + l2, l2 >= 0, z = 1 + y1 + l''
   inter(z, z') -{ 2 }-> ifinter(0, x, l1, 0) :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' = 0
   inter(z, z') -{ 2 }-> ifinter(0, x, l2, 0) :|: x >= 0, z' = 1 + x + l2, z = 0, l2 >= 0
   inter(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   inter(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(eq(z - 1, y'), 1 + (z - 1), l) :|: z' = 1 + (1 + y') + l, z - 1 >= 0, y' >= 0, l >= 0
   mem(z, z') -{ 2 }-> ifmem(1, 0, z' - 1) :|: z' - 1 >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 0, l) :|: x' >= 0, l >= 0, z' = 1 + (1 + x') + l, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 1 + (z - 1), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   mem(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {if}, {app}, {eq}, {ifmem,mem}, {inter,ifinter}
Previous analysis results are:
  if: runtime: ?, size: O(n^1) [z' + z'']

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(l1, z') :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' >= 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   ifinter(z, z', z'', z1) -{ 1 }-> inter(z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   ifinter(z, z', z'', z1) -{ 1 }-> 1 + z' + inter(z'', z1) :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0
   ifmem(z, z', z'') -{ 1 }-> mem(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   ifmem(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y''), x, l'), x, l1, 1 + y'' + l') :|: x >= 0, z = 1 + x + l1, l' >= 0, y'' >= 0, z' = 1 + y'' + l', l1 >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y1), x, l''), x, l2, 1 + y1 + l'') :|: y1 >= 0, l'' >= 0, x >= 0, z' = 1 + x + l2, l2 >= 0, z = 1 + y1 + l''
   inter(z, z') -{ 2 }-> ifinter(0, x, l1, 0) :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' = 0
   inter(z, z') -{ 2 }-> ifinter(0, x, l2, 0) :|: x >= 0, z' = 1 + x + l2, z = 0, l2 >= 0
   inter(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   inter(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(eq(z - 1, y'), 1 + (z - 1), l) :|: z' = 1 + (1 + y') + l, z - 1 >= 0, y' >= 0, l >= 0
   mem(z, z') -{ 2 }-> ifmem(1, 0, z' - 1) :|: z' - 1 >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 0, l) :|: x' >= 0, l >= 0, z' = 1 + (1 + x') + l, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 1 + (z - 1), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   mem(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {app}, {eq}, {ifmem,mem}, {inter,ifinter}
Previous analysis results are:
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']

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

(25) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(l1, z') :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' >= 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   ifinter(z, z', z'', z1) -{ 1 }-> inter(z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   ifinter(z, z', z'', z1) -{ 1 }-> 1 + z' + inter(z'', z1) :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0
   ifmem(z, z', z'') -{ 1 }-> mem(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   ifmem(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y''), x, l'), x, l1, 1 + y'' + l') :|: x >= 0, z = 1 + x + l1, l' >= 0, y'' >= 0, z' = 1 + y'' + l', l1 >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y1), x, l''), x, l2, 1 + y1 + l'') :|: y1 >= 0, l'' >= 0, x >= 0, z' = 1 + x + l2, l2 >= 0, z = 1 + y1 + l''
   inter(z, z') -{ 2 }-> ifinter(0, x, l1, 0) :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' = 0
   inter(z, z') -{ 2 }-> ifinter(0, x, l2, 0) :|: x >= 0, z' = 1 + x + l2, z = 0, l2 >= 0
   inter(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   inter(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(eq(z - 1, y'), 1 + (z - 1), l) :|: z' = 1 + (1 + y') + l, z - 1 >= 0, y' >= 0, l >= 0
   mem(z, z') -{ 2 }-> ifmem(1, 0, z' - 1) :|: z' - 1 >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 0, l) :|: x' >= 0, l >= 0, z' = 1 + (1 + x') + l, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 1 + (z - 1), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   mem(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {app}, {eq}, {ifmem,mem}, {inter,ifinter}
Previous analysis results are:
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: app
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z + z'

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

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(l1, z') :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' >= 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   ifinter(z, z', z'', z1) -{ 1 }-> inter(z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   ifinter(z, z', z'', z1) -{ 1 }-> 1 + z' + inter(z'', z1) :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0
   ifmem(z, z', z'') -{ 1 }-> mem(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   ifmem(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y''), x, l'), x, l1, 1 + y'' + l') :|: x >= 0, z = 1 + x + l1, l' >= 0, y'' >= 0, z' = 1 + y'' + l', l1 >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y1), x, l''), x, l2, 1 + y1 + l'') :|: y1 >= 0, l'' >= 0, x >= 0, z' = 1 + x + l2, l2 >= 0, z = 1 + y1 + l''
   inter(z, z') -{ 2 }-> ifinter(0, x, l1, 0) :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' = 0
   inter(z, z') -{ 2 }-> ifinter(0, x, l2, 0) :|: x >= 0, z' = 1 + x + l2, z = 0, l2 >= 0
   inter(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   inter(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(eq(z - 1, y'), 1 + (z - 1), l) :|: z' = 1 + (1 + y') + l, z - 1 >= 0, y' >= 0, l >= 0
   mem(z, z') -{ 2 }-> ifmem(1, 0, z' - 1) :|: z' - 1 >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 0, l) :|: x' >= 0, l >= 0, z' = 1 + (1 + x') + l, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 1 + (z - 1), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   mem(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {app}, {eq}, {ifmem,mem}, {inter,ifinter}
Previous analysis results are:
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  app: runtime: ?, size: O(n^1) [z + z']

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: app
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z

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

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(l1, z') :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' >= 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   ifinter(z, z', z'', z1) -{ 1 }-> inter(z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   ifinter(z, z', z'', z1) -{ 1 }-> 1 + z' + inter(z'', z1) :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0
   ifmem(z, z', z'') -{ 1 }-> mem(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   ifmem(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y''), x, l'), x, l1, 1 + y'' + l') :|: x >= 0, z = 1 + x + l1, l' >= 0, y'' >= 0, z' = 1 + y'' + l', l1 >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y1), x, l''), x, l2, 1 + y1 + l'') :|: y1 >= 0, l'' >= 0, x >= 0, z' = 1 + x + l2, l2 >= 0, z = 1 + y1 + l''
   inter(z, z') -{ 2 }-> ifinter(0, x, l1, 0) :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' = 0
   inter(z, z') -{ 2 }-> ifinter(0, x, l2, 0) :|: x >= 0, z' = 1 + x + l2, z = 0, l2 >= 0
   inter(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   inter(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(eq(z - 1, y'), 1 + (z - 1), l) :|: z' = 1 + (1 + y') + l, z - 1 >= 0, y' >= 0, l >= 0
   mem(z, z') -{ 2 }-> ifmem(1, 0, z' - 1) :|: z' - 1 >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 0, l) :|: x' >= 0, l >= 0, z' = 1 + (1 + x') + l, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 1 + (z - 1), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   mem(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {eq}, {ifmem,mem}, {inter,ifinter}
Previous analysis results are:
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

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

(31) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + l1 }-> 1 + x + s :|: s >= 0, s <= l1 + z', x >= 0, z = 1 + x + l1, l1 >= 0, z' >= 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   ifinter(z, z', z'', z1) -{ 1 }-> inter(z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   ifinter(z, z', z'', z1) -{ 1 }-> 1 + z' + inter(z'', z1) :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0
   ifmem(z, z', z'') -{ 1 }-> mem(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   ifmem(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y''), x, l'), x, l1, 1 + y'' + l') :|: x >= 0, z = 1 + x + l1, l' >= 0, y'' >= 0, z' = 1 + y'' + l', l1 >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y1), x, l''), x, l2, 1 + y1 + l'') :|: y1 >= 0, l'' >= 0, x >= 0, z' = 1 + x + l2, l2 >= 0, z = 1 + y1 + l''
   inter(z, z') -{ 2 }-> ifinter(0, x, l1, 0) :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' = 0
   inter(z, z') -{ 2 }-> ifinter(0, x, l2, 0) :|: x >= 0, z' = 1 + x + l2, z = 0, l2 >= 0
   inter(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   inter(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(eq(z - 1, y'), 1 + (z - 1), l) :|: z' = 1 + (1 + y') + l, z - 1 >= 0, y' >= 0, l >= 0
   mem(z, z') -{ 2 }-> ifmem(1, 0, z' - 1) :|: z' - 1 >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 0, l) :|: x' >= 0, l >= 0, z' = 1 + (1 + x') + l, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 1 + (z - 1), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   mem(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {eq}, {ifmem,mem}, {inter,ifinter}
Previous analysis results are:
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: eq
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + l1 }-> 1 + x + s :|: s >= 0, s <= l1 + z', x >= 0, z = 1 + x + l1, l1 >= 0, z' >= 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   ifinter(z, z', z'', z1) -{ 1 }-> inter(z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   ifinter(z, z', z'', z1) -{ 1 }-> 1 + z' + inter(z'', z1) :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0
   ifmem(z, z', z'') -{ 1 }-> mem(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   ifmem(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y''), x, l'), x, l1, 1 + y'' + l') :|: x >= 0, z = 1 + x + l1, l' >= 0, y'' >= 0, z' = 1 + y'' + l', l1 >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y1), x, l''), x, l2, 1 + y1 + l'') :|: y1 >= 0, l'' >= 0, x >= 0, z' = 1 + x + l2, l2 >= 0, z = 1 + y1 + l''
   inter(z, z') -{ 2 }-> ifinter(0, x, l1, 0) :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' = 0
   inter(z, z') -{ 2 }-> ifinter(0, x, l2, 0) :|: x >= 0, z' = 1 + x + l2, z = 0, l2 >= 0
   inter(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   inter(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(eq(z - 1, y'), 1 + (z - 1), l) :|: z' = 1 + (1 + y') + l, z - 1 >= 0, y' >= 0, l >= 0
   mem(z, z') -{ 2 }-> ifmem(1, 0, z' - 1) :|: z' - 1 >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 0, l) :|: x' >= 0, l >= 0, z' = 1 + (1 + x') + l, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 1 + (z - 1), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   mem(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {eq}, {ifmem,mem}, {inter,ifinter}
Previous analysis results are:
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  eq: runtime: ?, size: O(1) [1]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: eq
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + z'

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

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + l1 }-> 1 + x + s :|: s >= 0, s <= l1 + z', x >= 0, z = 1 + x + l1, l1 >= 0, z' >= 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   ifinter(z, z', z'', z1) -{ 1 }-> inter(z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   ifinter(z, z', z'', z1) -{ 1 }-> 1 + z' + inter(z'', z1) :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0
   ifmem(z, z', z'') -{ 1 }-> mem(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   ifmem(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y''), x, l'), x, l1, 1 + y'' + l') :|: x >= 0, z = 1 + x + l1, l' >= 0, y'' >= 0, z' = 1 + y'' + l', l1 >= 0
   inter(z, z') -{ 2 }-> ifinter(ifmem(eq(x, y1), x, l''), x, l2, 1 + y1 + l'') :|: y1 >= 0, l'' >= 0, x >= 0, z' = 1 + x + l2, l2 >= 0, z = 1 + y1 + l''
   inter(z, z') -{ 2 }-> ifinter(0, x, l1, 0) :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' = 0
   inter(z, z') -{ 2 }-> ifinter(0, x, l2, 0) :|: x >= 0, z' = 1 + x + l2, z = 0, l2 >= 0
   inter(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   inter(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(eq(z - 1, y'), 1 + (z - 1), l) :|: z' = 1 + (1 + y') + l, z - 1 >= 0, y' >= 0, l >= 0
   mem(z, z') -{ 2 }-> ifmem(1, 0, z' - 1) :|: z' - 1 >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 0, l) :|: x' >= 0, l >= 0, z' = 1 + (1 + x') + l, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 1 + (z - 1), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   mem(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {ifmem,mem}, {inter,ifinter}
Previous analysis results are:
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]

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

(37) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + l1 }-> 1 + x + s :|: s >= 0, s <= l1 + z', x >= 0, z = 1 + x + l1, l1 >= 0, z' >= 0
   eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   ifinter(z, z', z'', z1) -{ 1 }-> inter(z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   ifinter(z, z', z'', z1) -{ 1 }-> 1 + z' + inter(z'', z1) :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0
   ifmem(z, z', z'') -{ 1 }-> mem(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   ifmem(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
   inter(z, z') -{ 5 + y'' }-> ifinter(ifmem(s1, x, l'), x, l1, 1 + y'' + l') :|: s1 >= 0, s1 <= 1, x >= 0, z = 1 + x + l1, l' >= 0, y'' >= 0, z' = 1 + y'' + l', l1 >= 0
   inter(z, z') -{ 5 + y1 }-> ifinter(ifmem(s2, x, l''), x, l2, 1 + y1 + l'') :|: s2 >= 0, s2 <= 1, y1 >= 0, l'' >= 0, x >= 0, z' = 1 + x + l2, l2 >= 0, z = 1 + y1 + l''
   inter(z, z') -{ 2 }-> ifinter(0, x, l1, 0) :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' = 0
   inter(z, z') -{ 2 }-> ifinter(0, x, l2, 0) :|: x >= 0, z' = 1 + x + l2, z = 0, l2 >= 0
   inter(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   inter(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mem(z, z') -{ 5 + y' }-> ifmem(s'', 1 + (z - 1), l) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + y') + l, z - 1 >= 0, y' >= 0, l >= 0
   mem(z, z') -{ 2 }-> ifmem(1, 0, z' - 1) :|: z' - 1 >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 0, l) :|: x' >= 0, l >= 0, z' = 1 + (1 + x') + l, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 1 + (z - 1), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   mem(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {ifmem,mem}, {inter,ifinter}
Previous analysis results are:
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: ifmem
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

Computed SIZE bound using CoFloCo for: mem
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + l1 }-> 1 + x + s :|: s >= 0, s <= l1 + z', x >= 0, z = 1 + x + l1, l1 >= 0, z' >= 0
   eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   ifinter(z, z', z'', z1) -{ 1 }-> inter(z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   ifinter(z, z', z'', z1) -{ 1 }-> 1 + z' + inter(z'', z1) :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0
   ifmem(z, z', z'') -{ 1 }-> mem(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   ifmem(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
   inter(z, z') -{ 5 + y'' }-> ifinter(ifmem(s1, x, l'), x, l1, 1 + y'' + l') :|: s1 >= 0, s1 <= 1, x >= 0, z = 1 + x + l1, l' >= 0, y'' >= 0, z' = 1 + y'' + l', l1 >= 0
   inter(z, z') -{ 5 + y1 }-> ifinter(ifmem(s2, x, l''), x, l2, 1 + y1 + l'') :|: s2 >= 0, s2 <= 1, y1 >= 0, l'' >= 0, x >= 0, z' = 1 + x + l2, l2 >= 0, z = 1 + y1 + l''
   inter(z, z') -{ 2 }-> ifinter(0, x, l1, 0) :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' = 0
   inter(z, z') -{ 2 }-> ifinter(0, x, l2, 0) :|: x >= 0, z' = 1 + x + l2, z = 0, l2 >= 0
   inter(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   inter(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mem(z, z') -{ 5 + y' }-> ifmem(s'', 1 + (z - 1), l) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + y') + l, z - 1 >= 0, y' >= 0, l >= 0
   mem(z, z') -{ 2 }-> ifmem(1, 0, z' - 1) :|: z' - 1 >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 0, l) :|: x' >= 0, l >= 0, z' = 1 + (1 + x') + l, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 1 + (z - 1), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   mem(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {ifmem,mem}, {inter,ifinter}
Previous analysis results are:
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  ifmem: runtime: ?, size: O(1) [1]
  mem: runtime: ?, size: O(1) [1]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: ifmem
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 7 + 11*z''

Computed RUNTIME bound using CoFloCo for: mem
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 13 + 11*z'

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

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + l1 }-> 1 + x + s :|: s >= 0, s <= l1 + z', x >= 0, z = 1 + x + l1, l1 >= 0, z' >= 0
   eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   ifinter(z, z', z'', z1) -{ 1 }-> inter(z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   ifinter(z, z', z'', z1) -{ 1 }-> 1 + z' + inter(z'', z1) :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0
   ifmem(z, z', z'') -{ 1 }-> mem(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   ifmem(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
   inter(z, z') -{ 5 + y'' }-> ifinter(ifmem(s1, x, l'), x, l1, 1 + y'' + l') :|: s1 >= 0, s1 <= 1, x >= 0, z = 1 + x + l1, l' >= 0, y'' >= 0, z' = 1 + y'' + l', l1 >= 0
   inter(z, z') -{ 5 + y1 }-> ifinter(ifmem(s2, x, l''), x, l2, 1 + y1 + l'') :|: s2 >= 0, s2 <= 1, y1 >= 0, l'' >= 0, x >= 0, z' = 1 + x + l2, l2 >= 0, z = 1 + y1 + l''
   inter(z, z') -{ 2 }-> ifinter(0, x, l1, 0) :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' = 0
   inter(z, z') -{ 2 }-> ifinter(0, x, l2, 0) :|: x >= 0, z' = 1 + x + l2, z = 0, l2 >= 0
   inter(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   inter(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mem(z, z') -{ 5 + y' }-> ifmem(s'', 1 + (z - 1), l) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + y') + l, z - 1 >= 0, y' >= 0, l >= 0
   mem(z, z') -{ 2 }-> ifmem(1, 0, z' - 1) :|: z' - 1 >= 0, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 0, l) :|: x' >= 0, l >= 0, z' = 1 + (1 + x') + l, z = 0
   mem(z, z') -{ 2 }-> ifmem(0, 1 + (z - 1), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   mem(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {inter,ifinter}
Previous analysis results are:
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  ifmem: runtime: O(n^1) [7 + 11*z''], size: O(1) [1]
  mem: runtime: O(n^1) [13 + 11*z'], size: O(1) [1]

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

(43) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + l1 }-> 1 + x + s :|: s >= 0, s <= l1 + z', x >= 0, z = 1 + x + l1, l1 >= 0, z' >= 0
   eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   ifinter(z, z', z'', z1) -{ 1 }-> inter(z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   ifinter(z, z', z'', z1) -{ 1 }-> 1 + z' + inter(z'', z1) :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0
   ifmem(z, z', z'') -{ 14 + 11*z'' }-> s7 :|: s7 >= 0, s7 <= 1, z' >= 0, z'' >= 0, z = 0
   ifmem(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
   inter(z, z') -{ 12 + 11*l' + y'' }-> ifinter(s8, x, l1, 1 + y'' + l') :|: s8 >= 0, s8 <= 1, s1 >= 0, s1 <= 1, x >= 0, z = 1 + x + l1, l' >= 0, y'' >= 0, z' = 1 + y'' + l', l1 >= 0
   inter(z, z') -{ 12 + 11*l'' + y1 }-> ifinter(s9, x, l2, 1 + y1 + l'') :|: s9 >= 0, s9 <= 1, s2 >= 0, s2 <= 1, y1 >= 0, l'' >= 0, x >= 0, z' = 1 + x + l2, l2 >= 0, z = 1 + y1 + l''
   inter(z, z') -{ 2 }-> ifinter(0, x, l1, 0) :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' = 0
   inter(z, z') -{ 2 }-> ifinter(0, x, l2, 0) :|: x >= 0, z' = 1 + x + l2, z = 0, l2 >= 0
   inter(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   inter(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mem(z, z') -{ -2 + 11*z' }-> s3 :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z = 0
   mem(z, z') -{ 9 + 11*l }-> s4 :|: s4 >= 0, s4 <= 1, x' >= 0, l >= 0, z' = 1 + (1 + x') + l, z = 0
   mem(z, z') -{ -2 + 11*z' }-> s5 :|: s5 >= 0, s5 <= 1, z' - 1 >= 0, z - 1 >= 0
   mem(z, z') -{ 12 + 11*l + y' }-> s6 :|: s6 >= 0, s6 <= 1, s'' >= 0, s'' <= 1, z' = 1 + (1 + y') + l, z - 1 >= 0, y' >= 0, l >= 0
   mem(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {inter,ifinter}
Previous analysis results are:
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  ifmem: runtime: O(n^1) [7 + 11*z''], size: O(1) [1]
  mem: runtime: O(n^1) [13 + 11*z'], size: O(1) [1]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: inter
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z + z'

Computed SIZE bound using CoFloCo for: ifinter
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z' + z'' + z1

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

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + l1 }-> 1 + x + s :|: s >= 0, s <= l1 + z', x >= 0, z = 1 + x + l1, l1 >= 0, z' >= 0
   eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   ifinter(z, z', z'', z1) -{ 1 }-> inter(z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   ifinter(z, z', z'', z1) -{ 1 }-> 1 + z' + inter(z'', z1) :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0
   ifmem(z, z', z'') -{ 14 + 11*z'' }-> s7 :|: s7 >= 0, s7 <= 1, z' >= 0, z'' >= 0, z = 0
   ifmem(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
   inter(z, z') -{ 12 + 11*l' + y'' }-> ifinter(s8, x, l1, 1 + y'' + l') :|: s8 >= 0, s8 <= 1, s1 >= 0, s1 <= 1, x >= 0, z = 1 + x + l1, l' >= 0, y'' >= 0, z' = 1 + y'' + l', l1 >= 0
   inter(z, z') -{ 12 + 11*l'' + y1 }-> ifinter(s9, x, l2, 1 + y1 + l'') :|: s9 >= 0, s9 <= 1, s2 >= 0, s2 <= 1, y1 >= 0, l'' >= 0, x >= 0, z' = 1 + x + l2, l2 >= 0, z = 1 + y1 + l''
   inter(z, z') -{ 2 }-> ifinter(0, x, l1, 0) :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' = 0
   inter(z, z') -{ 2 }-> ifinter(0, x, l2, 0) :|: x >= 0, z' = 1 + x + l2, z = 0, l2 >= 0
   inter(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   inter(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mem(z, z') -{ -2 + 11*z' }-> s3 :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z = 0
   mem(z, z') -{ 9 + 11*l }-> s4 :|: s4 >= 0, s4 <= 1, x' >= 0, l >= 0, z' = 1 + (1 + x') + l, z = 0
   mem(z, z') -{ -2 + 11*z' }-> s5 :|: s5 >= 0, s5 <= 1, z' - 1 >= 0, z - 1 >= 0
   mem(z, z') -{ 12 + 11*l + y' }-> s6 :|: s6 >= 0, s6 <= 1, s'' >= 0, s'' <= 1, z' = 1 + (1 + y') + l, z - 1 >= 0, y' >= 0, l >= 0
   mem(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {inter,ifinter}
Previous analysis results are:
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  ifmem: runtime: O(n^1) [7 + 11*z''], size: O(1) [1]
  mem: runtime: O(n^1) [13 + 11*z'], size: O(1) [1]
  inter: runtime: ?, size: O(n^1) [z + z']
  ifinter: runtime: ?, size: O(n^1) [1 + z' + z'' + z1]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: inter
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 2 + 60*z + 96*z*z' + 48*z^2 + 60*z' + 48*z'^2

Computed RUNTIME bound using KoAT for: ifinter
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 6 + 120*z'' + 192*z''*z1 + 96*z''^2 + 120*z1 + 96*z1^2

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

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + l1 }-> 1 + x + s :|: s >= 0, s <= l1 + z', x >= 0, z = 1 + x + l1, l1 >= 0, z' >= 0
   eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   ifinter(z, z', z'', z1) -{ 1 }-> inter(z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   ifinter(z, z', z'', z1) -{ 1 }-> 1 + z' + inter(z'', z1) :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0
   ifmem(z, z', z'') -{ 14 + 11*z'' }-> s7 :|: s7 >= 0, s7 <= 1, z' >= 0, z'' >= 0, z = 0
   ifmem(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
   inter(z, z') -{ 12 + 11*l' + y'' }-> ifinter(s8, x, l1, 1 + y'' + l') :|: s8 >= 0, s8 <= 1, s1 >= 0, s1 <= 1, x >= 0, z = 1 + x + l1, l' >= 0, y'' >= 0, z' = 1 + y'' + l', l1 >= 0
   inter(z, z') -{ 12 + 11*l'' + y1 }-> ifinter(s9, x, l2, 1 + y1 + l'') :|: s9 >= 0, s9 <= 1, s2 >= 0, s2 <= 1, y1 >= 0, l'' >= 0, x >= 0, z' = 1 + x + l2, l2 >= 0, z = 1 + y1 + l''
   inter(z, z') -{ 2 }-> ifinter(0, x, l1, 0) :|: x >= 0, z = 1 + x + l1, l1 >= 0, z' = 0
   inter(z, z') -{ 2 }-> ifinter(0, x, l2, 0) :|: x >= 0, z' = 1 + x + l2, z = 0, l2 >= 0
   inter(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   inter(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mem(z, z') -{ -2 + 11*z' }-> s3 :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z = 0
   mem(z, z') -{ 9 + 11*l }-> s4 :|: s4 >= 0, s4 <= 1, x' >= 0, l >= 0, z' = 1 + (1 + x') + l, z = 0
   mem(z, z') -{ -2 + 11*z' }-> s5 :|: s5 >= 0, s5 <= 1, z' - 1 >= 0, z - 1 >= 0
   mem(z, z') -{ 12 + 11*l + y' }-> s6 :|: s6 >= 0, s6 <= 1, s'' >= 0, s'' <= 1, z' = 1 + (1 + y') + l, z - 1 >= 0, y' >= 0, l >= 0
   mem(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: 
Previous analysis results are:
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  ifmem: runtime: O(n^1) [7 + 11*z''], size: O(1) [1]
  mem: runtime: O(n^1) [13 + 11*z'], size: O(1) [1]
  inter: runtime: O(n^2) [2 + 60*z + 96*z*z' + 48*z^2 + 60*z' + 48*z'^2], size: O(n^1) [z + z']
  ifinter: runtime: O(n^2) [6 + 120*z'' + 192*z''*z1 + 96*z''^2 + 120*z1 + 96*z1^2], size: O(n^1) [1 + z' + z'' + z1]

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

(49) FinalProof (FINISHED)
Computed overall runtime complexity
----------------------------------------

(50)
BOUNDS(1, n^2)

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

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

(52)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   if(true, x, y) -> x
   if(false, x, y) -> y
   eq(0', 0') -> true
   eq(0', s(x)) -> false
   eq(s(x), 0') -> false
   eq(s(x), s(y)) -> eq(x, y)
   app(nil, l) -> l
   app(cons(x, l1), l2) -> cons(x, app(l1, l2))
   app(app(l1, l2), l3) -> app(l1, app(l2, l3))
   mem(x, nil) -> false
   mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l)
   ifmem(true, x, l) -> true
   ifmem(false, x, l) -> mem(x, l)
   inter(x, nil) -> nil
   inter(nil, x) -> nil
   inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
   inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
   inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
   inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
   ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
   ifinter(false, x, l1, l2) -> inter(l1, l2)

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

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

(54)
Obligation:
TRS:
Rules:
if(true, x, y) -> x
if(false, x, y) -> y
eq(0', 0') -> true
eq(0', s(x)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
app(app(l1, l2), l3) -> app(l1, app(l2, l3))
mem(x, nil) -> false
mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l)
ifmem(true, x, l) -> true
ifmem(false, x, l) -> mem(x, l)
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

Types:
if :: true:false -> if -> if -> if
true :: true:false
false :: true:false
eq :: 0':s -> 0':s -> true:false
0' :: 0':s
s :: 0':s -> 0':s
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
mem :: 0':s -> nil:cons -> true:false
ifmem :: true:false -> 0':s -> nil:cons -> true:false
inter :: nil:cons -> nil:cons -> nil:cons
ifinter :: true:false -> 0':s -> nil:cons -> nil:cons -> nil:cons
hole_if1_0 :: if
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
hole_nil:cons4_0 :: nil:cons
gen_0':s5_0 :: Nat -> 0':s
gen_nil:cons6_0 :: Nat -> nil:cons

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

(55) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
eq, app, mem, inter

They will be analysed ascendingly in the following order:
eq < mem
app < inter
mem < inter

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

(56)
Obligation:
TRS:
Rules:
if(true, x, y) -> x
if(false, x, y) -> y
eq(0', 0') -> true
eq(0', s(x)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
app(app(l1, l2), l3) -> app(l1, app(l2, l3))
mem(x, nil) -> false
mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l)
ifmem(true, x, l) -> true
ifmem(false, x, l) -> mem(x, l)
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

Types:
if :: true:false -> if -> if -> if
true :: true:false
false :: true:false
eq :: 0':s -> 0':s -> true:false
0' :: 0':s
s :: 0':s -> 0':s
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
mem :: 0':s -> nil:cons -> true:false
ifmem :: true:false -> 0':s -> nil:cons -> true:false
inter :: nil:cons -> nil:cons -> nil:cons
ifinter :: true:false -> 0':s -> nil:cons -> nil:cons -> nil:cons
hole_if1_0 :: if
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
hole_nil:cons4_0 :: nil:cons
gen_0':s5_0 :: Nat -> 0':s
gen_nil:cons6_0 :: Nat -> nil:cons


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:cons6_0(0) <=> nil
gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x))


The following defined symbols remain to be analysed:
eq, app, mem, inter

They will be analysed ascendingly in the following order:
eq < mem
app < inter
mem < inter

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

(57) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
eq(gen_0':s5_0(n8_0), gen_0':s5_0(n8_0)) -> true, rt in Omega(1 + n8_0)

Induction Base:
eq(gen_0':s5_0(0), gen_0':s5_0(0)) ->_R^Omega(1)
true

Induction Step:
eq(gen_0':s5_0(+(n8_0, 1)), gen_0':s5_0(+(n8_0, 1))) ->_R^Omega(1)
eq(gen_0':s5_0(n8_0), gen_0':s5_0(n8_0)) ->_IH
true

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

(58)
Complex Obligation (BEST)

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

(59)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
if(true, x, y) -> x
if(false, x, y) -> y
eq(0', 0') -> true
eq(0', s(x)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
app(app(l1, l2), l3) -> app(l1, app(l2, l3))
mem(x, nil) -> false
mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l)
ifmem(true, x, l) -> true
ifmem(false, x, l) -> mem(x, l)
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

Types:
if :: true:false -> if -> if -> if
true :: true:false
false :: true:false
eq :: 0':s -> 0':s -> true:false
0' :: 0':s
s :: 0':s -> 0':s
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
mem :: 0':s -> nil:cons -> true:false
ifmem :: true:false -> 0':s -> nil:cons -> true:false
inter :: nil:cons -> nil:cons -> nil:cons
ifinter :: true:false -> 0':s -> nil:cons -> nil:cons -> nil:cons
hole_if1_0 :: if
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
hole_nil:cons4_0 :: nil:cons
gen_0':s5_0 :: Nat -> 0':s
gen_nil:cons6_0 :: Nat -> nil:cons


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:cons6_0(0) <=> nil
gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x))


The following defined symbols remain to be analysed:
eq, app, mem, inter

They will be analysed ascendingly in the following order:
eq < mem
app < inter
mem < inter

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

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

(61)
BOUNDS(n^1, INF)

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

(62)
Obligation:
TRS:
Rules:
if(true, x, y) -> x
if(false, x, y) -> y
eq(0', 0') -> true
eq(0', s(x)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
app(app(l1, l2), l3) -> app(l1, app(l2, l3))
mem(x, nil) -> false
mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l)
ifmem(true, x, l) -> true
ifmem(false, x, l) -> mem(x, l)
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

Types:
if :: true:false -> if -> if -> if
true :: true:false
false :: true:false
eq :: 0':s -> 0':s -> true:false
0' :: 0':s
s :: 0':s -> 0':s
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
mem :: 0':s -> nil:cons -> true:false
ifmem :: true:false -> 0':s -> nil:cons -> true:false
inter :: nil:cons -> nil:cons -> nil:cons
ifinter :: true:false -> 0':s -> nil:cons -> nil:cons -> nil:cons
hole_if1_0 :: if
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
hole_nil:cons4_0 :: nil:cons
gen_0':s5_0 :: Nat -> 0':s
gen_nil:cons6_0 :: Nat -> nil:cons


Lemmas:
eq(gen_0':s5_0(n8_0), gen_0':s5_0(n8_0)) -> true, rt in Omega(1 + n8_0)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:cons6_0(0) <=> nil
gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x))


The following defined symbols remain to be analysed:
app, mem, inter

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

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

(63) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
app(gen_nil:cons6_0(n537_0), gen_nil:cons6_0(b)) -> gen_nil:cons6_0(+(n537_0, b)), rt in Omega(1 + n537_0)

Induction Base:
app(gen_nil:cons6_0(0), gen_nil:cons6_0(b)) ->_R^Omega(1)
gen_nil:cons6_0(b)

Induction Step:
app(gen_nil:cons6_0(+(n537_0, 1)), gen_nil:cons6_0(b)) ->_R^Omega(1)
cons(0', app(gen_nil:cons6_0(n537_0), gen_nil:cons6_0(b))) ->_IH
cons(0', gen_nil:cons6_0(+(b, c538_0)))

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

(64)
Obligation:
TRS:
Rules:
if(true, x, y) -> x
if(false, x, y) -> y
eq(0', 0') -> true
eq(0', s(x)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
app(app(l1, l2), l3) -> app(l1, app(l2, l3))
mem(x, nil) -> false
mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l)
ifmem(true, x, l) -> true
ifmem(false, x, l) -> mem(x, l)
inter(x, nil) -> nil
inter(nil, x) -> nil
inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
ifinter(false, x, l1, l2) -> inter(l1, l2)

Types:
if :: true:false -> if -> if -> if
true :: true:false
false :: true:false
eq :: 0':s -> 0':s -> true:false
0' :: 0':s
s :: 0':s -> 0':s
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
mem :: 0':s -> nil:cons -> true:false
ifmem :: true:false -> 0':s -> nil:cons -> true:false
inter :: nil:cons -> nil:cons -> nil:cons
ifinter :: true:false -> 0':s -> nil:cons -> nil:cons -> nil:cons
hole_if1_0 :: if
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
hole_nil:cons4_0 :: nil:cons
gen_0':s5_0 :: Nat -> 0':s
gen_nil:cons6_0 :: Nat -> nil:cons


Lemmas:
eq(gen_0':s5_0(n8_0), gen_0':s5_0(n8_0)) -> true, rt in Omega(1 + n8_0)
app(gen_nil:cons6_0(n537_0), gen_nil:cons6_0(b)) -> gen_nil:cons6_0(+(n537_0, b)), rt in Omega(1 + n537_0)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:cons6_0(0) <=> nil
gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x))


The following defined symbols remain to be analysed:
mem, inter

They will be analysed ascendingly in the following order:
mem < inter