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


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


WORST_CASE(Omega(n^1), O(n^1))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1).

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) CpxWeightedTrsRenamingProof [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), 292 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 110 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 868 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 456 ms]
    (30) CpxRNTS
    (31) FinalProof [FINISHED, 0 ms]
    (32) BOUNDS(1, n^1)
(33) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(34) CpxTRS
    (35) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) typed CpxTrs
    (37) OrderProof [LOWER BOUND(ID), 0 ms]
    (38) typed CpxTrs
    (39) RewriteLemmaProof [LOWER BOUND(ID), 364 ms]
    (40) proven lower bound
    (41) LowerBoundPropagationProof [FINISHED, 0 ms]
    (42) BOUNDS(n^1, INF)


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

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


The TRS R consists of the following rules:

   del(.(x, .(y, z))) -> f(=(x, y), x, y, z)
   f(true, x, y, z) -> del(.(y, z))
   f(false, x, y, z) -> .(x, del(.(y, z)))
   =(nil, nil) -> true
   =(.(x, y), nil) -> false
   =(nil, .(y, z)) -> false
   =(.(x, y), .(u, v)) -> and(=(x, u), =(y, v))

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

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

The duplicating contexts are:
del(.([], .(y, z)))
del(.(x, .([], z)))


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


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

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


The TRS R consists of the following rules:

   del(.(x, .(y, z))) -> f(=(x, y), x, y, z)
   f(true, x, y, z) -> del(.(y, z))
   f(false, x, y, z) -> .(x, del(.(y, z)))
   =(nil, nil) -> true
   =(.(x, y), nil) -> false
   =(nil, .(y, z)) -> false
   =(.(x, y), .(u, v)) -> and(=(x, u), =(y, v))

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

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

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


The TRS R consists of the following rules:

   del(.(x, .(y, z))) -> f(=(x, y), x, y, z) [1]
   f(true, x, y, z) -> del(.(y, z)) [1]
   f(false, x, y, z) -> .(x, del(.(y, z))) [1]
   =(nil, nil) -> true [1]
   =(.(x, y), nil) -> false [1]
   =(nil, .(y, z)) -> false [1]
   =(.(x, y), .(u, v)) -> and(=(x, u), =(y, v)) [1]

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

(5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

= => eq

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

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


The TRS R consists of the following rules:

   del(.(x, .(y, z))) -> f(eq(x, y), x, y, z) [1]
   f(true, x, y, z) -> del(.(y, z)) [1]
   f(false, x, y, z) -> .(x, del(.(y, z))) [1]
   eq(nil, nil) -> true [1]
   eq(.(x, y), nil) -> false [1]
   eq(nil, .(y, z)) -> false [1]
   eq(.(x, y), .(u, v)) -> and(eq(x, u), eq(y, v)) [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:

   del(.(x, .(y, z))) -> f(eq(x, y), x, y, z) [1]
   f(true, x, y, z) -> del(.(y, z)) [1]
   f(false, x, y, z) -> .(x, del(.(y, z))) [1]
   eq(nil, nil) -> true [1]
   eq(.(x, y), nil) -> false [1]
   eq(nil, .(y, z)) -> false [1]
   eq(.(x, y), .(u, v)) -> and(eq(x, u), eq(y, v)) [1]

The TRS has the following type information:
del :: .:nil:u:v -> .:nil:u:v
. :: .:nil:u:v -> .:nil:u:v -> .:nil:u:v
f :: true:false:and -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v
eq :: .:nil:u:v -> .:nil:u:v -> true:false:and
true :: true:false:and
false :: true:false:and
nil :: .:nil:u:v
u :: .:nil:u:v
v :: .:nil:u:v
and :: true:false:and -> true:false:and -> true:false:and

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:

   del_1
   f_4

(c) The following functions are completely defined:

   eq_2

Due to the following rules being added:

   eq(v0, v1) -> null_eq [0]

And the following fresh constants:    null_eq

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

(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:

   del(.(x, .(y, z))) -> f(eq(x, y), x, y, z) [1]
   f(true, x, y, z) -> del(.(y, z)) [1]
   f(false, x, y, z) -> .(x, del(.(y, z))) [1]
   eq(nil, nil) -> true [1]
   eq(.(x, y), nil) -> false [1]
   eq(nil, .(y, z)) -> false [1]
   eq(.(x, y), .(u, v)) -> and(eq(x, u), eq(y, v)) [1]
   eq(v0, v1) -> null_eq [0]

The TRS has the following type information:
del :: .:nil:u:v -> .:nil:u:v
. :: .:nil:u:v -> .:nil:u:v -> .:nil:u:v
f :: true:false:and:null_eq -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v
eq :: .:nil:u:v -> .:nil:u:v -> true:false:and:null_eq
true :: true:false:and:null_eq
false :: true:false:and:null_eq
nil :: .:nil:u:v
u :: .:nil:u:v
v :: .:nil:u:v
and :: true:false:and:null_eq -> true:false:and:null_eq -> true:false:and:null_eq
null_eq :: true:false:and:null_eq

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:

   del(.(nil, .(nil, z))) -> f(true, nil, nil, z) [2]
   del(.(.(x', y'), .(nil, z))) -> f(false, .(x', y'), nil, z) [2]
   del(.(nil, .(.(y'', z'), z))) -> f(false, nil, .(y'', z'), z) [2]
   del(.(.(x'', y1), .(.(u, v), z))) -> f(and(eq(x'', u), eq(y1, v)), .(x'', y1), .(u, v), z) [2]
   del(.(x, .(y, z))) -> f(null_eq, x, y, z) [1]
   f(true, x, y, z) -> del(.(y, z)) [1]
   f(false, x, y, z) -> .(x, del(.(y, z))) [1]
   eq(nil, nil) -> true [1]
   eq(.(x, y), nil) -> false [1]
   eq(nil, .(y, z)) -> false [1]
   eq(.(x, y), .(u, v)) -> and(eq(x, u), eq(y, v)) [1]
   eq(v0, v1) -> null_eq [0]

The TRS has the following type information:
del :: .:nil:u:v -> .:nil:u:v
. :: .:nil:u:v -> .:nil:u:v -> .:nil:u:v
f :: true:false:and:null_eq -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v
eq :: .:nil:u:v -> .:nil:u:v -> true:false:and:null_eq
true :: true:false:and:null_eq
false :: true:false:and:null_eq
nil :: .:nil:u:v
u :: .:nil:u:v
v :: .:nil:u:v
and :: true:false:and:null_eq -> true:false:and:null_eq -> true:false:and:null_eq
null_eq :: true:false:and:null_eq

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
   nil => 0
   u => 1
   v => 2
   null_eq => 0

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

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

   del(z'') -{ 2 }-> f(1, 0, 0, z) :|: z >= 0, z'' = 1 + 0 + (1 + 0 + z)
   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
   del(z'') -{ 2 }-> f(1 + eq(x'', 1) + eq(y1, 2), 1 + x'' + y1, 1 + 1 + 2, z) :|: y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   eq(z'', z1) -{ 1 }-> 1 + eq(x, 1) + eq(y, 2) :|: z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
   f(z'', z1, z2, z3) -{ 1 }-> del(1 + y + z) :|: z2 = y, z >= 0, z3 = z, x >= 0, y >= 0, z'' = 1, z1 = x
   f(z'', z1, z2, z3) -{ 1 }-> 1 + x + del(1 + y + z) :|: z'' = 0, z2 = y, z >= 0, z3 = z, x >= 0, y >= 0, z1 = x


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

(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:

   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
   del(z'') -{ 2 }-> f(1 + eq(x'', 1) + eq(y1, 2), 1 + x'' + y1, 1 + 1 + 2, z) :|: y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   eq(z'', z1) -{ 1 }-> 1 + eq(x, 1) + eq(y, 2) :|: z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0


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

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

   { eq }
   { del, f }

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

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

   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
   del(z'') -{ 2 }-> f(1 + eq(x'', 1) + eq(y1, 2), 1 + x'' + y1, 1 + 1 + 2, z) :|: y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   eq(z'', z1) -{ 1 }-> 1 + eq(x, 1) + eq(y, 2) :|: z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0

Function symbols to be analyzed: {eq}, {del,f}

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

(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:

   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
   del(z'') -{ 2 }-> f(1 + eq(x'', 1) + eq(y1, 2), 1 + x'' + y1, 1 + 1 + 2, z) :|: y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   eq(z'', z1) -{ 1 }-> 1 + eq(x, 1) + eq(y, 2) :|: z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0

Function symbols to be analyzed: {eq}, {del,f}

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

(21) 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

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

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

   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
   del(z'') -{ 2 }-> f(1 + eq(x'', 1) + eq(y1, 2), 1 + x'' + y1, 1 + 1 + 2, z) :|: y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   eq(z'', z1) -{ 1 }-> 1 + eq(x, 1) + eq(y, 2) :|: z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0

Function symbols to be analyzed: {eq}, {del,f}
Previous analysis results are:
  eq: runtime: ?, size: O(1) [1]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: eq
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 3

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

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

   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
   del(z'') -{ 2 }-> f(1 + eq(x'', 1) + eq(y1, 2), 1 + x'' + y1, 1 + 1 + 2, z) :|: y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   eq(z'', z1) -{ 1 }-> 1 + eq(x, 1) + eq(y, 2) :|: z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0

Function symbols to be analyzed: {del,f}
Previous analysis results are:
  eq: runtime: O(1) [3], size: O(1) [1]

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

(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:

   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
   del(z'') -{ 8 }-> f(1 + s + s', 1 + x'' + y1, 1 + 1 + 2, z) :|: s >= 0, s <= 1, s' >= 0, s' <= 1, y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   eq(z'', z1) -{ 7 }-> 1 + s'' + s1 :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0

Function symbols to be analyzed: {del,f}
Previous analysis results are:
  eq: runtime: O(1) [3], size: O(1) [1]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: del
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

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

   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
   del(z'') -{ 8 }-> f(1 + s + s', 1 + x'' + y1, 1 + 1 + 2, z) :|: s >= 0, s <= 1, s' >= 0, s' <= 1, y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   eq(z'', z1) -{ 7 }-> 1 + s'' + s1 :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0

Function symbols to be analyzed: {del,f}
Previous analysis results are:
  eq: runtime: O(1) [3], size: O(1) [1]
  del: runtime: ?, size: O(1) [0]
  f: runtime: ?, size: O(n^1) [1 + z1]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: del
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 34*z''

Computed RUNTIME bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 35 + 34*z2 + 34*z3

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

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

   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
   del(z'') -{ 8 }-> f(1 + s + s', 1 + x'' + y1, 1 + 1 + 2, z) :|: s >= 0, s <= 1, s' >= 0, s' <= 1, y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   eq(z'', z1) -{ 7 }-> 1 + s'' + s1 :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  eq: runtime: O(1) [3], size: O(1) [1]
  del: runtime: O(n^1) [34*z''], size: O(1) [0]
  f: runtime: O(n^1) [35 + 34*z2 + 34*z3], size: O(n^1) [1 + z1]

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

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

(32)
BOUNDS(1, n^1)

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

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

(34)
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:

   del(.(x, .(y, z))) -> f(='(x, y), x, y, z)
   f(true, x, y, z) -> del(.(y, z))
   f(false, x, y, z) -> .(x, del(.(y, z)))
   ='(nil, nil) -> true
   ='(.(x, y), nil) -> false
   ='(nil, .(y, z)) -> false
   ='(.(x, y), .(u, v)) -> and(='(x, u), ='(y, v))

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

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

(36)
Obligation:
TRS:
Rules:
del(.(x, .(y, z))) -> f(='(x, y), x, y, z)
f(true, x, y, z) -> del(.(y, z))
f(false, x, y, z) -> .(x, del(.(y, z)))
='(nil, nil) -> true
='(.(x, y), nil) -> false
='(nil, .(y, z)) -> false
='(.(x, y), .(u, v)) -> and(='(x, u), ='(y, v))

Types:
del :: .:nil:u:v -> .:nil:u:v
. :: .:nil:u:v -> .:nil:u:v -> .:nil:u:v
f :: true:false:and -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v
=' :: .:nil:u:v -> .:nil:u:v -> true:false:and
true :: true:false:and
false :: true:false:and
nil :: .:nil:u:v
u :: .:nil:u:v
v :: .:nil:u:v
and :: true:false:and -> true:false:and -> true:false:and
hole_.:nil:u:v1_0 :: .:nil:u:v
hole_true:false:and2_0 :: true:false:and
gen_.:nil:u:v3_0 :: Nat -> .:nil:u:v
gen_true:false:and4_0 :: Nat -> true:false:and

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

(37) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
del, ='

They will be analysed ascendingly in the following order:
=' < del

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

(38)
Obligation:
TRS:
Rules:
del(.(x, .(y, z))) -> f(='(x, y), x, y, z)
f(true, x, y, z) -> del(.(y, z))
f(false, x, y, z) -> .(x, del(.(y, z)))
='(nil, nil) -> true
='(.(x, y), nil) -> false
='(nil, .(y, z)) -> false
='(.(x, y), .(u, v)) -> and(='(x, u), ='(y, v))

Types:
del :: .:nil:u:v -> .:nil:u:v
. :: .:nil:u:v -> .:nil:u:v -> .:nil:u:v
f :: true:false:and -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v
=' :: .:nil:u:v -> .:nil:u:v -> true:false:and
true :: true:false:and
false :: true:false:and
nil :: .:nil:u:v
u :: .:nil:u:v
v :: .:nil:u:v
and :: true:false:and -> true:false:and -> true:false:and
hole_.:nil:u:v1_0 :: .:nil:u:v
hole_true:false:and2_0 :: true:false:and
gen_.:nil:u:v3_0 :: Nat -> .:nil:u:v
gen_true:false:and4_0 :: Nat -> true:false:and


Generator Equations:
gen_.:nil:u:v3_0(0) <=> nil
gen_.:nil:u:v3_0(+(x, 1)) <=> .(nil, gen_.:nil:u:v3_0(x))
gen_true:false:and4_0(0) <=> false
gen_true:false:and4_0(+(x, 1)) <=> and(false, gen_true:false:and4_0(x))


The following defined symbols remain to be analysed:
=', del

They will be analysed ascendingly in the following order:
=' < del

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

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
del(gen_.:nil:u:v3_0(+(2, n56_0))) -> *5_0, rt in Omega(n56_0)

Induction Base:
del(gen_.:nil:u:v3_0(+(2, 0)))

Induction Step:
del(gen_.:nil:u:v3_0(+(2, +(n56_0, 1)))) ->_R^Omega(1)
f(='(nil, nil), nil, nil, gen_.:nil:u:v3_0(+(1, n56_0))) ->_R^Omega(1)
f(true, nil, nil, gen_.:nil:u:v3_0(+(1, n56_0))) ->_R^Omega(1)
del(.(nil, gen_.:nil:u:v3_0(+(1, n56_0)))) ->_IH
*5_0

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

(40)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
del(.(x, .(y, z))) -> f(='(x, y), x, y, z)
f(true, x, y, z) -> del(.(y, z))
f(false, x, y, z) -> .(x, del(.(y, z)))
='(nil, nil) -> true
='(.(x, y), nil) -> false
='(nil, .(y, z)) -> false
='(.(x, y), .(u, v)) -> and(='(x, u), ='(y, v))

Types:
del :: .:nil:u:v -> .:nil:u:v
. :: .:nil:u:v -> .:nil:u:v -> .:nil:u:v
f :: true:false:and -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v
=' :: .:nil:u:v -> .:nil:u:v -> true:false:and
true :: true:false:and
false :: true:false:and
nil :: .:nil:u:v
u :: .:nil:u:v
v :: .:nil:u:v
and :: true:false:and -> true:false:and -> true:false:and
hole_.:nil:u:v1_0 :: .:nil:u:v
hole_true:false:and2_0 :: true:false:and
gen_.:nil:u:v3_0 :: Nat -> .:nil:u:v
gen_true:false:and4_0 :: Nat -> true:false:and


Generator Equations:
gen_.:nil:u:v3_0(0) <=> nil
gen_.:nil:u:v3_0(+(x, 1)) <=> .(nil, gen_.:nil:u:v3_0(x))
gen_true:false:and4_0(0) <=> false
gen_true:false:and4_0(+(x, 1)) <=> and(false, gen_true:false:and4_0(x))


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

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

(42)
BOUNDS(n^1, INF)