WORST_CASE(?,O(n^1))
proof of input_OXf4v3QZwr.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
(3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CdtProblem
(5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
(7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CdtProblem
(9) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxRelTRS
(11) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxWeightedTrs
(13) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxWeightedTrs
(15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxTypedWeightedTrs
(17) CompletionProof [UPPER BOUND(ID), 0 ms]
(18) CpxTypedWeightedCompleteTrs
(19) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CpxTypedWeightedCompleteTrs
(21) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(22) CpxRNTS
(23) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(24) CpxRNTS
(25) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(26) CpxRNTS
(27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 266 ms]
(30) CpxRNTS
(31) IntTrsBoundProof [UPPER BOUND(ID), 119 ms]
(32) CpxRNTS
(33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(34) CpxRNTS
(35) IntTrsBoundProof [UPPER BOUND(ID), 780 ms]
(36) CpxRNTS
(37) IntTrsBoundProof [UPPER BOUND(ID), 422 ms]
(38) CpxRNTS
(39) FinalProof [FINISHED, 0 ms]
(40) BOUNDS(1, n^1)


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

(0)
Obligation:
The Runtime Complexity (parallel-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: PARALLEL_INNERMOST
----------------------------------------

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(.(z0, .(z1, z2))) -> f(=(z0, z1), z0, z1, z2)
   f(true, z0, z1, z2) -> del(.(z1, z2))
   f(false, z0, z1, z2) -> .(z0, del(.(z1, z2)))
   =(nil, nil) -> true
   =(.(z0, z1), nil) -> false
   =(nil, .(z0, z1)) -> false
   =(.(z0, z1), .(u, v)) -> and(=(z0, u), =(z1, v))
Tuples:
   DEL(.(z0, .(z1, z2))) -> c(F(=(z0, z1), z0, z1, z2), ='(z0, z1))
   F(true, z0, z1, z2) -> c1(DEL(.(z1, z2)))
   F(false, z0, z1, z2) -> c2(DEL(.(z1, z2)))
   ='(nil, nil) -> c3
   ='(.(z0, z1), nil) -> c4
   ='(nil, .(z0, z1)) -> c5
   ='(.(z0, z1), .(u, v)) -> c6(='(z0, u))
   ='(.(z0, z1), .(u, v)) -> c7(='(z1, v))
S tuples:
   DEL(.(z0, .(z1, z2))) -> c(F(=(z0, z1), z0, z1, z2), ='(z0, z1))
   F(true, z0, z1, z2) -> c1(DEL(.(z1, z2)))
   F(false, z0, z1, z2) -> c2(DEL(.(z1, z2)))
   ='(nil, nil) -> c3
   ='(.(z0, z1), nil) -> c4
   ='(nil, .(z0, z1)) -> c5
   ='(.(z0, z1), .(u, v)) -> c6(='(z0, u))
   ='(.(z0, z1), .(u, v)) -> c7(='(z1, v))
K tuples:none
Defined Rule Symbols:   del_1, f_4, =_2

Defined Pair Symbols:   DEL_1, F_4, ='_2

Compound Symbols:   c_2, c1_1, c2_1, c3, c4, c5, c6_1, c7_1


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing nodes:
   ='(.(z0, z1), .(u, v)) -> c6(='(z0, u))
   ='(nil, .(z0, z1)) -> c5
   ='(.(z0, z1), .(u, v)) -> c7(='(z1, v))
   ='(nil, nil) -> c3
   ='(.(z0, z1), nil) -> c4

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(.(z0, .(z1, z2))) -> f(=(z0, z1), z0, z1, z2)
   f(true, z0, z1, z2) -> del(.(z1, z2))
   f(false, z0, z1, z2) -> .(z0, del(.(z1, z2)))
   =(nil, nil) -> true
   =(.(z0, z1), nil) -> false
   =(nil, .(z0, z1)) -> false
   =(.(z0, z1), .(u, v)) -> and(=(z0, u), =(z1, v))
Tuples:
   DEL(.(z0, .(z1, z2))) -> c(F(=(z0, z1), z0, z1, z2), ='(z0, z1))
   F(true, z0, z1, z2) -> c1(DEL(.(z1, z2)))
   F(false, z0, z1, z2) -> c2(DEL(.(z1, z2)))
S tuples:
   DEL(.(z0, .(z1, z2))) -> c(F(=(z0, z1), z0, z1, z2), ='(z0, z1))
   F(true, z0, z1, z2) -> c1(DEL(.(z1, z2)))
   F(false, z0, z1, z2) -> c2(DEL(.(z1, z2)))
K tuples:none
Defined Rule Symbols:   del_1, f_4, =_2

Defined Pair Symbols:   DEL_1, F_4

Compound Symbols:   c_2, c1_1, c2_1


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

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(.(z0, .(z1, z2))) -> f(=(z0, z1), z0, z1, z2)
   f(true, z0, z1, z2) -> del(.(z1, z2))
   f(false, z0, z1, z2) -> .(z0, del(.(z1, z2)))
   =(nil, nil) -> true
   =(.(z0, z1), nil) -> false
   =(nil, .(z0, z1)) -> false
   =(.(z0, z1), .(u, v)) -> and(=(z0, u), =(z1, v))
Tuples:
   F(true, z0, z1, z2) -> c1(DEL(.(z1, z2)))
   F(false, z0, z1, z2) -> c2(DEL(.(z1, z2)))
   DEL(.(z0, .(z1, z2))) -> c(F(=(z0, z1), z0, z1, z2))
S tuples:
   F(true, z0, z1, z2) -> c1(DEL(.(z1, z2)))
   F(false, z0, z1, z2) -> c2(DEL(.(z1, z2)))
   DEL(.(z0, .(z1, z2))) -> c(F(=(z0, z1), z0, z1, z2))
K tuples:none
Defined Rule Symbols:   del_1, f_4, =_2

Defined Pair Symbols:   F_4, DEL_1

Compound Symbols:   c1_1, c2_1, c_1


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   del(.(z0, .(z1, z2))) -> f(=(z0, z1), z0, z1, z2)
   f(true, z0, z1, z2) -> del(.(z1, z2))
   f(false, z0, z1, z2) -> .(z0, del(.(z1, z2)))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   =(nil, nil) -> true
   =(.(z0, z1), nil) -> false
   =(nil, .(z0, z1)) -> false
   =(.(z0, z1), .(u, v)) -> and(=(z0, u), =(z1, v))
Tuples:
   F(true, z0, z1, z2) -> c1(DEL(.(z1, z2)))
   F(false, z0, z1, z2) -> c2(DEL(.(z1, z2)))
   DEL(.(z0, .(z1, z2))) -> c(F(=(z0, z1), z0, z1, z2))
S tuples:
   F(true, z0, z1, z2) -> c1(DEL(.(z1, z2)))
   F(false, z0, z1, z2) -> c2(DEL(.(z1, z2)))
   DEL(.(z0, .(z1, z2))) -> c(F(=(z0, z1), z0, z1, z2))
K tuples:none
Defined Rule Symbols:   =_2

Defined Pair Symbols:   F_4, DEL_1

Compound Symbols:   c1_1, c2_1, c_1


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

(9) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   F(true, z0, z1, z2) -> c1(DEL(.(z1, z2)))
   F(false, z0, z1, z2) -> c2(DEL(.(z1, z2)))
   DEL(.(z0, .(z1, z2))) -> c(F(=(z0, z1), z0, z1, z2))

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

   =(nil, nil) -> true
   =(.(z0, z1), nil) -> false
   =(nil, .(z0, z1)) -> false
   =(.(z0, z1), .(u, v)) -> and(=(z0, u), =(z1, v))

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

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

(12)
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:

   F(true, z0, z1, z2) -> c1(DEL(.(z1, z2))) [1]
   F(false, z0, z1, z2) -> c2(DEL(.(z1, z2))) [1]
   DEL(.(z0, .(z1, z2))) -> c(F(=(z0, z1), z0, z1, z2)) [1]
   =(nil, nil) -> true [0]
   =(.(z0, z1), nil) -> false [0]
   =(nil, .(z0, z1)) -> false [0]
   =(.(z0, z1), .(u, v)) -> and(=(z0, u), =(z1, v)) [0]

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

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

= => eq

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

(14)
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:

   F(true, z0, z1, z2) -> c1(DEL(.(z1, z2))) [1]
   F(false, z0, z1, z2) -> c2(DEL(.(z1, z2))) [1]
   DEL(.(z0, .(z1, z2))) -> c(F(eq(z0, z1), z0, z1, z2)) [1]
   eq(nil, nil) -> true [0]
   eq(.(z0, z1), nil) -> false [0]
   eq(nil, .(z0, z1)) -> false [0]
   eq(.(z0, z1), .(u, v)) -> and(eq(z0, u), eq(z1, v)) [0]

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

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

(16)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   F(true, z0, z1, z2) -> c1(DEL(.(z1, z2))) [1]
   F(false, z0, z1, z2) -> c2(DEL(.(z1, z2))) [1]
   DEL(.(z0, .(z1, z2))) -> c(F(eq(z0, z1), z0, z1, z2)) [1]
   eq(nil, nil) -> true [0]
   eq(.(z0, z1), nil) -> false [0]
   eq(nil, .(z0, z1)) -> false [0]
   eq(.(z0, z1), .(u, v)) -> and(eq(z0, u), eq(z1, v)) [0]

The TRS has the following type information:
F :: true:false:and -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v -> c1:c2
true :: true:false:and
c1 :: c -> c1:c2
DEL :: .:nil:u:v -> c
. :: .:nil:u:v -> .:nil:u:v -> .:nil:u:v
false :: true:false:and
c2 :: c -> c1:c2
c :: c1:c2 -> c
eq :: .:nil:u:v -> .:nil:u:v -> 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
----------------------------------------

(17) 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:

   F_4
   DEL_1

(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, const, const1

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

(18)
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:

   F(true, z0, z1, z2) -> c1(DEL(.(z1, z2))) [1]
   F(false, z0, z1, z2) -> c2(DEL(.(z1, z2))) [1]
   DEL(.(z0, .(z1, z2))) -> c(F(eq(z0, z1), z0, z1, z2)) [1]
   eq(nil, nil) -> true [0]
   eq(.(z0, z1), nil) -> false [0]
   eq(nil, .(z0, z1)) -> false [0]
   eq(.(z0, z1), .(u, v)) -> and(eq(z0, u), eq(z1, v)) [0]
   eq(v0, v1) -> null_eq [0]

The TRS has the following type information:
F :: true:false:and:null_eq -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v -> c1:c2
true :: true:false:and:null_eq
c1 :: c -> c1:c2
DEL :: .:nil:u:v -> c
. :: .:nil:u:v -> .:nil:u:v -> .:nil:u:v
false :: true:false:and:null_eq
c2 :: c -> c1:c2
c :: c1:c2 -> c
eq :: .:nil:u:v -> .:nil:u:v -> 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
const :: c1:c2
const1 :: c

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

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

(20)
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:

   F(true, z0, z1, z2) -> c1(DEL(.(z1, z2))) [1]
   F(false, z0, z1, z2) -> c2(DEL(.(z1, z2))) [1]
   DEL(.(nil, .(nil, z2))) -> c(F(true, nil, nil, z2)) [1]
   DEL(.(.(z0', z1'), .(nil, z2))) -> c(F(false, .(z0', z1'), nil, z2)) [1]
   DEL(.(nil, .(.(z0'', z1''), z2))) -> c(F(false, nil, .(z0'', z1''), z2)) [1]
   DEL(.(.(z01, z11), .(.(u, v), z2))) -> c(F(and(eq(z01, u), eq(z11, v)), .(z01, z11), .(u, v), z2)) [1]
   DEL(.(z0, .(z1, z2))) -> c(F(null_eq, z0, z1, z2)) [1]
   eq(nil, nil) -> true [0]
   eq(.(z0, z1), nil) -> false [0]
   eq(nil, .(z0, z1)) -> false [0]
   eq(.(z0, z1), .(u, v)) -> and(eq(z0, u), eq(z1, v)) [0]
   eq(v0, v1) -> null_eq [0]

The TRS has the following type information:
F :: true:false:and:null_eq -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v -> c1:c2
true :: true:false:and:null_eq
c1 :: c -> c1:c2
DEL :: .:nil:u:v -> c
. :: .:nil:u:v -> .:nil:u:v -> .:nil:u:v
false :: true:false:and:null_eq
c2 :: c -> c1:c2
c :: c1:c2 -> c
eq :: .:nil:u:v -> .:nil:u:v -> 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
const :: c1:c2
const1 :: c

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

(21) 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
   const => 0
   const1 => 0

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

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

   DEL(z) -{ 1 }-> 1 + F(1, 0, 0, z2) :|: z = 1 + 0 + (1 + 0 + z2), z2 >= 0
   DEL(z) -{ 1 }-> 1 + F(0, z0, z1, z2) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   DEL(z) -{ 1 }-> 1 + F(0, 0, 1 + z0'' + z1'', z2) :|: z = 1 + 0 + (1 + (1 + z0'' + z1'') + z2), z0'' >= 0, z2 >= 0, z1'' >= 0
   DEL(z) -{ 1 }-> 1 + F(0, 1 + z0' + z1', 0, z2) :|: z0' >= 0, z1' >= 0, z = 1 + (1 + z0' + z1') + (1 + 0 + z2), z2 >= 0
   DEL(z) -{ 1 }-> 1 + F(1 + eq(z01, 1) + eq(z11, 2), 1 + z01 + z11, 1 + 1 + 2, z2) :|: z11 >= 0, z01 >= 0, z = 1 + (1 + z01 + z11) + (1 + (1 + 1 + 2) + z2), z2 >= 0
   F(z, z', z'', z3) -{ 1 }-> 1 + DEL(1 + z1 + z2) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   F(z, z', z'', z3) -{ 1 }-> 1 + DEL(1 + z1 + z2) :|: z1 >= 0, z0 >= 0, z = 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   eq(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 0
   eq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   eq(z, z') -{ 0 }-> 1 + eq(z0, 1) + eq(z1, 2) :|: z1 >= 0, z0 >= 0, z' = 1 + 1 + 2, z = 1 + z0 + z1


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

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

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

   DEL(z) -{ 1 }-> 1 + F(1, 0, 0, z - 2) :|: z - 2 >= 0
   DEL(z) -{ 1 }-> 1 + F(0, z0, z1, z2) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   DEL(z) -{ 1 }-> 1 + F(0, 0, 1 + z0'' + z1'', z2) :|: z = 1 + 0 + (1 + (1 + z0'' + z1'') + z2), z0'' >= 0, z2 >= 0, z1'' >= 0
   DEL(z) -{ 1 }-> 1 + F(0, 1 + z0' + z1', 0, z2) :|: z0' >= 0, z1' >= 0, z = 1 + (1 + z0' + z1') + (1 + 0 + z2), z2 >= 0
   DEL(z) -{ 1 }-> 1 + F(1 + eq(z01, 1) + eq(z11, 2), 1 + z01 + z11, 1 + 1 + 2, z2) :|: z11 >= 0, z01 >= 0, z = 1 + (1 + z01 + z11) + (1 + (1 + 1 + 2) + z2), z2 >= 0
   F(z, z', z'', z3) -{ 1 }-> 1 + DEL(1 + z'' + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   F(z, z', z'', z3) -{ 1 }-> 1 + DEL(1 + z'' + z3) :|: z'' >= 0, z' >= 0, z = 0, z3 >= 0
   eq(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eq(z, z') -{ 0 }-> 1 + eq(z0, 1) + eq(z1, 2) :|: z1 >= 0, z0 >= 0, z' = 1 + 1 + 2, z = 1 + z0 + z1


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

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

   { eq }
   { DEL, F }

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

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

   DEL(z) -{ 1 }-> 1 + F(1, 0, 0, z - 2) :|: z - 2 >= 0
   DEL(z) -{ 1 }-> 1 + F(0, z0, z1, z2) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   DEL(z) -{ 1 }-> 1 + F(0, 0, 1 + z0'' + z1'', z2) :|: z = 1 + 0 + (1 + (1 + z0'' + z1'') + z2), z0'' >= 0, z2 >= 0, z1'' >= 0
   DEL(z) -{ 1 }-> 1 + F(0, 1 + z0' + z1', 0, z2) :|: z0' >= 0, z1' >= 0, z = 1 + (1 + z0' + z1') + (1 + 0 + z2), z2 >= 0
   DEL(z) -{ 1 }-> 1 + F(1 + eq(z01, 1) + eq(z11, 2), 1 + z01 + z11, 1 + 1 + 2, z2) :|: z11 >= 0, z01 >= 0, z = 1 + (1 + z01 + z11) + (1 + (1 + 1 + 2) + z2), z2 >= 0
   F(z, z', z'', z3) -{ 1 }-> 1 + DEL(1 + z'' + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   F(z, z', z'', z3) -{ 1 }-> 1 + DEL(1 + z'' + z3) :|: z'' >= 0, z' >= 0, z = 0, z3 >= 0
   eq(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eq(z, z') -{ 0 }-> 1 + eq(z0, 1) + eq(z1, 2) :|: z1 >= 0, z0 >= 0, z' = 1 + 1 + 2, z = 1 + z0 + z1

Function symbols to be analyzed: {eq}, {DEL,F}

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

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

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

   DEL(z) -{ 1 }-> 1 + F(1, 0, 0, z - 2) :|: z - 2 >= 0
   DEL(z) -{ 1 }-> 1 + F(0, z0, z1, z2) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   DEL(z) -{ 1 }-> 1 + F(0, 0, 1 + z0'' + z1'', z2) :|: z = 1 + 0 + (1 + (1 + z0'' + z1'') + z2), z0'' >= 0, z2 >= 0, z1'' >= 0
   DEL(z) -{ 1 }-> 1 + F(0, 1 + z0' + z1', 0, z2) :|: z0' >= 0, z1' >= 0, z = 1 + (1 + z0' + z1') + (1 + 0 + z2), z2 >= 0
   DEL(z) -{ 1 }-> 1 + F(1 + eq(z01, 1) + eq(z11, 2), 1 + z01 + z11, 1 + 1 + 2, z2) :|: z11 >= 0, z01 >= 0, z = 1 + (1 + z01 + z11) + (1 + (1 + 1 + 2) + z2), z2 >= 0
   F(z, z', z'', z3) -{ 1 }-> 1 + DEL(1 + z'' + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   F(z, z', z'', z3) -{ 1 }-> 1 + DEL(1 + z'' + z3) :|: z'' >= 0, z' >= 0, z = 0, z3 >= 0
   eq(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eq(z, z') -{ 0 }-> 1 + eq(z0, 1) + eq(z1, 2) :|: z1 >= 0, z0 >= 0, z' = 1 + 1 + 2, z = 1 + z0 + z1

Function symbols to be analyzed: {eq}, {DEL,F}

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

(29) 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

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

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

   DEL(z) -{ 1 }-> 1 + F(1, 0, 0, z - 2) :|: z - 2 >= 0
   DEL(z) -{ 1 }-> 1 + F(0, z0, z1, z2) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   DEL(z) -{ 1 }-> 1 + F(0, 0, 1 + z0'' + z1'', z2) :|: z = 1 + 0 + (1 + (1 + z0'' + z1'') + z2), z0'' >= 0, z2 >= 0, z1'' >= 0
   DEL(z) -{ 1 }-> 1 + F(0, 1 + z0' + z1', 0, z2) :|: z0' >= 0, z1' >= 0, z = 1 + (1 + z0' + z1') + (1 + 0 + z2), z2 >= 0
   DEL(z) -{ 1 }-> 1 + F(1 + eq(z01, 1) + eq(z11, 2), 1 + z01 + z11, 1 + 1 + 2, z2) :|: z11 >= 0, z01 >= 0, z = 1 + (1 + z01 + z11) + (1 + (1 + 1 + 2) + z2), z2 >= 0
   F(z, z', z'', z3) -{ 1 }-> 1 + DEL(1 + z'' + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   F(z, z', z'', z3) -{ 1 }-> 1 + DEL(1 + z'' + z3) :|: z'' >= 0, z' >= 0, z = 0, z3 >= 0
   eq(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eq(z, z') -{ 0 }-> 1 + eq(z0, 1) + eq(z1, 2) :|: z1 >= 0, z0 >= 0, z' = 1 + 1 + 2, z = 1 + z0 + z1

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

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

(31) 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: 0

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

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

   DEL(z) -{ 1 }-> 1 + F(1, 0, 0, z - 2) :|: z - 2 >= 0
   DEL(z) -{ 1 }-> 1 + F(0, z0, z1, z2) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   DEL(z) -{ 1 }-> 1 + F(0, 0, 1 + z0'' + z1'', z2) :|: z = 1 + 0 + (1 + (1 + z0'' + z1'') + z2), z0'' >= 0, z2 >= 0, z1'' >= 0
   DEL(z) -{ 1 }-> 1 + F(0, 1 + z0' + z1', 0, z2) :|: z0' >= 0, z1' >= 0, z = 1 + (1 + z0' + z1') + (1 + 0 + z2), z2 >= 0
   DEL(z) -{ 1 }-> 1 + F(1 + eq(z01, 1) + eq(z11, 2), 1 + z01 + z11, 1 + 1 + 2, z2) :|: z11 >= 0, z01 >= 0, z = 1 + (1 + z01 + z11) + (1 + (1 + 1 + 2) + z2), z2 >= 0
   F(z, z', z'', z3) -{ 1 }-> 1 + DEL(1 + z'' + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   F(z, z', z'', z3) -{ 1 }-> 1 + DEL(1 + z'' + z3) :|: z'' >= 0, z' >= 0, z = 0, z3 >= 0
   eq(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eq(z, z') -{ 0 }-> 1 + eq(z0, 1) + eq(z1, 2) :|: z1 >= 0, z0 >= 0, z' = 1 + 1 + 2, z = 1 + z0 + z1

Function symbols to be analyzed: {DEL,F}
Previous analysis results are:
  eq: runtime: O(1) [0], size: O(1) [1]

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

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

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

   DEL(z) -{ 1 }-> 1 + F(1, 0, 0, z - 2) :|: z - 2 >= 0
   DEL(z) -{ 1 }-> 1 + F(0, z0, z1, z2) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   DEL(z) -{ 1 }-> 1 + F(0, 0, 1 + z0'' + z1'', z2) :|: z = 1 + 0 + (1 + (1 + z0'' + z1'') + z2), z0'' >= 0, z2 >= 0, z1'' >= 0
   DEL(z) -{ 1 }-> 1 + F(0, 1 + z0' + z1', 0, z2) :|: z0' >= 0, z1' >= 0, z = 1 + (1 + z0' + z1') + (1 + 0 + z2), z2 >= 0
   DEL(z) -{ 1 }-> 1 + F(1 + s + s', 1 + z01 + z11, 1 + 1 + 2, z2) :|: s >= 0, s <= 1, s' >= 0, s' <= 1, z11 >= 0, z01 >= 0, z = 1 + (1 + z01 + z11) + (1 + (1 + 1 + 2) + z2), z2 >= 0
   F(z, z', z'', z3) -{ 1 }-> 1 + DEL(1 + z'' + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   F(z, z', z'', z3) -{ 1 }-> 1 + DEL(1 + z'' + z3) :|: z'' >= 0, z' >= 0, z = 0, z3 >= 0
   eq(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eq(z, z') -{ 0 }-> 1 + s'' + s1 :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, z1 >= 0, z0 >= 0, z' = 1 + 1 + 2, z = 1 + z0 + z1

Function symbols to be analyzed: {DEL,F}
Previous analysis results are:
  eq: runtime: O(1) [0], size: O(1) [1]

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

(35) 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(1) with polynomial bound: 1

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

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

   DEL(z) -{ 1 }-> 1 + F(1, 0, 0, z - 2) :|: z - 2 >= 0
   DEL(z) -{ 1 }-> 1 + F(0, z0, z1, z2) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   DEL(z) -{ 1 }-> 1 + F(0, 0, 1 + z0'' + z1'', z2) :|: z = 1 + 0 + (1 + (1 + z0'' + z1'') + z2), z0'' >= 0, z2 >= 0, z1'' >= 0
   DEL(z) -{ 1 }-> 1 + F(0, 1 + z0' + z1', 0, z2) :|: z0' >= 0, z1' >= 0, z = 1 + (1 + z0' + z1') + (1 + 0 + z2), z2 >= 0
   DEL(z) -{ 1 }-> 1 + F(1 + s + s', 1 + z01 + z11, 1 + 1 + 2, z2) :|: s >= 0, s <= 1, s' >= 0, s' <= 1, z11 >= 0, z01 >= 0, z = 1 + (1 + z01 + z11) + (1 + (1 + 1 + 2) + z2), z2 >= 0
   F(z, z', z'', z3) -{ 1 }-> 1 + DEL(1 + z'' + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   F(z, z', z'', z3) -{ 1 }-> 1 + DEL(1 + z'' + z3) :|: z'' >= 0, z' >= 0, z = 0, z3 >= 0
   eq(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eq(z, z') -{ 0 }-> 1 + s'' + s1 :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, z1 >= 0, z0 >= 0, z' = 1 + 1 + 2, z = 1 + z0 + z1

Function symbols to be analyzed: {DEL,F}
Previous analysis results are:
  eq: runtime: O(1) [0], size: O(1) [1]
  DEL: runtime: ?, size: O(1) [0]
  F: runtime: ?, size: O(1) [1]

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

(37) 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: 14*z

Computed RUNTIME bound using CoFloCo for: F
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 15 + 14*z'' + 14*z3

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

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

   DEL(z) -{ 1 }-> 1 + F(1, 0, 0, z - 2) :|: z - 2 >= 0
   DEL(z) -{ 1 }-> 1 + F(0, z0, z1, z2) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   DEL(z) -{ 1 }-> 1 + F(0, 0, 1 + z0'' + z1'', z2) :|: z = 1 + 0 + (1 + (1 + z0'' + z1'') + z2), z0'' >= 0, z2 >= 0, z1'' >= 0
   DEL(z) -{ 1 }-> 1 + F(0, 1 + z0' + z1', 0, z2) :|: z0' >= 0, z1' >= 0, z = 1 + (1 + z0' + z1') + (1 + 0 + z2), z2 >= 0
   DEL(z) -{ 1 }-> 1 + F(1 + s + s', 1 + z01 + z11, 1 + 1 + 2, z2) :|: s >= 0, s <= 1, s' >= 0, s' <= 1, z11 >= 0, z01 >= 0, z = 1 + (1 + z01 + z11) + (1 + (1 + 1 + 2) + z2), z2 >= 0
   F(z, z', z'', z3) -{ 1 }-> 1 + DEL(1 + z'' + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   F(z, z', z'', z3) -{ 1 }-> 1 + DEL(1 + z'' + z3) :|: z'' >= 0, z' >= 0, z = 0, z3 >= 0
   eq(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eq(z, z') -{ 0 }-> 1 + s'' + s1 :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, z1 >= 0, z0 >= 0, z' = 1 + 1 + 2, z = 1 + z0 + z1

Function symbols to be analyzed: 
Previous analysis results are:
  eq: runtime: O(1) [0], size: O(1) [1]
  DEL: runtime: O(n^1) [14*z], size: O(1) [0]
  F: runtime: O(n^1) [15 + 14*z'' + 14*z3], size: O(1) [1]

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

(40)
BOUNDS(1, n^1)