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


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WORST_CASE(Omega(n^1), O(n^3))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 4 ms]
    (10) CpxRNTS
    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 6 ms]
    (12) CpxRNTS
    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 339 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 158 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 815 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 275 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 1101 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 22 ms]
    (32) CpxRNTS
    (33) FinalProof [FINISHED, 0 ms]
    (34) BOUNDS(1, n^3)
(35) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(36) CpxTRS
    (37) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) typed CpxTrs
    (39) OrderProof [LOWER BOUND(ID), 0 ms]
    (40) typed CpxTrs
    (41) RewriteLemmaProof [LOWER BOUND(ID), 266 ms]
    (42) BEST
        (43) proven lower bound
            (44) LowerBoundPropagationProof [FINISHED, 0 ms]
            (45) BOUNDS(n^1, INF)
        (46) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   minus(0, y) -> 0
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
   if_minus(true, s(x), y) -> 0
   if_minus(false, s(x), y) -> s(minus(x, y))
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
   if_mod(false, s(x), s(y)) -> s(x)

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

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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   mod(0, y) -> 0 [1]
   mod(s(x), 0) -> 0 [1]
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) [1]
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) [1]
   if_mod(false, s(x), s(y)) -> s(x) [1]

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

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

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

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   mod(0, y) -> 0 [1]
   mod(s(x), 0) -> 0 [1]
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) [1]
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) [1]
   if_mod(false, s(x), s(y)) -> s(x) [1]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
minus :: 0:s -> 0:s -> 0:s
if_minus :: true:false -> 0:s -> 0:s -> 0:s
mod :: 0:s -> 0:s -> 0:s
if_mod :: true:false -> 0:s -> 0:s -> 0:s

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

(5) 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:

   mod_2
   if_mod_3

(c) The following functions are completely defined:

   le_2
   minus_2
   if_minus_3

Due to the following rules being added:

   if_minus(v0, v1, v2) -> 0 [0]

And the following fresh constants: none

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

(6)
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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   mod(0, y) -> 0 [1]
   mod(s(x), 0) -> 0 [1]
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) [1]
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) [1]
   if_mod(false, s(x), s(y)) -> s(x) [1]
   if_minus(v0, v1, v2) -> 0 [0]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
minus :: 0:s -> 0:s -> 0:s
if_minus :: true:false -> 0:s -> 0:s -> 0:s
mod :: 0:s -> 0:s -> 0:s
if_mod :: true:false -> 0:s -> 0:s -> 0:s

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

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

(8)
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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), 0) -> if_minus(false, s(x), 0) [2]
   minus(s(x), s(y')) -> if_minus(le(x, y'), s(x), s(y')) [2]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   mod(0, y) -> 0 [1]
   mod(s(x), 0) -> 0 [1]
   mod(s(x), s(0)) -> if_mod(true, s(x), s(0)) [2]
   mod(s(0), s(s(x'))) -> if_mod(false, s(0), s(s(x'))) [2]
   mod(s(s(y'')), s(s(x''))) -> if_mod(le(x'', y''), s(s(y'')), s(s(x''))) [2]
   if_mod(true, s(0), s(y)) -> mod(0, s(y)) [2]
   if_mod(true, s(s(x1)), s(y)) -> mod(if_minus(le(s(x1), y), s(x1), y), s(y)) [2]
   if_mod(false, s(x), s(y)) -> s(x) [1]
   if_minus(v0, v1, v2) -> 0 [0]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
minus :: 0:s -> 0:s -> 0:s
if_minus :: true:false -> 0:s -> 0:s -> 0:s
mod :: 0:s -> 0:s -> 0:s
if_mod :: true:false -> 0:s -> 0:s -> 0:s

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

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   0 => 0
   true => 1
   false => 0

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z' = 1 + x, z'' = y, z = 1, x >= 0, y >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0, z = 0
   if_mod(z, z', z'') -{ 2 }-> mod(if_minus(le(1 + x1, y), 1 + x1, y), 1 + y) :|: x1 >= 0, z' = 1 + (1 + x1), z = 1, y >= 0, z'' = 1 + y
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + y) :|: z = 1, y >= 0, z'' = 1 + y, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> 1 + x :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y, z = 0
   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
   le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   minus(z, z') -{ 2 }-> if_minus(le(x, y'), 1 + x, 1 + y') :|: x >= 0, y' >= 0, z = 1 + x, z' = 1 + y'
   minus(z, z') -{ 2 }-> if_minus(0, 1 + x, 0) :|: x >= 0, z = 1 + x, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   mod(z, z') -{ 2 }-> if_mod(le(x'', y''), 1 + (1 + y''), 1 + (1 + x'')) :|: z' = 1 + (1 + x''), z = 1 + (1 + y''), y'' >= 0, x'' >= 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + x, 1 + 0) :|: x >= 0, z' = 1 + 0, z = 1 + x
   mod(z, z') -{ 2 }-> if_mod(0, 1 + 0, 1 + (1 + x')) :|: z' = 1 + (1 + x'), z = 1 + 0, x' >= 0
   mod(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   mod(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0


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

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
   if_mod(z, z', z'') -{ 2 }-> mod(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 2 }-> if_mod(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0


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

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

   { le }
   { minus, if_minus }
   { mod, if_mod }

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

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

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
   if_mod(z, z', z'') -{ 2 }-> mod(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 2 }-> if_mod(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0

Function symbols to be analyzed: {le}, {minus,if_minus}, {mod,if_mod}

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

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

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

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
   if_mod(z, z', z'') -{ 2 }-> mod(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 2 }-> if_mod(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0

Function symbols to be analyzed: {le}, {minus,if_minus}, {mod,if_mod}

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

(17) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
   if_mod(z, z', z'') -{ 2 }-> mod(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 2 }-> if_mod(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0

Function symbols to be analyzed: {le}, {minus,if_minus}, {mod,if_mod}
Previous analysis results are:
  le: runtime: ?, size: O(1) [1]

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: le
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z'

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

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

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
   if_mod(z, z', z'') -{ 2 }-> mod(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 2 }-> if_mod(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0

Function symbols to be analyzed: {minus,if_minus}, {mod,if_mod}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

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

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

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
   if_mod(z, z', z'') -{ 3 + z'' }-> mod(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 2 + z }-> if_mod(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0

Function symbols to be analyzed: {minus,if_minus}, {mod,if_mod}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
   if_mod(z, z', z'') -{ 3 + z'' }-> mod(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 2 + z }-> if_mod(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0

Function symbols to be analyzed: {minus,if_minus}, {mod,if_mod}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  minus: runtime: ?, size: O(n^1) [z]
  if_minus: runtime: ?, size: O(n^1) [z']

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 8 + 4*z + z*z' + 2*z'

Computed RUNTIME bound using KoAT for: if_minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 6 + 4*z' + z'*z'' + z''

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

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

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
   if_mod(z, z', z'') -{ 3 + z'' }-> mod(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 2 + z }-> if_mod(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0

Function symbols to be analyzed: {mod,if_mod}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']

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

(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:

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s4 :|: s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z'' >= 0, z = 0
   if_mod(z, z', z'') -{ 5 + 3*z' + z'*z'' + z'' }-> mod(s5, 1 + (z'' - 1)) :|: s5 >= 0, s5 <= 1 + (z' - 2), s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 2 + z }-> if_mod(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0

Function symbols to be analyzed: {mod,if_mod}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s4 :|: s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z'' >= 0, z = 0
   if_mod(z, z', z'') -{ 5 + 3*z' + z'*z'' + z'' }-> mod(s5, 1 + (z'' - 1)) :|: s5 >= 0, s5 <= 1 + (z' - 2), s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 2 + z }-> if_mod(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0

Function symbols to be analyzed: {mod,if_mod}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: ?, size: O(n^1) [z]
  if_mod: runtime: ?, size: O(n^1) [z']

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: mod
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 3 + 28*z + 2*z*z' + 10*z^2 + 2*z^2*z'

Computed RUNTIME bound using KoAT for: if_mod
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 6 + 11*z' + 10*z'^2 + 2*z'^2*z'' + z''

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

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

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s4 :|: s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z'' >= 0, z = 0
   if_mod(z, z', z'') -{ 5 + 3*z' + z'*z'' + z'' }-> mod(s5, 1 + (z'' - 1)) :|: s5 >= 0, s5 <= 1 + (z' - 2), s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
   if_mod(z, z', z'') -{ 2 }-> mod(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
   if_mod(z, z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 2 + z }-> if_mod(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
   mod(z, z') -{ 2 }-> if_mod(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   mod(z, z') -{ 2 }-> if_mod(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   mod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0

Function symbols to be analyzed: 
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  mod: runtime: O(n^3) [3 + 28*z + 2*z*z' + 10*z^2 + 2*z^2*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^3) [6 + 11*z' + 10*z'^2 + 2*z'^2*z'' + z''], size: O(n^1) [z']

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

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

(34)
BOUNDS(1, n^3)

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

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

(36)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   minus(0', y) -> 0'
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
   if_minus(true, s(x), y) -> 0'
   if_minus(false, s(x), y) -> s(minus(x, y))
   mod(0', y) -> 0'
   mod(s(x), 0') -> 0'
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
   if_mod(false, s(x), s(y)) -> s(x)

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

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

(38)
Obligation:
Innermost TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(0', y) -> 0'
minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
if_minus(true, s(x), y) -> 0'
if_minus(false, s(x), y) -> s(minus(x, y))
mod(0', y) -> 0'
mod(s(x), 0') -> 0'
mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
if_mod(false, s(x), s(y)) -> s(x)

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
minus :: 0':s -> 0':s -> 0':s
if_minus :: true:false -> 0':s -> 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s

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

(39) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
le, minus, mod

They will be analysed ascendingly in the following order:
le < minus
le < mod
minus < mod

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

(40)
Obligation:
Innermost TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(0', y) -> 0'
minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
if_minus(true, s(x), y) -> 0'
if_minus(false, s(x), y) -> s(minus(x, y))
mod(0', y) -> 0'
mod(s(x), 0') -> 0'
mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
if_mod(false, s(x), s(y)) -> s(x)

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
minus :: 0':s -> 0':s -> 0':s
if_minus :: true:false -> 0':s -> 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
le, minus, mod

They will be analysed ascendingly in the following order:
le < minus
le < mod
minus < mod

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(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)

Induction Base:
le(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
true

Induction Step:
le(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
true

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(42)
Complex Obligation (BEST)

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(43)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(0', y) -> 0'
minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
if_minus(true, s(x), y) -> 0'
if_minus(false, s(x), y) -> s(minus(x, y))
mod(0', y) -> 0'
mod(s(x), 0') -> 0'
mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
if_mod(false, s(x), s(y)) -> s(x)

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
minus :: 0':s -> 0':s -> 0':s
if_minus :: true:false -> 0':s -> 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
le, minus, mod

They will be analysed ascendingly in the following order:
le < minus
le < mod
minus < mod

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(44) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(45)
BOUNDS(n^1, INF)

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(46)
Obligation:
Innermost TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(0', y) -> 0'
minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
if_minus(true, s(x), y) -> 0'
if_minus(false, s(x), y) -> s(minus(x, y))
mod(0', y) -> 0'
mod(s(x), 0') -> 0'
mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
if_mod(false, s(x), s(y)) -> s(x)

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
minus :: 0':s -> 0':s -> 0':s
if_minus :: true:false -> 0':s -> 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
minus, mod

They will be analysed ascendingly in the following order:
minus < mod