WORST_CASE(Omega(n^1),O(n^1))
proof of input_wcb2EBzcLK.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 3 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) InliningProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 248 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 97 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 162 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 74 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 114 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 124 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 318 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 230 ms]
    (40) CpxRNTS
    (41) FinalProof [FINISHED, 0 ms]
    (42) BOUNDS(1, n^1)
(43) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(44) CdtProblem
    (45) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CpxRelTRS
    (47) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CpxRelTRS
    (49) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) typed CpxTrs
    (51) OrderProof [LOWER BOUND(ID), 17 ms]
    (52) typed CpxTrs
    (53) RewriteLemmaProof [LOWER BOUND(ID), 370 ms]
    (54) BEST
        (55) proven lower bound
            (56) LowerBoundPropagationProof [FINISHED, 0 ms]
            (57) BOUNDS(n^1, INF)
        (58) typed CpxTrs
            (59) RewriteLemmaProof [LOWER BOUND(ID), 135 ms]
            (60) typed CpxTrs
            (61) RewriteLemmaProof [LOWER BOUND(ID), 57 ms]
            (62) typed CpxTrs
            (63) RewriteLemmaProof [LOWER BOUND(ID), 38 ms]
            (64) typed CpxTrs


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

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


The TRS R consists of the following rules:

   minus(0, Y) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   geq(X, 0) -> true
   geq(0, s(Y)) -> false
   geq(s(X), s(Y)) -> geq(X, Y)
   div(0, s(Y)) -> 0
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

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

(1) RelTrsToWeightedTrsProof (UPPER BOUND(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^1).


The TRS R consists of the following rules:

   minus(0, Y) -> 0 [1]
   minus(s(X), s(Y)) -> minus(X, Y) [1]
   geq(X, 0) -> true [1]
   geq(0, s(Y)) -> false [1]
   geq(s(X), s(Y)) -> geq(X, Y) [1]
   div(0, s(Y)) -> 0 [1]
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [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:

   minus(0, Y) -> 0 [1]
   minus(s(X), s(Y)) -> minus(X, Y) [1]
   geq(X, 0) -> true [1]
   geq(0, s(Y)) -> false [1]
   geq(s(X), s(Y)) -> geq(X, Y) [1]
   div(0, s(Y)) -> 0 [1]
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
geq :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
div :: 0:s -> 0:s -> 0:s
if :: 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:
none

(c) The following functions are completely defined:

   geq_2
   div_2
   minus_2
   if_3

Due to the following rules being added:

   div(v0, v1) -> 0 [0]
   minus(v0, v1) -> 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:

   minus(0, Y) -> 0 [1]
   minus(s(X), s(Y)) -> minus(X, Y) [1]
   geq(X, 0) -> true [1]
   geq(0, s(Y)) -> false [1]
   geq(s(X), s(Y)) -> geq(X, Y) [1]
   div(0, s(Y)) -> 0 [1]
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]
   div(v0, v1) -> 0 [0]
   minus(v0, v1) -> 0 [0]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
geq :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
div :: 0:s -> 0:s -> 0:s
if :: 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:

   minus(0, Y) -> 0 [1]
   minus(s(X), s(Y)) -> minus(X, Y) [1]
   geq(X, 0) -> true [1]
   geq(0, s(Y)) -> false [1]
   geq(s(X), s(Y)) -> geq(X, Y) [1]
   div(0, s(Y)) -> 0 [1]
   div(s(0), s(0)) -> if(true, s(div(0, s(0))), 0) [3]
   div(s(X), s(0)) -> if(true, s(div(0, s(0))), 0) [2]
   div(s(0), s(s(Y'))) -> if(false, s(div(0, s(s(Y')))), 0) [3]
   div(s(0), s(s(Y'))) -> if(false, s(div(0, s(s(Y')))), 0) [2]
   div(s(s(X')), s(s(Y''))) -> if(geq(X', Y''), s(div(minus(X', Y''), s(s(Y'')))), 0) [3]
   div(s(s(X')), s(s(Y''))) -> if(geq(X', Y''), s(div(0, s(s(Y'')))), 0) [2]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]
   div(v0, v1) -> 0 [0]
   minus(v0, v1) -> 0 [0]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
geq :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
div :: 0:s -> 0:s -> 0:s
if :: 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:

   div(z, z') -{ 3 }-> if(geq(X', Y''), 1 + div(minus(X', Y''), 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X')
   div(z, z') -{ 2 }-> if(geq(X', Y''), 1 + div(0, 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X')
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + X, X >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0
   div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   geq(z, z') -{ 1 }-> geq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   geq(z, z') -{ 1 }-> 1 :|: X >= 0, z = X, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0
   if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
   if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0
   minus(z, z') -{ 1 }-> minus(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

(11) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
   if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0

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

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

   div(z, z') -{ 3 }-> if(geq(X', Y''), 1 + div(minus(X', Y''), 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X')
   div(z, z') -{ 2 }-> if(geq(X', Y''), 1 + div(0, 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X')
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + X, X >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0
   div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   geq(z, z') -{ 1 }-> geq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   geq(z, z') -{ 1 }-> 1 :|: X >= 0, z = X, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0
   if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
   if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0
   minus(z, z') -{ 1 }-> minus(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

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

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

   div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0


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

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

   { minus }
   { if }
   { geq }
   { div }

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

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

   div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus}, {if}, {geq}, {div}

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

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

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

   div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus}, {if}, {geq}, {div}

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus}, {if}, {geq}, {div}
Previous analysis results are:
  minus: runtime: ?, size: O(1) [0]

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {if}, {geq}, {div}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]

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

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

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

   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {if}, {geq}, {div}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]

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

(25) 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''

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

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

   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {if}, {geq}, {div}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
  if: runtime: ?, size: O(n^1) [z' + z'']

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

(27) 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

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

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

   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {geq}, {div}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']

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

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

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

   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {geq}, {div}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {geq}, {div}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  geq: runtime: ?, size: O(1) [1]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {div}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  geq: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

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

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

   div(z, z') -{ 2 + 2*z' }-> if(s1, 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 + z' }-> if(s2, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {div}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  geq: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 2 + 2*z' }-> if(s1, 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 + z' }-> if(s2, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {div}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  geq: runtime: O(n^1) [2 + z'], size: O(1) [1]
  div: runtime: ?, size: O(1) [1]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: div
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 5 + 2*z'

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

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

   div(z, z') -{ 2 + 2*z' }-> if(s1, 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 + z' }-> if(s2, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  geq: runtime: O(n^1) [2 + z'], size: O(1) [1]
  div: runtime: O(n^1) [5 + 2*z'], size: O(1) [1]

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

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

(42)
BOUNDS(1, n^1)

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

(43) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   geq(z0, 0) -> true
   geq(0, s(z0)) -> false
   geq(s(z0), s(z1)) -> geq(z0, z1)
   div(0, s(z0)) -> 0
   div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
Tuples:
   MINUS(0, z0) -> c
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   GEQ(z0, 0) -> c2
   GEQ(0, s(z0)) -> c3
   GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
   DIV(0, s(z0)) -> c5
   DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF(true, z0, z1) -> c8
   IF(false, z0, z1) -> c9
S tuples:
   MINUS(0, z0) -> c
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   GEQ(z0, 0) -> c2
   GEQ(0, s(z0)) -> c3
   GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
   DIV(0, s(z0)) -> c5
   DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF(true, z0, z1) -> c8
   IF(false, z0, z1) -> c9
K tuples:none
Defined Rule Symbols:   minus_2, geq_2, div_2, if_3

Defined Pair Symbols:   MINUS_2, GEQ_2, DIV_2, IF_3

Compound Symbols:   c, c1_1, c2, c3, c4_1, c5, c6_2, c7_3, c8, c9


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

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

(46)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   MINUS(0, z0) -> c
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   GEQ(z0, 0) -> c2
   GEQ(0, s(z0)) -> c3
   GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
   DIV(0, s(z0)) -> c5
   DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF(true, z0, z1) -> c8
   IF(false, z0, z1) -> c9

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

   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   geq(z0, 0) -> true
   geq(0, s(z0)) -> false
   geq(s(z0), s(z1)) -> geq(z0, z1)
   div(0, s(z0)) -> 0
   div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1

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

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

(48)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   MINUS(0', z0) -> c
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   GEQ(z0, 0') -> c2
   GEQ(0', s(z0)) -> c3
   GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
   DIV(0', s(z0)) -> c5
   DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), GEQ(z0, z1))
   DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF(true, z0, z1) -> c8
   IF(false, z0, z1) -> c9

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

   minus(0', z0) -> 0'
   minus(s(z0), s(z1)) -> minus(z0, z1)
   geq(z0, 0') -> true
   geq(0', s(z0)) -> false
   geq(s(z0), s(z1)) -> geq(z0, z1)
   div(0', s(z0)) -> 0'
   div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0')
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1

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

(49) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(50)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
GEQ(z0, 0') -> c2
GEQ(0', s(z0)) -> c3
GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
DIV(0', s(z0)) -> c5
DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), GEQ(z0, z1))
DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(true, z0, z1) -> c8
IF(false, z0, z1) -> c9
minus(0', z0) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
geq(z0, 0') -> true
geq(0', s(z0)) -> false
geq(s(z0), s(z1)) -> geq(z0, z1)
div(0', s(z0)) -> 0'
div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0')
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
GEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
DIV :: 0':s -> 0':s -> c5:c6:c7
c5 :: c5:c6:c7
c6 :: c8:c9 -> c2:c3:c4 -> c5:c6:c7
IF :: true:false -> 0':s -> 0':s -> c8:c9
geq :: 0':s -> 0':s -> true:false
div :: 0':s -> 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
c7 :: c8:c9 -> c5:c6:c7 -> c:c1 -> c5:c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c8:c9
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_10 :: c:c1
hole_0':s2_10 :: 0':s
hole_c2:c3:c43_10 :: c2:c3:c4
hole_c5:c6:c74_10 :: c5:c6:c7
hole_c8:c95_10 :: c8:c9
hole_true:false6_10 :: true:false
gen_c:c17_10 :: Nat -> c:c1
gen_0':s8_10 :: Nat -> 0':s
gen_c2:c3:c49_10 :: Nat -> c2:c3:c4
gen_c5:c6:c710_10 :: Nat -> c5:c6:c7

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

(51) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
MINUS, GEQ, DIV, geq, div, minus

They will be analysed ascendingly in the following order:
MINUS < DIV
GEQ < DIV
geq < DIV
div < DIV
minus < DIV
geq < div
minus < div

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

(52)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
GEQ(z0, 0') -> c2
GEQ(0', s(z0)) -> c3
GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
DIV(0', s(z0)) -> c5
DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), GEQ(z0, z1))
DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(true, z0, z1) -> c8
IF(false, z0, z1) -> c9
minus(0', z0) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
geq(z0, 0') -> true
geq(0', s(z0)) -> false
geq(s(z0), s(z1)) -> geq(z0, z1)
div(0', s(z0)) -> 0'
div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0')
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
GEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
DIV :: 0':s -> 0':s -> c5:c6:c7
c5 :: c5:c6:c7
c6 :: c8:c9 -> c2:c3:c4 -> c5:c6:c7
IF :: true:false -> 0':s -> 0':s -> c8:c9
geq :: 0':s -> 0':s -> true:false
div :: 0':s -> 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
c7 :: c8:c9 -> c5:c6:c7 -> c:c1 -> c5:c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c8:c9
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_10 :: c:c1
hole_0':s2_10 :: 0':s
hole_c2:c3:c43_10 :: c2:c3:c4
hole_c5:c6:c74_10 :: c5:c6:c7
hole_c8:c95_10 :: c8:c9
hole_true:false6_10 :: true:false
gen_c:c17_10 :: Nat -> c:c1
gen_0':s8_10 :: Nat -> 0':s
gen_c2:c3:c49_10 :: Nat -> c2:c3:c4
gen_c5:c6:c710_10 :: Nat -> c5:c6:c7


Generator Equations:
gen_c:c17_10(0) <=> c
gen_c:c17_10(+(x, 1)) <=> c1(gen_c:c17_10(x))
gen_0':s8_10(0) <=> 0'
gen_0':s8_10(+(x, 1)) <=> s(gen_0':s8_10(x))
gen_c2:c3:c49_10(0) <=> c2
gen_c2:c3:c49_10(+(x, 1)) <=> c4(gen_c2:c3:c49_10(x))
gen_c5:c6:c710_10(0) <=> c5
gen_c5:c6:c710_10(+(x, 1)) <=> c7(c8, gen_c5:c6:c710_10(x), c)


The following defined symbols remain to be analysed:
MINUS, GEQ, DIV, geq, div, minus

They will be analysed ascendingly in the following order:
MINUS < DIV
GEQ < DIV
geq < DIV
div < DIV
minus < DIV
geq < div
minus < div

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

(53) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MINUS(gen_0':s8_10(n12_10), gen_0':s8_10(n12_10)) -> gen_c:c17_10(n12_10), rt in Omega(1 + n12_10)

Induction Base:
MINUS(gen_0':s8_10(0), gen_0':s8_10(0)) ->_R^Omega(1)
c

Induction Step:
MINUS(gen_0':s8_10(+(n12_10, 1)), gen_0':s8_10(+(n12_10, 1))) ->_R^Omega(1)
c1(MINUS(gen_0':s8_10(n12_10), gen_0':s8_10(n12_10))) ->_IH
c1(gen_c:c17_10(c13_10))

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

(54)
Complex Obligation (BEST)

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

(55)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
GEQ(z0, 0') -> c2
GEQ(0', s(z0)) -> c3
GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
DIV(0', s(z0)) -> c5
DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), GEQ(z0, z1))
DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(true, z0, z1) -> c8
IF(false, z0, z1) -> c9
minus(0', z0) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
geq(z0, 0') -> true
geq(0', s(z0)) -> false
geq(s(z0), s(z1)) -> geq(z0, z1)
div(0', s(z0)) -> 0'
div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0')
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
GEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
DIV :: 0':s -> 0':s -> c5:c6:c7
c5 :: c5:c6:c7
c6 :: c8:c9 -> c2:c3:c4 -> c5:c6:c7
IF :: true:false -> 0':s -> 0':s -> c8:c9
geq :: 0':s -> 0':s -> true:false
div :: 0':s -> 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
c7 :: c8:c9 -> c5:c6:c7 -> c:c1 -> c5:c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c8:c9
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_10 :: c:c1
hole_0':s2_10 :: 0':s
hole_c2:c3:c43_10 :: c2:c3:c4
hole_c5:c6:c74_10 :: c5:c6:c7
hole_c8:c95_10 :: c8:c9
hole_true:false6_10 :: true:false
gen_c:c17_10 :: Nat -> c:c1
gen_0':s8_10 :: Nat -> 0':s
gen_c2:c3:c49_10 :: Nat -> c2:c3:c4
gen_c5:c6:c710_10 :: Nat -> c5:c6:c7


Generator Equations:
gen_c:c17_10(0) <=> c
gen_c:c17_10(+(x, 1)) <=> c1(gen_c:c17_10(x))
gen_0':s8_10(0) <=> 0'
gen_0':s8_10(+(x, 1)) <=> s(gen_0':s8_10(x))
gen_c2:c3:c49_10(0) <=> c2
gen_c2:c3:c49_10(+(x, 1)) <=> c4(gen_c2:c3:c49_10(x))
gen_c5:c6:c710_10(0) <=> c5
gen_c5:c6:c710_10(+(x, 1)) <=> c7(c8, gen_c5:c6:c710_10(x), c)


The following defined symbols remain to be analysed:
MINUS, GEQ, DIV, geq, div, minus

They will be analysed ascendingly in the following order:
MINUS < DIV
GEQ < DIV
geq < DIV
div < DIV
minus < DIV
geq < div
minus < div

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

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

(57)
BOUNDS(n^1, INF)

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

(58)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
GEQ(z0, 0') -> c2
GEQ(0', s(z0)) -> c3
GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
DIV(0', s(z0)) -> c5
DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), GEQ(z0, z1))
DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(true, z0, z1) -> c8
IF(false, z0, z1) -> c9
minus(0', z0) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
geq(z0, 0') -> true
geq(0', s(z0)) -> false
geq(s(z0), s(z1)) -> geq(z0, z1)
div(0', s(z0)) -> 0'
div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0')
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
GEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
DIV :: 0':s -> 0':s -> c5:c6:c7
c5 :: c5:c6:c7
c6 :: c8:c9 -> c2:c3:c4 -> c5:c6:c7
IF :: true:false -> 0':s -> 0':s -> c8:c9
geq :: 0':s -> 0':s -> true:false
div :: 0':s -> 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
c7 :: c8:c9 -> c5:c6:c7 -> c:c1 -> c5:c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c8:c9
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_10 :: c:c1
hole_0':s2_10 :: 0':s
hole_c2:c3:c43_10 :: c2:c3:c4
hole_c5:c6:c74_10 :: c5:c6:c7
hole_c8:c95_10 :: c8:c9
hole_true:false6_10 :: true:false
gen_c:c17_10 :: Nat -> c:c1
gen_0':s8_10 :: Nat -> 0':s
gen_c2:c3:c49_10 :: Nat -> c2:c3:c4
gen_c5:c6:c710_10 :: Nat -> c5:c6:c7


Lemmas:
MINUS(gen_0':s8_10(n12_10), gen_0':s8_10(n12_10)) -> gen_c:c17_10(n12_10), rt in Omega(1 + n12_10)


Generator Equations:
gen_c:c17_10(0) <=> c
gen_c:c17_10(+(x, 1)) <=> c1(gen_c:c17_10(x))
gen_0':s8_10(0) <=> 0'
gen_0':s8_10(+(x, 1)) <=> s(gen_0':s8_10(x))
gen_c2:c3:c49_10(0) <=> c2
gen_c2:c3:c49_10(+(x, 1)) <=> c4(gen_c2:c3:c49_10(x))
gen_c5:c6:c710_10(0) <=> c5
gen_c5:c6:c710_10(+(x, 1)) <=> c7(c8, gen_c5:c6:c710_10(x), c)


The following defined symbols remain to be analysed:
GEQ, DIV, geq, div, minus

They will be analysed ascendingly in the following order:
GEQ < DIV
geq < DIV
div < DIV
minus < DIV
geq < div
minus < div

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

(59) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
GEQ(gen_0':s8_10(n402_10), gen_0':s8_10(n402_10)) -> gen_c2:c3:c49_10(n402_10), rt in Omega(1 + n402_10)

Induction Base:
GEQ(gen_0':s8_10(0), gen_0':s8_10(0)) ->_R^Omega(1)
c2

Induction Step:
GEQ(gen_0':s8_10(+(n402_10, 1)), gen_0':s8_10(+(n402_10, 1))) ->_R^Omega(1)
c4(GEQ(gen_0':s8_10(n402_10), gen_0':s8_10(n402_10))) ->_IH
c4(gen_c2:c3:c49_10(c403_10))

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

(60)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
GEQ(z0, 0') -> c2
GEQ(0', s(z0)) -> c3
GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
DIV(0', s(z0)) -> c5
DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), GEQ(z0, z1))
DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(true, z0, z1) -> c8
IF(false, z0, z1) -> c9
minus(0', z0) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
geq(z0, 0') -> true
geq(0', s(z0)) -> false
geq(s(z0), s(z1)) -> geq(z0, z1)
div(0', s(z0)) -> 0'
div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0')
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
GEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
DIV :: 0':s -> 0':s -> c5:c6:c7
c5 :: c5:c6:c7
c6 :: c8:c9 -> c2:c3:c4 -> c5:c6:c7
IF :: true:false -> 0':s -> 0':s -> c8:c9
geq :: 0':s -> 0':s -> true:false
div :: 0':s -> 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
c7 :: c8:c9 -> c5:c6:c7 -> c:c1 -> c5:c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c8:c9
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_10 :: c:c1
hole_0':s2_10 :: 0':s
hole_c2:c3:c43_10 :: c2:c3:c4
hole_c5:c6:c74_10 :: c5:c6:c7
hole_c8:c95_10 :: c8:c9
hole_true:false6_10 :: true:false
gen_c:c17_10 :: Nat -> c:c1
gen_0':s8_10 :: Nat -> 0':s
gen_c2:c3:c49_10 :: Nat -> c2:c3:c4
gen_c5:c6:c710_10 :: Nat -> c5:c6:c7


Lemmas:
MINUS(gen_0':s8_10(n12_10), gen_0':s8_10(n12_10)) -> gen_c:c17_10(n12_10), rt in Omega(1 + n12_10)
GEQ(gen_0':s8_10(n402_10), gen_0':s8_10(n402_10)) -> gen_c2:c3:c49_10(n402_10), rt in Omega(1 + n402_10)


Generator Equations:
gen_c:c17_10(0) <=> c
gen_c:c17_10(+(x, 1)) <=> c1(gen_c:c17_10(x))
gen_0':s8_10(0) <=> 0'
gen_0':s8_10(+(x, 1)) <=> s(gen_0':s8_10(x))
gen_c2:c3:c49_10(0) <=> c2
gen_c2:c3:c49_10(+(x, 1)) <=> c4(gen_c2:c3:c49_10(x))
gen_c5:c6:c710_10(0) <=> c5
gen_c5:c6:c710_10(+(x, 1)) <=> c7(c8, gen_c5:c6:c710_10(x), c)


The following defined symbols remain to be analysed:
geq, DIV, div, minus

They will be analysed ascendingly in the following order:
geq < DIV
div < DIV
minus < DIV
geq < div
minus < div

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

(61) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
geq(gen_0':s8_10(n1051_10), gen_0':s8_10(n1051_10)) -> true, rt in Omega(0)

Induction Base:
geq(gen_0':s8_10(0), gen_0':s8_10(0)) ->_R^Omega(0)
true

Induction Step:
geq(gen_0':s8_10(+(n1051_10, 1)), gen_0':s8_10(+(n1051_10, 1))) ->_R^Omega(0)
geq(gen_0':s8_10(n1051_10), gen_0':s8_10(n1051_10)) ->_IH
true

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(62)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
GEQ(z0, 0') -> c2
GEQ(0', s(z0)) -> c3
GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
DIV(0', s(z0)) -> c5
DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), GEQ(z0, z1))
DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(true, z0, z1) -> c8
IF(false, z0, z1) -> c9
minus(0', z0) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
geq(z0, 0') -> true
geq(0', s(z0)) -> false
geq(s(z0), s(z1)) -> geq(z0, z1)
div(0', s(z0)) -> 0'
div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0')
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
GEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
DIV :: 0':s -> 0':s -> c5:c6:c7
c5 :: c5:c6:c7
c6 :: c8:c9 -> c2:c3:c4 -> c5:c6:c7
IF :: true:false -> 0':s -> 0':s -> c8:c9
geq :: 0':s -> 0':s -> true:false
div :: 0':s -> 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
c7 :: c8:c9 -> c5:c6:c7 -> c:c1 -> c5:c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c8:c9
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_10 :: c:c1
hole_0':s2_10 :: 0':s
hole_c2:c3:c43_10 :: c2:c3:c4
hole_c5:c6:c74_10 :: c5:c6:c7
hole_c8:c95_10 :: c8:c9
hole_true:false6_10 :: true:false
gen_c:c17_10 :: Nat -> c:c1
gen_0':s8_10 :: Nat -> 0':s
gen_c2:c3:c49_10 :: Nat -> c2:c3:c4
gen_c5:c6:c710_10 :: Nat -> c5:c6:c7


Lemmas:
MINUS(gen_0':s8_10(n12_10), gen_0':s8_10(n12_10)) -> gen_c:c17_10(n12_10), rt in Omega(1 + n12_10)
GEQ(gen_0':s8_10(n402_10), gen_0':s8_10(n402_10)) -> gen_c2:c3:c49_10(n402_10), rt in Omega(1 + n402_10)
geq(gen_0':s8_10(n1051_10), gen_0':s8_10(n1051_10)) -> true, rt in Omega(0)


Generator Equations:
gen_c:c17_10(0) <=> c
gen_c:c17_10(+(x, 1)) <=> c1(gen_c:c17_10(x))
gen_0':s8_10(0) <=> 0'
gen_0':s8_10(+(x, 1)) <=> s(gen_0':s8_10(x))
gen_c2:c3:c49_10(0) <=> c2
gen_c2:c3:c49_10(+(x, 1)) <=> c4(gen_c2:c3:c49_10(x))
gen_c5:c6:c710_10(0) <=> c5
gen_c5:c6:c710_10(+(x, 1)) <=> c7(c8, gen_c5:c6:c710_10(x), c)


The following defined symbols remain to be analysed:
minus, DIV, div

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

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

(63) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s8_10(n1411_10), gen_0':s8_10(n1411_10)) -> gen_0':s8_10(0), rt in Omega(0)

Induction Base:
minus(gen_0':s8_10(0), gen_0':s8_10(0)) ->_R^Omega(0)
0'

Induction Step:
minus(gen_0':s8_10(+(n1411_10, 1)), gen_0':s8_10(+(n1411_10, 1))) ->_R^Omega(0)
minus(gen_0':s8_10(n1411_10), gen_0':s8_10(n1411_10)) ->_IH
gen_0':s8_10(0)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(64)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
GEQ(z0, 0') -> c2
GEQ(0', s(z0)) -> c3
GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1))
DIV(0', s(z0)) -> c5
DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), GEQ(z0, z1))
DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(true, z0, z1) -> c8
IF(false, z0, z1) -> c9
minus(0', z0) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
geq(z0, 0') -> true
geq(0', s(z0)) -> false
geq(s(z0), s(z1)) -> geq(z0, z1)
div(0', s(z0)) -> 0'
div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0')
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
GEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
DIV :: 0':s -> 0':s -> c5:c6:c7
c5 :: c5:c6:c7
c6 :: c8:c9 -> c2:c3:c4 -> c5:c6:c7
IF :: true:false -> 0':s -> 0':s -> c8:c9
geq :: 0':s -> 0':s -> true:false
div :: 0':s -> 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
c7 :: c8:c9 -> c5:c6:c7 -> c:c1 -> c5:c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c8:c9
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_10 :: c:c1
hole_0':s2_10 :: 0':s
hole_c2:c3:c43_10 :: c2:c3:c4
hole_c5:c6:c74_10 :: c5:c6:c7
hole_c8:c95_10 :: c8:c9
hole_true:false6_10 :: true:false
gen_c:c17_10 :: Nat -> c:c1
gen_0':s8_10 :: Nat -> 0':s
gen_c2:c3:c49_10 :: Nat -> c2:c3:c4
gen_c5:c6:c710_10 :: Nat -> c5:c6:c7


Lemmas:
MINUS(gen_0':s8_10(n12_10), gen_0':s8_10(n12_10)) -> gen_c:c17_10(n12_10), rt in Omega(1 + n12_10)
GEQ(gen_0':s8_10(n402_10), gen_0':s8_10(n402_10)) -> gen_c2:c3:c49_10(n402_10), rt in Omega(1 + n402_10)
geq(gen_0':s8_10(n1051_10), gen_0':s8_10(n1051_10)) -> true, rt in Omega(0)
minus(gen_0':s8_10(n1411_10), gen_0':s8_10(n1411_10)) -> gen_0':s8_10(0), rt in Omega(0)


Generator Equations:
gen_c:c17_10(0) <=> c
gen_c:c17_10(+(x, 1)) <=> c1(gen_c:c17_10(x))
gen_0':s8_10(0) <=> 0'
gen_0':s8_10(+(x, 1)) <=> s(gen_0':s8_10(x))
gen_c2:c3:c49_10(0) <=> c2
gen_c2:c3:c49_10(+(x, 1)) <=> c4(gen_c2:c3:c49_10(x))
gen_c5:c6:c710_10(0) <=> c5
gen_c5:c6:c710_10(+(x, 1)) <=> c7(c8, gen_c5:c6:c710_10(x), c)


The following defined symbols remain to be analysed:
div, DIV

They will be analysed ascendingly in the following order:
div < DIV