WORST_CASE(Omega(n^1),O(n^2))
proof of input_ZV2zExKfT7.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# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished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^2).

(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), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 4 ms]
    (12) CpxRNTS
    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 2 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 250 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 9 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 246 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 26 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 193 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 117 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 131 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 15 ms]
    (38) CpxRNTS
    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 3358 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 1071 ms]
    (44) CpxRNTS
    (45) FinalProof [FINISHED, 0 ms]
    (46) BOUNDS(1, n^2)
(47) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(48) CdtProblem
    (49) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CpxRelTRS
    (51) RenamingProof [BOTH BOUNDS(ID, ID), 1 ms]
    (52) CpxRelTRS
    (53) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) typed CpxTrs
    (55) OrderProof [LOWER BOUND(ID), 0 ms]
    (56) typed CpxTrs
    (57) RewriteLemmaProof [LOWER BOUND(ID), 439 ms]
    (58) BEST
        (59) proven lower bound
            (60) LowerBoundPropagationProof [FINISHED, 0 ms]
            (61) BOUNDS(n^1, INF)
        (62) typed CpxTrs
            (63) RewriteLemmaProof [LOWER BOUND(ID), 56 ms]
            (64) typed CpxTrs
            (65) RewriteLemmaProof [LOWER BOUND(ID), 53 ms]
            (66) typed CpxTrs
            (67) RewriteLemmaProof [LOWER BOUND(ID), 68 ms]
            (68) typed CpxTrs
            (69) RewriteLemmaProof [LOWER BOUND(ID), 45 ms]
            (70) typed CpxTrs
            (71) RewriteLemmaProof [LOWER BOUND(ID), 67 ms]
            (72) typed CpxTrs


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

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


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)
   zero(0) -> true
   zero(s(x)) -> false
   id(0) -> 0
   id(s(x)) -> s(id(x))
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
   if_mod(true, b1, b2, x, y) -> 0
   if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y)
   if2(true, b2, x, y) -> 0
   if2(false, b2, x, y) -> if3(b2, x, y)
   if3(true, x, y) -> mod(minus(x, y), s(y))
   if3(false, x, y) -> x

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^2).


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]
   zero(0) -> true [1]
   zero(s(x)) -> false [1]
   id(0) -> 0 [1]
   id(s(x)) -> s(id(x)) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y)) [1]
   if_mod(true, b1, b2, x, y) -> 0 [1]
   if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y) [1]
   if2(true, b2, x, y) -> 0 [1]
   if2(false, b2, x, y) -> if3(b2, x, y) [1]
   if3(true, x, y) -> mod(minus(x, y), s(y)) [1]
   if3(false, x, y) -> 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]
   zero(0) -> true [1]
   zero(s(x)) -> false [1]
   id(0) -> 0 [1]
   id(s(x)) -> s(id(x)) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y)) [1]
   if_mod(true, b1, b2, x, y) -> 0 [1]
   if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y) [1]
   if2(true, b2, x, y) -> 0 [1]
   if2(false, b2, x, y) -> if3(b2, x, y) [1]
   if3(true, x, y) -> mod(minus(x, y), s(y)) [1]
   if3(false, x, y) -> 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
zero :: 0:s -> true:false
id :: 0:s -> 0:s
minus :: 0:s -> 0:s -> 0:s
mod :: 0:s -> 0:s -> 0:s
if_mod :: true:false -> true:false -> true:false -> 0:s -> 0:s -> 0:s
if2 :: true:false -> true:false -> 0:s -> 0:s -> 0:s
if3 :: 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_5
   if2_4
   if3_3

(c) The following functions are completely defined:

   minus_2
   zero_1
   le_2
   id_1

Due to the following rules being added:

   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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   zero(0) -> true [1]
   zero(s(x)) -> false [1]
   id(0) -> 0 [1]
   id(s(x)) -> s(id(x)) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y)) [1]
   if_mod(true, b1, b2, x, y) -> 0 [1]
   if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y) [1]
   if2(true, b2, x, y) -> 0 [1]
   if2(false, b2, x, y) -> if3(b2, x, y) [1]
   if3(true, x, y) -> mod(minus(x, y), s(y)) [1]
   if3(false, x, y) -> x [1]
   minus(v0, v1) -> 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
zero :: 0:s -> true:false
id :: 0:s -> 0:s
minus :: 0:s -> 0:s -> 0:s
mod :: 0:s -> 0:s -> 0:s
if_mod :: true:false -> true:false -> true:false -> 0:s -> 0:s -> 0:s
if2 :: true:false -> true:false -> 0:s -> 0:s -> 0:s
if3 :: 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]
   zero(0) -> true [1]
   zero(s(x)) -> false [1]
   id(0) -> 0 [1]
   id(s(x)) -> s(id(x)) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   mod(0, 0) -> if_mod(true, true, true, 0, 0) [6]
   mod(0, s(x'')) -> if_mod(true, false, false, 0, s(id(x''))) [6]
   mod(s(x'), 0) -> if_mod(false, true, true, s(id(x')), 0) [6]
   mod(s(x'), s(x1)) -> if_mod(false, false, le(x1, x'), s(id(x')), s(id(x1))) [6]
   if_mod(true, b1, b2, x, y) -> 0 [1]
   if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y) [1]
   if2(true, b2, x, y) -> 0 [1]
   if2(false, b2, x, y) -> if3(b2, x, y) [1]
   if3(true, x, 0) -> mod(x, s(0)) [2]
   if3(true, s(x2), s(y')) -> mod(minus(x2, y'), s(s(y'))) [2]
   if3(true, x, y) -> mod(0, s(y)) [1]
   if3(false, x, y) -> x [1]
   minus(v0, v1) -> 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
zero :: 0:s -> true:false
id :: 0:s -> 0:s
minus :: 0:s -> 0:s -> 0:s
mod :: 0:s -> 0:s -> 0:s
if_mod :: true:false -> true:false -> true:false -> 0:s -> 0:s -> 0:s
if2 :: true:false -> true:false -> 0:s -> 0:s -> 0:s
if3 :: 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:

   id(z) -{ 1 }-> 0 :|: z = 0
   id(z) -{ 1 }-> 1 + id(x) :|: x >= 0, z = 1 + x
   if2(z, z', z'', z1) -{ 1 }-> if3(b2, x, y) :|: b2 >= 0, z1 = y, x >= 0, y >= 0, z' = b2, z'' = x, z = 0
   if2(z, z', z'', z1) -{ 1 }-> 0 :|: b2 >= 0, z1 = y, z = 1, x >= 0, y >= 0, z' = b2, z'' = x
   if3(z, z', z'') -{ 1 }-> x :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   if3(z, z', z'') -{ 2 }-> mod(x, 1 + 0) :|: z'' = 0, z' = x, z = 1, x >= 0
   if3(z, z', z'') -{ 2 }-> mod(minus(x2, y'), 1 + (1 + y')) :|: z' = 1 + x2, z = 1, y' >= 0, x2 >= 0, z'' = 1 + y'
   if3(z, z', z'') -{ 1 }-> mod(0, 1 + y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> if2(b1, b2, x, y) :|: b2 >= 0, z2 = y, b1 >= 0, x >= 0, y >= 0, z' = b1, z = 0, z1 = x, z'' = b2
   if_mod(z, z', z'', z1, z2) -{ 1 }-> 0 :|: b2 >= 0, z2 = y, b1 >= 0, z = 1, x >= 0, y >= 0, z' = b1, z1 = x, z'' = b2
   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') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   mod(z, z') -{ 6 }-> if_mod(1, 1, 1, 0, 0) :|: z = 0, z' = 0
   mod(z, z') -{ 6 }-> if_mod(1, 0, 0, 0, 1 + id(x'')) :|: z' = 1 + x'', x'' >= 0, z = 0
   mod(z, z') -{ 6 }-> if_mod(0, 1, 1, 1 + id(x'), 0) :|: z = 1 + x', x' >= 0, z' = 0
   mod(z, z') -{ 6 }-> if_mod(0, 0, le(x1, x'), 1 + id(x'), 1 + id(x1)) :|: z = 1 + x', x1 >= 0, x' >= 0, z' = 1 + x1
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: x >= 0, z = 1 + x


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

(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:

   id(z) -{ 1 }-> 0 :|: z = 0
   id(z) -{ 1 }-> 1 + id(z - 1) :|: z - 1 >= 0
   if2(z, z', z'', z1) -{ 1 }-> if3(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if3(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 0
   if3(z, z', z'') -{ 2 }-> mod(z', 1 + 0) :|: z'' = 0, z = 1, z' >= 0
   if3(z, z', z'') -{ 2 }-> mod(minus(z' - 1, z'' - 1), 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
   if3(z, z', z'') -{ 1 }-> mod(0, 1 + z'') :|: z = 1, z' >= 0, z'' >= 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> if2(z', z'', z1, z2) :|: z'' >= 0, z' >= 0, z1 >= 0, z2 >= 0, z = 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' >= 0, z' >= 0, z = 1, z1 >= 0, z2 >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 6 }-> if_mod(1, 1, 1, 0, 0) :|: z = 0, z' = 0
   mod(z, z') -{ 6 }-> if_mod(1, 0, 0, 0, 1 + id(z' - 1)) :|: z' - 1 >= 0, z = 0
   mod(z, z') -{ 6 }-> if_mod(0, 1, 1, 1 + id(z - 1), 0) :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 6 }-> if_mod(0, 0, le(z' - 1, z - 1), 1 + id(z - 1), 1 + id(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0


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

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

   { id }
   { minus }
   { le }
   { zero }
   { mod, if_mod, if2, if3 }

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

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

   id(z) -{ 1 }-> 0 :|: z = 0
   id(z) -{ 1 }-> 1 + id(z - 1) :|: z - 1 >= 0
   if2(z, z', z'', z1) -{ 1 }-> if3(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if3(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 0
   if3(z, z', z'') -{ 2 }-> mod(z', 1 + 0) :|: z'' = 0, z = 1, z' >= 0
   if3(z, z', z'') -{ 2 }-> mod(minus(z' - 1, z'' - 1), 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
   if3(z, z', z'') -{ 1 }-> mod(0, 1 + z'') :|: z = 1, z' >= 0, z'' >= 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> if2(z', z'', z1, z2) :|: z'' >= 0, z' >= 0, z1 >= 0, z2 >= 0, z = 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' >= 0, z' >= 0, z = 1, z1 >= 0, z2 >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 6 }-> if_mod(1, 1, 1, 0, 0) :|: z = 0, z' = 0
   mod(z, z') -{ 6 }-> if_mod(1, 0, 0, 0, 1 + id(z' - 1)) :|: z' - 1 >= 0, z = 0
   mod(z, z') -{ 6 }-> if_mod(0, 1, 1, 1 + id(z - 1), 0) :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 6 }-> if_mod(0, 0, le(z' - 1, z - 1), 1 + id(z - 1), 1 + id(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {id}, {minus}, {le}, {zero}, {mod,if_mod,if2,if3}

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

(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:

   id(z) -{ 1 }-> 0 :|: z = 0
   id(z) -{ 1 }-> 1 + id(z - 1) :|: z - 1 >= 0
   if2(z, z', z'', z1) -{ 1 }-> if3(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if3(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 0
   if3(z, z', z'') -{ 2 }-> mod(z', 1 + 0) :|: z'' = 0, z = 1, z' >= 0
   if3(z, z', z'') -{ 2 }-> mod(minus(z' - 1, z'' - 1), 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
   if3(z, z', z'') -{ 1 }-> mod(0, 1 + z'') :|: z = 1, z' >= 0, z'' >= 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> if2(z', z'', z1, z2) :|: z'' >= 0, z' >= 0, z1 >= 0, z2 >= 0, z = 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' >= 0, z' >= 0, z = 1, z1 >= 0, z2 >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 6 }-> if_mod(1, 1, 1, 0, 0) :|: z = 0, z' = 0
   mod(z, z') -{ 6 }-> if_mod(1, 0, 0, 0, 1 + id(z' - 1)) :|: z' - 1 >= 0, z = 0
   mod(z, z') -{ 6 }-> if_mod(0, 1, 1, 1 + id(z - 1), 0) :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 6 }-> if_mod(0, 0, le(z' - 1, z - 1), 1 + id(z - 1), 1 + id(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {id}, {minus}, {le}, {zero}, {mod,if_mod,if2,if3}

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

(17) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   id(z) -{ 1 }-> 0 :|: z = 0
   id(z) -{ 1 }-> 1 + id(z - 1) :|: z - 1 >= 0
   if2(z, z', z'', z1) -{ 1 }-> if3(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if3(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 0
   if3(z, z', z'') -{ 2 }-> mod(z', 1 + 0) :|: z'' = 0, z = 1, z' >= 0
   if3(z, z', z'') -{ 2 }-> mod(minus(z' - 1, z'' - 1), 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
   if3(z, z', z'') -{ 1 }-> mod(0, 1 + z'') :|: z = 1, z' >= 0, z'' >= 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> if2(z', z'', z1, z2) :|: z'' >= 0, z' >= 0, z1 >= 0, z2 >= 0, z = 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' >= 0, z' >= 0, z = 1, z1 >= 0, z2 >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 6 }-> if_mod(1, 1, 1, 0, 0) :|: z = 0, z' = 0
   mod(z, z') -{ 6 }-> if_mod(1, 0, 0, 0, 1 + id(z' - 1)) :|: z' - 1 >= 0, z = 0
   mod(z, z') -{ 6 }-> if_mod(0, 1, 1, 1 + id(z - 1), 0) :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 6 }-> if_mod(0, 0, le(z' - 1, z - 1), 1 + id(z - 1), 1 + id(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {id}, {minus}, {le}, {zero}, {mod,if_mod,if2,if3}
Previous analysis results are:
  id: runtime: ?, size: O(n^1) [z]

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   id(z) -{ 1 }-> 0 :|: z = 0
   id(z) -{ 1 }-> 1 + id(z - 1) :|: z - 1 >= 0
   if2(z, z', z'', z1) -{ 1 }-> if3(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if3(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 0
   if3(z, z', z'') -{ 2 }-> mod(z', 1 + 0) :|: z'' = 0, z = 1, z' >= 0
   if3(z, z', z'') -{ 2 }-> mod(minus(z' - 1, z'' - 1), 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
   if3(z, z', z'') -{ 1 }-> mod(0, 1 + z'') :|: z = 1, z' >= 0, z'' >= 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> if2(z', z'', z1, z2) :|: z'' >= 0, z' >= 0, z1 >= 0, z2 >= 0, z = 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' >= 0, z' >= 0, z = 1, z1 >= 0, z2 >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 6 }-> if_mod(1, 1, 1, 0, 0) :|: z = 0, z' = 0
   mod(z, z') -{ 6 }-> if_mod(1, 0, 0, 0, 1 + id(z' - 1)) :|: z' - 1 >= 0, z = 0
   mod(z, z') -{ 6 }-> if_mod(0, 1, 1, 1 + id(z - 1), 0) :|: z - 1 >= 0, z' = 0
   mod(z, z') -{ 6 }-> if_mod(0, 0, le(z' - 1, z - 1), 1 + id(z - 1), 1 + id(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {minus}, {le}, {zero}, {mod,if_mod,if2,if3}
Previous analysis results are:
  id: runtime: O(n^1) [1 + z], size: O(n^1) [z]

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

(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:

   id(z) -{ 1 }-> 0 :|: z = 0
   id(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   if2(z, z', z'', z1) -{ 1 }-> if3(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if3(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 0
   if3(z, z', z'') -{ 2 }-> mod(z', 1 + 0) :|: z'' = 0, z = 1, z' >= 0
   if3(z, z', z'') -{ 2 }-> mod(minus(z' - 1, z'' - 1), 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
   if3(z, z', z'') -{ 1 }-> mod(0, 1 + z'') :|: z = 1, z' >= 0, z'' >= 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> if2(z', z'', z1, z2) :|: z'' >= 0, z' >= 0, z1 >= 0, z2 >= 0, z = 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' >= 0, z' >= 0, z = 1, z1 >= 0, z2 >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 6 }-> if_mod(1, 1, 1, 0, 0) :|: z = 0, z' = 0
   mod(z, z') -{ 6 + z' }-> if_mod(1, 0, 0, 0, 1 + s') :|: s' >= 0, s' <= z' - 1, z' - 1 >= 0, z = 0
   mod(z, z') -{ 6 + z }-> if_mod(0, 1, 1, 1 + s'', 0) :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' = 0
   mod(z, z') -{ 6 + z + z' }-> if_mod(0, 0, le(z' - 1, z - 1), 1 + s1, 1 + s2) :|: s1 >= 0, s1 <= z - 1, s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {minus}, {le}, {zero}, {mod,if_mod,if2,if3}
Previous analysis results are:
  id: runtime: O(n^1) [1 + z], size: O(n^1) [z]

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

(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

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

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

   id(z) -{ 1 }-> 0 :|: z = 0
   id(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   if2(z, z', z'', z1) -{ 1 }-> if3(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if3(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 0
   if3(z, z', z'') -{ 2 }-> mod(z', 1 + 0) :|: z'' = 0, z = 1, z' >= 0
   if3(z, z', z'') -{ 2 }-> mod(minus(z' - 1, z'' - 1), 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
   if3(z, z', z'') -{ 1 }-> mod(0, 1 + z'') :|: z = 1, z' >= 0, z'' >= 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> if2(z', z'', z1, z2) :|: z'' >= 0, z' >= 0, z1 >= 0, z2 >= 0, z = 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' >= 0, z' >= 0, z = 1, z1 >= 0, z2 >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 6 }-> if_mod(1, 1, 1, 0, 0) :|: z = 0, z' = 0
   mod(z, z') -{ 6 + z' }-> if_mod(1, 0, 0, 0, 1 + s') :|: s' >= 0, s' <= z' - 1, z' - 1 >= 0, z = 0
   mod(z, z') -{ 6 + z }-> if_mod(0, 1, 1, 1 + s'', 0) :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' = 0
   mod(z, z') -{ 6 + z + z' }-> if_mod(0, 0, le(z' - 1, z - 1), 1 + s1, 1 + s2) :|: s1 >= 0, s1 <= z - 1, s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {minus}, {le}, {zero}, {mod,if_mod,if2,if3}
Previous analysis results are:
  id: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  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^1) with polynomial bound: 1 + z'

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

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

   id(z) -{ 1 }-> 0 :|: z = 0
   id(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   if2(z, z', z'', z1) -{ 1 }-> if3(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if3(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 0
   if3(z, z', z'') -{ 2 }-> mod(z', 1 + 0) :|: z'' = 0, z = 1, z' >= 0
   if3(z, z', z'') -{ 2 }-> mod(minus(z' - 1, z'' - 1), 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
   if3(z, z', z'') -{ 1 }-> mod(0, 1 + z'') :|: z = 1, z' >= 0, z'' >= 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> if2(z', z'', z1, z2) :|: z'' >= 0, z' >= 0, z1 >= 0, z2 >= 0, z = 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' >= 0, z' >= 0, z = 1, z1 >= 0, z2 >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 6 }-> if_mod(1, 1, 1, 0, 0) :|: z = 0, z' = 0
   mod(z, z') -{ 6 + z' }-> if_mod(1, 0, 0, 0, 1 + s') :|: s' >= 0, s' <= z' - 1, z' - 1 >= 0, z = 0
   mod(z, z') -{ 6 + z }-> if_mod(0, 1, 1, 1 + s'', 0) :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' = 0
   mod(z, z') -{ 6 + z + z' }-> if_mod(0, 0, le(z' - 1, z - 1), 1 + s1, 1 + s2) :|: s1 >= 0, s1 <= z - 1, s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {le}, {zero}, {mod,if_mod,if2,if3}
Previous analysis results are:
  id: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  minus: runtime: O(n^1) [1 + 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:

   id(z) -{ 1 }-> 0 :|: z = 0
   id(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   if2(z, z', z'', z1) -{ 1 }-> if3(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if3(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 0
   if3(z, z', z'') -{ 2 + z'' }-> mod(s4, 1 + (1 + (z'' - 1))) :|: s4 >= 0, s4 <= z' - 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0
   if3(z, z', z'') -{ 2 }-> mod(z', 1 + 0) :|: z'' = 0, z = 1, z' >= 0
   if3(z, z', z'') -{ 1 }-> mod(0, 1 + z'') :|: z = 1, z' >= 0, z'' >= 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> if2(z', z'', z1, z2) :|: z'' >= 0, z' >= 0, z1 >= 0, z2 >= 0, z = 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' >= 0, z' >= 0, z = 1, z1 >= 0, z2 >= 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') -{ 1 + z' }-> s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 6 }-> if_mod(1, 1, 1, 0, 0) :|: z = 0, z' = 0
   mod(z, z') -{ 6 + z' }-> if_mod(1, 0, 0, 0, 1 + s') :|: s' >= 0, s' <= z' - 1, z' - 1 >= 0, z = 0
   mod(z, z') -{ 6 + z }-> if_mod(0, 1, 1, 1 + s'', 0) :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' = 0
   mod(z, z') -{ 6 + z + z' }-> if_mod(0, 0, le(z' - 1, z - 1), 1 + s1, 1 + s2) :|: s1 >= 0, s1 <= z - 1, s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {le}, {zero}, {mod,if_mod,if2,if3}
Previous analysis results are:
  id: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]

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

(29) 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

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

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

   id(z) -{ 1 }-> 0 :|: z = 0
   id(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   if2(z, z', z'', z1) -{ 1 }-> if3(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if3(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 0
   if3(z, z', z'') -{ 2 + z'' }-> mod(s4, 1 + (1 + (z'' - 1))) :|: s4 >= 0, s4 <= z' - 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0
   if3(z, z', z'') -{ 2 }-> mod(z', 1 + 0) :|: z'' = 0, z = 1, z' >= 0
   if3(z, z', z'') -{ 1 }-> mod(0, 1 + z'') :|: z = 1, z' >= 0, z'' >= 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> if2(z', z'', z1, z2) :|: z'' >= 0, z' >= 0, z1 >= 0, z2 >= 0, z = 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' >= 0, z' >= 0, z = 1, z1 >= 0, z2 >= 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') -{ 1 + z' }-> s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 6 }-> if_mod(1, 1, 1, 0, 0) :|: z = 0, z' = 0
   mod(z, z') -{ 6 + z' }-> if_mod(1, 0, 0, 0, 1 + s') :|: s' >= 0, s' <= z' - 1, z' - 1 >= 0, z = 0
   mod(z, z') -{ 6 + z }-> if_mod(0, 1, 1, 1 + s'', 0) :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' = 0
   mod(z, z') -{ 6 + z + z' }-> if_mod(0, 0, le(z' - 1, z - 1), 1 + s1, 1 + s2) :|: s1 >= 0, s1 <= z - 1, s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {le}, {zero}, {mod,if_mod,if2,if3}
Previous analysis results are:
  id: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  le: runtime: ?, size: O(1) [1]

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

(31) 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'

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

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

   id(z) -{ 1 }-> 0 :|: z = 0
   id(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   if2(z, z', z'', z1) -{ 1 }-> if3(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if3(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 0
   if3(z, z', z'') -{ 2 + z'' }-> mod(s4, 1 + (1 + (z'' - 1))) :|: s4 >= 0, s4 <= z' - 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0
   if3(z, z', z'') -{ 2 }-> mod(z', 1 + 0) :|: z'' = 0, z = 1, z' >= 0
   if3(z, z', z'') -{ 1 }-> mod(0, 1 + z'') :|: z = 1, z' >= 0, z'' >= 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> if2(z', z'', z1, z2) :|: z'' >= 0, z' >= 0, z1 >= 0, z2 >= 0, z = 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' >= 0, z' >= 0, z = 1, z1 >= 0, z2 >= 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') -{ 1 + z' }-> s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 6 }-> if_mod(1, 1, 1, 0, 0) :|: z = 0, z' = 0
   mod(z, z') -{ 6 + z' }-> if_mod(1, 0, 0, 0, 1 + s') :|: s' >= 0, s' <= z' - 1, z' - 1 >= 0, z = 0
   mod(z, z') -{ 6 + z }-> if_mod(0, 1, 1, 1 + s'', 0) :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' = 0
   mod(z, z') -{ 6 + z + z' }-> if_mod(0, 0, le(z' - 1, z - 1), 1 + s1, 1 + s2) :|: s1 >= 0, s1 <= z - 1, s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {zero}, {mod,if_mod,if2,if3}
Previous analysis results are:
  id: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

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

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

   id(z) -{ 1 }-> 0 :|: z = 0
   id(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   if2(z, z', z'', z1) -{ 1 }-> if3(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if3(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 0
   if3(z, z', z'') -{ 2 + z'' }-> mod(s4, 1 + (1 + (z'' - 1))) :|: s4 >= 0, s4 <= z' - 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0
   if3(z, z', z'') -{ 2 }-> mod(z', 1 + 0) :|: z'' = 0, z = 1, z' >= 0
   if3(z, z', z'') -{ 1 }-> mod(0, 1 + z'') :|: z = 1, z' >= 0, z'' >= 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> if2(z', z'', z1, z2) :|: z'' >= 0, z' >= 0, z1 >= 0, z2 >= 0, z = 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' >= 0, z' >= 0, z = 1, z1 >= 0, z2 >= 0
   le(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 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') -{ 1 + z' }-> s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 6 }-> if_mod(1, 1, 1, 0, 0) :|: z = 0, z' = 0
   mod(z, z') -{ 6 + z' }-> if_mod(1, 0, 0, 0, 1 + s') :|: s' >= 0, s' <= z' - 1, z' - 1 >= 0, z = 0
   mod(z, z') -{ 6 + z }-> if_mod(0, 1, 1, 1 + s'', 0) :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' = 0
   mod(z, z') -{ 7 + 2*z + z' }-> if_mod(0, 0, s6, 1 + s1, 1 + s2) :|: s6 >= 0, s6 <= 1, s1 >= 0, s1 <= z - 1, s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {zero}, {mod,if_mod,if2,if3}
Previous analysis results are:
  id: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   id(z) -{ 1 }-> 0 :|: z = 0
   id(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   if2(z, z', z'', z1) -{ 1 }-> if3(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if3(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 0
   if3(z, z', z'') -{ 2 + z'' }-> mod(s4, 1 + (1 + (z'' - 1))) :|: s4 >= 0, s4 <= z' - 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0
   if3(z, z', z'') -{ 2 }-> mod(z', 1 + 0) :|: z'' = 0, z = 1, z' >= 0
   if3(z, z', z'') -{ 1 }-> mod(0, 1 + z'') :|: z = 1, z' >= 0, z'' >= 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> if2(z', z'', z1, z2) :|: z'' >= 0, z' >= 0, z1 >= 0, z2 >= 0, z = 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' >= 0, z' >= 0, z = 1, z1 >= 0, z2 >= 0
   le(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 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') -{ 1 + z' }-> s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 6 }-> if_mod(1, 1, 1, 0, 0) :|: z = 0, z' = 0
   mod(z, z') -{ 6 + z' }-> if_mod(1, 0, 0, 0, 1 + s') :|: s' >= 0, s' <= z' - 1, z' - 1 >= 0, z = 0
   mod(z, z') -{ 6 + z }-> if_mod(0, 1, 1, 1 + s'', 0) :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' = 0
   mod(z, z') -{ 7 + 2*z + z' }-> if_mod(0, 0, s6, 1 + s1, 1 + s2) :|: s6 >= 0, s6 <= 1, s1 >= 0, s1 <= z - 1, s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {zero}, {mod,if_mod,if2,if3}
Previous analysis results are:
  id: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  zero: runtime: ?, size: O(1) [1]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: zero
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:

   id(z) -{ 1 }-> 0 :|: z = 0
   id(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   if2(z, z', z'', z1) -{ 1 }-> if3(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if3(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 0
   if3(z, z', z'') -{ 2 + z'' }-> mod(s4, 1 + (1 + (z'' - 1))) :|: s4 >= 0, s4 <= z' - 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0
   if3(z, z', z'') -{ 2 }-> mod(z', 1 + 0) :|: z'' = 0, z = 1, z' >= 0
   if3(z, z', z'') -{ 1 }-> mod(0, 1 + z'') :|: z = 1, z' >= 0, z'' >= 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> if2(z', z'', z1, z2) :|: z'' >= 0, z' >= 0, z1 >= 0, z2 >= 0, z = 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' >= 0, z' >= 0, z = 1, z1 >= 0, z2 >= 0
   le(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 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') -{ 1 + z' }-> s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 6 }-> if_mod(1, 1, 1, 0, 0) :|: z = 0, z' = 0
   mod(z, z') -{ 6 + z' }-> if_mod(1, 0, 0, 0, 1 + s') :|: s' >= 0, s' <= z' - 1, z' - 1 >= 0, z = 0
   mod(z, z') -{ 6 + z }-> if_mod(0, 1, 1, 1 + s'', 0) :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' = 0
   mod(z, z') -{ 7 + 2*z + z' }-> if_mod(0, 0, s6, 1 + s1, 1 + s2) :|: s6 >= 0, s6 <= 1, s1 >= 0, s1 <= z - 1, s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {mod,if_mod,if2,if3}
Previous analysis results are:
  id: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  zero: runtime: O(1) [1], size: O(1) [1]

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

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

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

   id(z) -{ 1 }-> 0 :|: z = 0
   id(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   if2(z, z', z'', z1) -{ 1 }-> if3(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if3(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 0
   if3(z, z', z'') -{ 2 + z'' }-> mod(s4, 1 + (1 + (z'' - 1))) :|: s4 >= 0, s4 <= z' - 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0
   if3(z, z', z'') -{ 2 }-> mod(z', 1 + 0) :|: z'' = 0, z = 1, z' >= 0
   if3(z, z', z'') -{ 1 }-> mod(0, 1 + z'') :|: z = 1, z' >= 0, z'' >= 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> if2(z', z'', z1, z2) :|: z'' >= 0, z' >= 0, z1 >= 0, z2 >= 0, z = 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' >= 0, z' >= 0, z = 1, z1 >= 0, z2 >= 0
   le(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 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') -{ 1 + z' }-> s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 6 }-> if_mod(1, 1, 1, 0, 0) :|: z = 0, z' = 0
   mod(z, z') -{ 6 + z' }-> if_mod(1, 0, 0, 0, 1 + s') :|: s' >= 0, s' <= z' - 1, z' - 1 >= 0, z = 0
   mod(z, z') -{ 6 + z }-> if_mod(0, 1, 1, 1 + s'', 0) :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' = 0
   mod(z, z') -{ 7 + 2*z + z' }-> if_mod(0, 0, s6, 1 + s1, 1 + s2) :|: s6 >= 0, s6 <= 1, s1 >= 0, s1 <= z - 1, s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {mod,if_mod,if2,if3}
Previous analysis results are:
  id: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  zero: runtime: O(1) [1], size: O(1) [1]

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

(41) 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: z1

Computed SIZE bound using KoAT for: if2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z''

Computed SIZE bound using KoAT for: if3
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z'

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

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

   id(z) -{ 1 }-> 0 :|: z = 0
   id(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   if2(z, z', z'', z1) -{ 1 }-> if3(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if3(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 0
   if3(z, z', z'') -{ 2 + z'' }-> mod(s4, 1 + (1 + (z'' - 1))) :|: s4 >= 0, s4 <= z' - 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0
   if3(z, z', z'') -{ 2 }-> mod(z', 1 + 0) :|: z'' = 0, z = 1, z' >= 0
   if3(z, z', z'') -{ 1 }-> mod(0, 1 + z'') :|: z = 1, z' >= 0, z'' >= 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> if2(z', z'', z1, z2) :|: z'' >= 0, z' >= 0, z1 >= 0, z2 >= 0, z = 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' >= 0, z' >= 0, z = 1, z1 >= 0, z2 >= 0
   le(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 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') -{ 1 + z' }-> s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 6 }-> if_mod(1, 1, 1, 0, 0) :|: z = 0, z' = 0
   mod(z, z') -{ 6 + z' }-> if_mod(1, 0, 0, 0, 1 + s') :|: s' >= 0, s' <= z' - 1, z' - 1 >= 0, z = 0
   mod(z, z') -{ 6 + z }-> if_mod(0, 1, 1, 1 + s'', 0) :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' = 0
   mod(z, z') -{ 7 + 2*z + z' }-> if_mod(0, 0, s6, 1 + s1, 1 + s2) :|: s6 >= 0, s6 <= 1, s1 >= 0, s1 <= z - 1, s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {mod,if_mod,if2,if3}
Previous analysis results are:
  id: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  zero: runtime: O(1) [1], size: O(1) [1]
  mod: runtime: ?, size: O(n^1) [z]
  if_mod: runtime: ?, size: O(n^1) [z1]
  if2: runtime: ?, size: O(n^1) [z'']
  if3: runtime: ?, size: O(n^1) [z']

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: mod
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 51 + 31*z + 3*z*z' + 5*z^2 + 4*z'

Computed RUNTIME bound using KoAT for: if_mod
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 146 + 58*z1 + 3*z1*z2 + 10*z1^2 + 6*z2

Computed RUNTIME bound using KoAT for: if2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 144 + 58*z'' + 3*z''*z1 + 10*z''^2 + 6*z1

Computed RUNTIME bound using KoAT for: if3
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 142 + 58*z' + 3*z'*z'' + 10*z'^2 + 6*z''

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

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

   id(z) -{ 1 }-> 0 :|: z = 0
   id(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   if2(z, z', z'', z1) -{ 1 }-> if3(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if3(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 0
   if3(z, z', z'') -{ 2 + z'' }-> mod(s4, 1 + (1 + (z'' - 1))) :|: s4 >= 0, s4 <= z' - 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0
   if3(z, z', z'') -{ 2 }-> mod(z', 1 + 0) :|: z'' = 0, z = 1, z' >= 0
   if3(z, z', z'') -{ 1 }-> mod(0, 1 + z'') :|: z = 1, z' >= 0, z'' >= 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> if2(z', z'', z1, z2) :|: z'' >= 0, z' >= 0, z1 >= 0, z2 >= 0, z = 0
   if_mod(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' >= 0, z' >= 0, z = 1, z1 >= 0, z2 >= 0
   le(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 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') -{ 1 + z' }-> s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   mod(z, z') -{ 6 }-> if_mod(1, 1, 1, 0, 0) :|: z = 0, z' = 0
   mod(z, z') -{ 6 + z' }-> if_mod(1, 0, 0, 0, 1 + s') :|: s' >= 0, s' <= z' - 1, z' - 1 >= 0, z = 0
   mod(z, z') -{ 6 + z }-> if_mod(0, 1, 1, 1 + s'', 0) :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' = 0
   mod(z, z') -{ 7 + 2*z + z' }-> if_mod(0, 0, s6, 1 + s1, 1 + s2) :|: s6 >= 0, s6 <= 1, s1 >= 0, s1 <= z - 1, s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  id: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  zero: runtime: O(1) [1], size: O(1) [1]
  mod: runtime: O(n^2) [51 + 31*z + 3*z*z' + 5*z^2 + 4*z'], size: O(n^1) [z]
  if_mod: runtime: O(n^2) [146 + 58*z1 + 3*z1*z2 + 10*z1^2 + 6*z2], size: O(n^1) [z1]
  if2: runtime: O(n^2) [144 + 58*z'' + 3*z''*z1 + 10*z''^2 + 6*z1], size: O(n^1) [z'']
  if3: runtime: O(n^2) [142 + 58*z' + 3*z'*z'' + 10*z'^2 + 6*z''], size: O(n^1) [z']

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

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

(46)
BOUNDS(1, n^2)

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

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   zero(0) -> true
   zero(s(z0)) -> false
   id(0) -> 0
   id(s(z0)) -> s(id(z0))
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   mod(z0, z1) -> if_mod(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1))
   if_mod(true, z0, z1, z2, z3) -> 0
   if_mod(false, z0, z1, z2, z3) -> if2(z0, z1, z2, z3)
   if2(true, z0, z1, z2) -> 0
   if2(false, z0, z1, z2) -> if3(z0, z1, z2)
   if3(true, z0, z1) -> mod(minus(z0, z1), s(z1))
   if3(false, z0, z1) -> z0
Tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   ZERO(0) -> c3
   ZERO(s(z0)) -> c4
   ID(0) -> c5
   ID(s(z0)) -> c6(ID(z0))
   MINUS(z0, 0) -> c7
   MINUS(s(z0), s(z1)) -> c8(MINUS(z0, z1))
   MOD(z0, z1) -> c9(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z0))
   MOD(z0, z1) -> c10(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z1))
   MOD(z0, z1) -> c11(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), LE(z1, z0))
   MOD(z0, z1) -> c12(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z0))
   MOD(z0, z1) -> c13(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z1))
   IF_MOD(true, z0, z1, z2, z3) -> c14
   IF_MOD(false, z0, z1, z2, z3) -> c15(IF2(z0, z1, z2, z3))
   IF2(true, z0, z1, z2) -> c16
   IF2(false, z0, z1, z2) -> c17(IF3(z0, z1, z2))
   IF3(true, z0, z1) -> c18(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF3(false, z0, z1) -> c19
S tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   ZERO(0) -> c3
   ZERO(s(z0)) -> c4
   ID(0) -> c5
   ID(s(z0)) -> c6(ID(z0))
   MINUS(z0, 0) -> c7
   MINUS(s(z0), s(z1)) -> c8(MINUS(z0, z1))
   MOD(z0, z1) -> c9(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z0))
   MOD(z0, z1) -> c10(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z1))
   MOD(z0, z1) -> c11(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), LE(z1, z0))
   MOD(z0, z1) -> c12(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z0))
   MOD(z0, z1) -> c13(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z1))
   IF_MOD(true, z0, z1, z2, z3) -> c14
   IF_MOD(false, z0, z1, z2, z3) -> c15(IF2(z0, z1, z2, z3))
   IF2(true, z0, z1, z2) -> c16
   IF2(false, z0, z1, z2) -> c17(IF3(z0, z1, z2))
   IF3(true, z0, z1) -> c18(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF3(false, z0, z1) -> c19
K tuples:none
Defined Rule Symbols:   le_2, zero_1, id_1, minus_2, mod_2, if_mod_5, if2_4, if3_3

Defined Pair Symbols:   LE_2, ZERO_1, ID_1, MINUS_2, MOD_2, IF_MOD_5, IF2_4, IF3_3

Compound Symbols:   c, c1, c2_1, c3, c4, c5, c6_1, c7, c8_1, c9_2, c10_2, c11_2, c12_2, c13_2, c14, c15_1, c16, c17_1, c18_2, c19


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

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

(50)
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:

   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   ZERO(0) -> c3
   ZERO(s(z0)) -> c4
   ID(0) -> c5
   ID(s(z0)) -> c6(ID(z0))
   MINUS(z0, 0) -> c7
   MINUS(s(z0), s(z1)) -> c8(MINUS(z0, z1))
   MOD(z0, z1) -> c9(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z0))
   MOD(z0, z1) -> c10(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z1))
   MOD(z0, z1) -> c11(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), LE(z1, z0))
   MOD(z0, z1) -> c12(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z0))
   MOD(z0, z1) -> c13(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z1))
   IF_MOD(true, z0, z1, z2, z3) -> c14
   IF_MOD(false, z0, z1, z2, z3) -> c15(IF2(z0, z1, z2, z3))
   IF2(true, z0, z1, z2) -> c16
   IF2(false, z0, z1, z2) -> c17(IF3(z0, z1, z2))
   IF3(true, z0, z1) -> c18(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF3(false, z0, z1) -> c19

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

   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   zero(0) -> true
   zero(s(z0)) -> false
   id(0) -> 0
   id(s(z0)) -> s(id(z0))
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   mod(z0, z1) -> if_mod(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1))
   if_mod(true, z0, z1, z2, z3) -> 0
   if_mod(false, z0, z1, z2, z3) -> if2(z0, z1, z2, z3)
   if2(true, z0, z1, z2) -> 0
   if2(false, z0, z1, z2) -> if3(z0, z1, z2)
   if3(true, z0, z1) -> mod(minus(z0, z1), s(z1))
   if3(false, z0, z1) -> z0

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

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

(52)
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:

   LE(0', z0) -> c
   LE(s(z0), 0') -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   ZERO(0') -> c3
   ZERO(s(z0)) -> c4
   ID(0') -> c5
   ID(s(z0)) -> c6(ID(z0))
   MINUS(z0, 0') -> c7
   MINUS(s(z0), s(z1)) -> c8(MINUS(z0, z1))
   MOD(z0, z1) -> c9(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z0))
   MOD(z0, z1) -> c10(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z1))
   MOD(z0, z1) -> c11(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), LE(z1, z0))
   MOD(z0, z1) -> c12(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z0))
   MOD(z0, z1) -> c13(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z1))
   IF_MOD(true, z0, z1, z2, z3) -> c14
   IF_MOD(false, z0, z1, z2, z3) -> c15(IF2(z0, z1, z2, z3))
   IF2(true, z0, z1, z2) -> c16
   IF2(false, z0, z1, z2) -> c17(IF3(z0, z1, z2))
   IF3(true, z0, z1) -> c18(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF3(false, z0, z1) -> c19

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

   le(0', z0) -> true
   le(s(z0), 0') -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   zero(0') -> true
   zero(s(z0)) -> false
   id(0') -> 0'
   id(s(z0)) -> s(id(z0))
   minus(z0, 0') -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   mod(z0, z1) -> if_mod(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1))
   if_mod(true, z0, z1, z2, z3) -> 0'
   if_mod(false, z0, z1, z2, z3) -> if2(z0, z1, z2, z3)
   if2(true, z0, z1, z2) -> 0'
   if2(false, z0, z1, z2) -> if3(z0, z1, z2)
   if3(true, z0, z1) -> mod(minus(z0, z1), s(z1))
   if3(false, z0, z1) -> z0

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

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

(54)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
ZERO(0') -> c3
ZERO(s(z0)) -> c4
ID(0') -> c5
ID(s(z0)) -> c6(ID(z0))
MINUS(z0, 0') -> c7
MINUS(s(z0), s(z1)) -> c8(MINUS(z0, z1))
MOD(z0, z1) -> c9(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z0))
MOD(z0, z1) -> c10(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z1))
MOD(z0, z1) -> c11(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), LE(z1, z0))
MOD(z0, z1) -> c12(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z0))
MOD(z0, z1) -> c13(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z1))
IF_MOD(true, z0, z1, z2, z3) -> c14
IF_MOD(false, z0, z1, z2, z3) -> c15(IF2(z0, z1, z2, z3))
IF2(true, z0, z1, z2) -> c16
IF2(false, z0, z1, z2) -> c17(IF3(z0, z1, z2))
IF3(true, z0, z1) -> c18(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF3(false, z0, z1) -> c19
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
zero(0') -> true
zero(s(z0)) -> false
id(0') -> 0'
id(s(z0)) -> s(id(z0))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
mod(z0, z1) -> if_mod(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1))
if_mod(true, z0, z1, z2, z3) -> 0'
if_mod(false, z0, z1, z2, z3) -> if2(z0, z1, z2, z3)
if2(true, z0, z1, z2) -> 0'
if2(false, z0, z1, z2) -> if3(z0, z1, z2)
if3(true, z0, z1) -> mod(minus(z0, z1), s(z1))
if3(false, z0, z1) -> z0

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
ZERO :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
ID :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
MINUS :: 0':s -> 0':s -> c7:c8
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
MOD :: 0':s -> 0':s -> c9:c10:c11:c12:c13
c9 :: c14:c15 -> c3:c4 -> c9:c10:c11:c12:c13
IF_MOD :: true:false -> true:false -> true:false -> 0':s -> 0':s -> c14:c15
zero :: 0':s -> true:false
le :: 0':s -> 0':s -> true:false
id :: 0':s -> 0':s
c10 :: c14:c15 -> c3:c4 -> c9:c10:c11:c12:c13
c11 :: c14:c15 -> c:c1:c2 -> c9:c10:c11:c12:c13
c12 :: c14:c15 -> c5:c6 -> c9:c10:c11:c12:c13
c13 :: c14:c15 -> c5:c6 -> c9:c10:c11:c12:c13
true :: true:false
c14 :: c14:c15
false :: true:false
c15 :: c16:c17 -> c14:c15
IF2 :: true:false -> true:false -> 0':s -> 0':s -> c16:c17
c16 :: c16:c17
c17 :: c18:c19 -> c16:c17
IF3 :: true:false -> 0':s -> 0':s -> c18:c19
c18 :: c9:c10:c11:c12:c13 -> c7:c8 -> c18:c19
minus :: 0':s -> 0':s -> 0':s
c19 :: c18:c19
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s
if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s
if3 :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_20 :: c:c1:c2
hole_0':s2_20 :: 0':s
hole_c3:c43_20 :: c3:c4
hole_c5:c64_20 :: c5:c6
hole_c7:c85_20 :: c7:c8
hole_c9:c10:c11:c12:c136_20 :: c9:c10:c11:c12:c13
hole_c14:c157_20 :: c14:c15
hole_true:false8_20 :: true:false
hole_c16:c179_20 :: c16:c17
hole_c18:c1910_20 :: c18:c19
gen_c:c1:c211_20 :: Nat -> c:c1:c2
gen_0':s12_20 :: Nat -> 0':s
gen_c5:c613_20 :: Nat -> c5:c6
gen_c7:c814_20 :: Nat -> c7:c8

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

(55) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
LE, ID, MINUS, MOD, le, id, minus, mod

They will be analysed ascendingly in the following order:
LE < MOD
ID < MOD
MINUS < MOD
le < MOD
id < MOD
minus < MOD
le < mod
id < mod
minus < mod

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

(56)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
ZERO(0') -> c3
ZERO(s(z0)) -> c4
ID(0') -> c5
ID(s(z0)) -> c6(ID(z0))
MINUS(z0, 0') -> c7
MINUS(s(z0), s(z1)) -> c8(MINUS(z0, z1))
MOD(z0, z1) -> c9(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z0))
MOD(z0, z1) -> c10(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z1))
MOD(z0, z1) -> c11(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), LE(z1, z0))
MOD(z0, z1) -> c12(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z0))
MOD(z0, z1) -> c13(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z1))
IF_MOD(true, z0, z1, z2, z3) -> c14
IF_MOD(false, z0, z1, z2, z3) -> c15(IF2(z0, z1, z2, z3))
IF2(true, z0, z1, z2) -> c16
IF2(false, z0, z1, z2) -> c17(IF3(z0, z1, z2))
IF3(true, z0, z1) -> c18(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF3(false, z0, z1) -> c19
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
zero(0') -> true
zero(s(z0)) -> false
id(0') -> 0'
id(s(z0)) -> s(id(z0))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
mod(z0, z1) -> if_mod(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1))
if_mod(true, z0, z1, z2, z3) -> 0'
if_mod(false, z0, z1, z2, z3) -> if2(z0, z1, z2, z3)
if2(true, z0, z1, z2) -> 0'
if2(false, z0, z1, z2) -> if3(z0, z1, z2)
if3(true, z0, z1) -> mod(minus(z0, z1), s(z1))
if3(false, z0, z1) -> z0

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
ZERO :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
ID :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
MINUS :: 0':s -> 0':s -> c7:c8
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
MOD :: 0':s -> 0':s -> c9:c10:c11:c12:c13
c9 :: c14:c15 -> c3:c4 -> c9:c10:c11:c12:c13
IF_MOD :: true:false -> true:false -> true:false -> 0':s -> 0':s -> c14:c15
zero :: 0':s -> true:false
le :: 0':s -> 0':s -> true:false
id :: 0':s -> 0':s
c10 :: c14:c15 -> c3:c4 -> c9:c10:c11:c12:c13
c11 :: c14:c15 -> c:c1:c2 -> c9:c10:c11:c12:c13
c12 :: c14:c15 -> c5:c6 -> c9:c10:c11:c12:c13
c13 :: c14:c15 -> c5:c6 -> c9:c10:c11:c12:c13
true :: true:false
c14 :: c14:c15
false :: true:false
c15 :: c16:c17 -> c14:c15
IF2 :: true:false -> true:false -> 0':s -> 0':s -> c16:c17
c16 :: c16:c17
c17 :: c18:c19 -> c16:c17
IF3 :: true:false -> 0':s -> 0':s -> c18:c19
c18 :: c9:c10:c11:c12:c13 -> c7:c8 -> c18:c19
minus :: 0':s -> 0':s -> 0':s
c19 :: c18:c19
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s
if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s
if3 :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_20 :: c:c1:c2
hole_0':s2_20 :: 0':s
hole_c3:c43_20 :: c3:c4
hole_c5:c64_20 :: c5:c6
hole_c7:c85_20 :: c7:c8
hole_c9:c10:c11:c12:c136_20 :: c9:c10:c11:c12:c13
hole_c14:c157_20 :: c14:c15
hole_true:false8_20 :: true:false
hole_c16:c179_20 :: c16:c17
hole_c18:c1910_20 :: c18:c19
gen_c:c1:c211_20 :: Nat -> c:c1:c2
gen_0':s12_20 :: Nat -> 0':s
gen_c5:c613_20 :: Nat -> c5:c6
gen_c7:c814_20 :: Nat -> c7:c8


Generator Equations:
gen_c:c1:c211_20(0) <=> c
gen_c:c1:c211_20(+(x, 1)) <=> c2(gen_c:c1:c211_20(x))
gen_0':s12_20(0) <=> 0'
gen_0':s12_20(+(x, 1)) <=> s(gen_0':s12_20(x))
gen_c5:c613_20(0) <=> c5
gen_c5:c613_20(+(x, 1)) <=> c6(gen_c5:c613_20(x))
gen_c7:c814_20(0) <=> c7
gen_c7:c814_20(+(x, 1)) <=> c8(gen_c7:c814_20(x))


The following defined symbols remain to be analysed:
LE, ID, MINUS, MOD, le, id, minus, mod

They will be analysed ascendingly in the following order:
LE < MOD
ID < MOD
MINUS < MOD
le < MOD
id < MOD
minus < MOD
le < mod
id < mod
minus < mod

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

(57) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LE(gen_0':s12_20(n16_20), gen_0':s12_20(n16_20)) -> gen_c:c1:c211_20(n16_20), rt in Omega(1 + n16_20)

Induction Base:
LE(gen_0':s12_20(0), gen_0':s12_20(0)) ->_R^Omega(1)
c

Induction Step:
LE(gen_0':s12_20(+(n16_20, 1)), gen_0':s12_20(+(n16_20, 1))) ->_R^Omega(1)
c2(LE(gen_0':s12_20(n16_20), gen_0':s12_20(n16_20))) ->_IH
c2(gen_c:c1:c211_20(c17_20))

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

(58)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
ZERO(0') -> c3
ZERO(s(z0)) -> c4
ID(0') -> c5
ID(s(z0)) -> c6(ID(z0))
MINUS(z0, 0') -> c7
MINUS(s(z0), s(z1)) -> c8(MINUS(z0, z1))
MOD(z0, z1) -> c9(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z0))
MOD(z0, z1) -> c10(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z1))
MOD(z0, z1) -> c11(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), LE(z1, z0))
MOD(z0, z1) -> c12(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z0))
MOD(z0, z1) -> c13(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z1))
IF_MOD(true, z0, z1, z2, z3) -> c14
IF_MOD(false, z0, z1, z2, z3) -> c15(IF2(z0, z1, z2, z3))
IF2(true, z0, z1, z2) -> c16
IF2(false, z0, z1, z2) -> c17(IF3(z0, z1, z2))
IF3(true, z0, z1) -> c18(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF3(false, z0, z1) -> c19
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
zero(0') -> true
zero(s(z0)) -> false
id(0') -> 0'
id(s(z0)) -> s(id(z0))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
mod(z0, z1) -> if_mod(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1))
if_mod(true, z0, z1, z2, z3) -> 0'
if_mod(false, z0, z1, z2, z3) -> if2(z0, z1, z2, z3)
if2(true, z0, z1, z2) -> 0'
if2(false, z0, z1, z2) -> if3(z0, z1, z2)
if3(true, z0, z1) -> mod(minus(z0, z1), s(z1))
if3(false, z0, z1) -> z0

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
ZERO :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
ID :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
MINUS :: 0':s -> 0':s -> c7:c8
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
MOD :: 0':s -> 0':s -> c9:c10:c11:c12:c13
c9 :: c14:c15 -> c3:c4 -> c9:c10:c11:c12:c13
IF_MOD :: true:false -> true:false -> true:false -> 0':s -> 0':s -> c14:c15
zero :: 0':s -> true:false
le :: 0':s -> 0':s -> true:false
id :: 0':s -> 0':s
c10 :: c14:c15 -> c3:c4 -> c9:c10:c11:c12:c13
c11 :: c14:c15 -> c:c1:c2 -> c9:c10:c11:c12:c13
c12 :: c14:c15 -> c5:c6 -> c9:c10:c11:c12:c13
c13 :: c14:c15 -> c5:c6 -> c9:c10:c11:c12:c13
true :: true:false
c14 :: c14:c15
false :: true:false
c15 :: c16:c17 -> c14:c15
IF2 :: true:false -> true:false -> 0':s -> 0':s -> c16:c17
c16 :: c16:c17
c17 :: c18:c19 -> c16:c17
IF3 :: true:false -> 0':s -> 0':s -> c18:c19
c18 :: c9:c10:c11:c12:c13 -> c7:c8 -> c18:c19
minus :: 0':s -> 0':s -> 0':s
c19 :: c18:c19
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s
if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s
if3 :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_20 :: c:c1:c2
hole_0':s2_20 :: 0':s
hole_c3:c43_20 :: c3:c4
hole_c5:c64_20 :: c5:c6
hole_c7:c85_20 :: c7:c8
hole_c9:c10:c11:c12:c136_20 :: c9:c10:c11:c12:c13
hole_c14:c157_20 :: c14:c15
hole_true:false8_20 :: true:false
hole_c16:c179_20 :: c16:c17
hole_c18:c1910_20 :: c18:c19
gen_c:c1:c211_20 :: Nat -> c:c1:c2
gen_0':s12_20 :: Nat -> 0':s
gen_c5:c613_20 :: Nat -> c5:c6
gen_c7:c814_20 :: Nat -> c7:c8


Generator Equations:
gen_c:c1:c211_20(0) <=> c
gen_c:c1:c211_20(+(x, 1)) <=> c2(gen_c:c1:c211_20(x))
gen_0':s12_20(0) <=> 0'
gen_0':s12_20(+(x, 1)) <=> s(gen_0':s12_20(x))
gen_c5:c613_20(0) <=> c5
gen_c5:c613_20(+(x, 1)) <=> c6(gen_c5:c613_20(x))
gen_c7:c814_20(0) <=> c7
gen_c7:c814_20(+(x, 1)) <=> c8(gen_c7:c814_20(x))


The following defined symbols remain to be analysed:
LE, ID, MINUS, MOD, le, id, minus, mod

They will be analysed ascendingly in the following order:
LE < MOD
ID < MOD
MINUS < MOD
le < MOD
id < MOD
minus < MOD
le < mod
id < mod
minus < mod

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

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

(61)
BOUNDS(n^1, INF)

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

(62)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
ZERO(0') -> c3
ZERO(s(z0)) -> c4
ID(0') -> c5
ID(s(z0)) -> c6(ID(z0))
MINUS(z0, 0') -> c7
MINUS(s(z0), s(z1)) -> c8(MINUS(z0, z1))
MOD(z0, z1) -> c9(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z0))
MOD(z0, z1) -> c10(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z1))
MOD(z0, z1) -> c11(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), LE(z1, z0))
MOD(z0, z1) -> c12(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z0))
MOD(z0, z1) -> c13(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z1))
IF_MOD(true, z0, z1, z2, z3) -> c14
IF_MOD(false, z0, z1, z2, z3) -> c15(IF2(z0, z1, z2, z3))
IF2(true, z0, z1, z2) -> c16
IF2(false, z0, z1, z2) -> c17(IF3(z0, z1, z2))
IF3(true, z0, z1) -> c18(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF3(false, z0, z1) -> c19
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
zero(0') -> true
zero(s(z0)) -> false
id(0') -> 0'
id(s(z0)) -> s(id(z0))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
mod(z0, z1) -> if_mod(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1))
if_mod(true, z0, z1, z2, z3) -> 0'
if_mod(false, z0, z1, z2, z3) -> if2(z0, z1, z2, z3)
if2(true, z0, z1, z2) -> 0'
if2(false, z0, z1, z2) -> if3(z0, z1, z2)
if3(true, z0, z1) -> mod(minus(z0, z1), s(z1))
if3(false, z0, z1) -> z0

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
ZERO :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
ID :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
MINUS :: 0':s -> 0':s -> c7:c8
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
MOD :: 0':s -> 0':s -> c9:c10:c11:c12:c13
c9 :: c14:c15 -> c3:c4 -> c9:c10:c11:c12:c13
IF_MOD :: true:false -> true:false -> true:false -> 0':s -> 0':s -> c14:c15
zero :: 0':s -> true:false
le :: 0':s -> 0':s -> true:false
id :: 0':s -> 0':s
c10 :: c14:c15 -> c3:c4 -> c9:c10:c11:c12:c13
c11 :: c14:c15 -> c:c1:c2 -> c9:c10:c11:c12:c13
c12 :: c14:c15 -> c5:c6 -> c9:c10:c11:c12:c13
c13 :: c14:c15 -> c5:c6 -> c9:c10:c11:c12:c13
true :: true:false
c14 :: c14:c15
false :: true:false
c15 :: c16:c17 -> c14:c15
IF2 :: true:false -> true:false -> 0':s -> 0':s -> c16:c17
c16 :: c16:c17
c17 :: c18:c19 -> c16:c17
IF3 :: true:false -> 0':s -> 0':s -> c18:c19
c18 :: c9:c10:c11:c12:c13 -> c7:c8 -> c18:c19
minus :: 0':s -> 0':s -> 0':s
c19 :: c18:c19
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s
if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s
if3 :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_20 :: c:c1:c2
hole_0':s2_20 :: 0':s
hole_c3:c43_20 :: c3:c4
hole_c5:c64_20 :: c5:c6
hole_c7:c85_20 :: c7:c8
hole_c9:c10:c11:c12:c136_20 :: c9:c10:c11:c12:c13
hole_c14:c157_20 :: c14:c15
hole_true:false8_20 :: true:false
hole_c16:c179_20 :: c16:c17
hole_c18:c1910_20 :: c18:c19
gen_c:c1:c211_20 :: Nat -> c:c1:c2
gen_0':s12_20 :: Nat -> 0':s
gen_c5:c613_20 :: Nat -> c5:c6
gen_c7:c814_20 :: Nat -> c7:c8


Lemmas:
LE(gen_0':s12_20(n16_20), gen_0':s12_20(n16_20)) -> gen_c:c1:c211_20(n16_20), rt in Omega(1 + n16_20)


Generator Equations:
gen_c:c1:c211_20(0) <=> c
gen_c:c1:c211_20(+(x, 1)) <=> c2(gen_c:c1:c211_20(x))
gen_0':s12_20(0) <=> 0'
gen_0':s12_20(+(x, 1)) <=> s(gen_0':s12_20(x))
gen_c5:c613_20(0) <=> c5
gen_c5:c613_20(+(x, 1)) <=> c6(gen_c5:c613_20(x))
gen_c7:c814_20(0) <=> c7
gen_c7:c814_20(+(x, 1)) <=> c8(gen_c7:c814_20(x))


The following defined symbols remain to be analysed:
ID, MINUS, MOD, le, id, minus, mod

They will be analysed ascendingly in the following order:
ID < MOD
MINUS < MOD
le < MOD
id < MOD
minus < MOD
le < mod
id < mod
minus < mod

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

(63) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ID(gen_0':s12_20(n740_20)) -> gen_c5:c613_20(n740_20), rt in Omega(1 + n740_20)

Induction Base:
ID(gen_0':s12_20(0)) ->_R^Omega(1)
c5

Induction Step:
ID(gen_0':s12_20(+(n740_20, 1))) ->_R^Omega(1)
c6(ID(gen_0':s12_20(n740_20))) ->_IH
c6(gen_c5:c613_20(c741_20))

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

(64)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
ZERO(0') -> c3
ZERO(s(z0)) -> c4
ID(0') -> c5
ID(s(z0)) -> c6(ID(z0))
MINUS(z0, 0') -> c7
MINUS(s(z0), s(z1)) -> c8(MINUS(z0, z1))
MOD(z0, z1) -> c9(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z0))
MOD(z0, z1) -> c10(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z1))
MOD(z0, z1) -> c11(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), LE(z1, z0))
MOD(z0, z1) -> c12(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z0))
MOD(z0, z1) -> c13(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z1))
IF_MOD(true, z0, z1, z2, z3) -> c14
IF_MOD(false, z0, z1, z2, z3) -> c15(IF2(z0, z1, z2, z3))
IF2(true, z0, z1, z2) -> c16
IF2(false, z0, z1, z2) -> c17(IF3(z0, z1, z2))
IF3(true, z0, z1) -> c18(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF3(false, z0, z1) -> c19
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
zero(0') -> true
zero(s(z0)) -> false
id(0') -> 0'
id(s(z0)) -> s(id(z0))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
mod(z0, z1) -> if_mod(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1))
if_mod(true, z0, z1, z2, z3) -> 0'
if_mod(false, z0, z1, z2, z3) -> if2(z0, z1, z2, z3)
if2(true, z0, z1, z2) -> 0'
if2(false, z0, z1, z2) -> if3(z0, z1, z2)
if3(true, z0, z1) -> mod(minus(z0, z1), s(z1))
if3(false, z0, z1) -> z0

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
ZERO :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
ID :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
MINUS :: 0':s -> 0':s -> c7:c8
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
MOD :: 0':s -> 0':s -> c9:c10:c11:c12:c13
c9 :: c14:c15 -> c3:c4 -> c9:c10:c11:c12:c13
IF_MOD :: true:false -> true:false -> true:false -> 0':s -> 0':s -> c14:c15
zero :: 0':s -> true:false
le :: 0':s -> 0':s -> true:false
id :: 0':s -> 0':s
c10 :: c14:c15 -> c3:c4 -> c9:c10:c11:c12:c13
c11 :: c14:c15 -> c:c1:c2 -> c9:c10:c11:c12:c13
c12 :: c14:c15 -> c5:c6 -> c9:c10:c11:c12:c13
c13 :: c14:c15 -> c5:c6 -> c9:c10:c11:c12:c13
true :: true:false
c14 :: c14:c15
false :: true:false
c15 :: c16:c17 -> c14:c15
IF2 :: true:false -> true:false -> 0':s -> 0':s -> c16:c17
c16 :: c16:c17
c17 :: c18:c19 -> c16:c17
IF3 :: true:false -> 0':s -> 0':s -> c18:c19
c18 :: c9:c10:c11:c12:c13 -> c7:c8 -> c18:c19
minus :: 0':s -> 0':s -> 0':s
c19 :: c18:c19
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s
if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s
if3 :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_20 :: c:c1:c2
hole_0':s2_20 :: 0':s
hole_c3:c43_20 :: c3:c4
hole_c5:c64_20 :: c5:c6
hole_c7:c85_20 :: c7:c8
hole_c9:c10:c11:c12:c136_20 :: c9:c10:c11:c12:c13
hole_c14:c157_20 :: c14:c15
hole_true:false8_20 :: true:false
hole_c16:c179_20 :: c16:c17
hole_c18:c1910_20 :: c18:c19
gen_c:c1:c211_20 :: Nat -> c:c1:c2
gen_0':s12_20 :: Nat -> 0':s
gen_c5:c613_20 :: Nat -> c5:c6
gen_c7:c814_20 :: Nat -> c7:c8


Lemmas:
LE(gen_0':s12_20(n16_20), gen_0':s12_20(n16_20)) -> gen_c:c1:c211_20(n16_20), rt in Omega(1 + n16_20)
ID(gen_0':s12_20(n740_20)) -> gen_c5:c613_20(n740_20), rt in Omega(1 + n740_20)


Generator Equations:
gen_c:c1:c211_20(0) <=> c
gen_c:c1:c211_20(+(x, 1)) <=> c2(gen_c:c1:c211_20(x))
gen_0':s12_20(0) <=> 0'
gen_0':s12_20(+(x, 1)) <=> s(gen_0':s12_20(x))
gen_c5:c613_20(0) <=> c5
gen_c5:c613_20(+(x, 1)) <=> c6(gen_c5:c613_20(x))
gen_c7:c814_20(0) <=> c7
gen_c7:c814_20(+(x, 1)) <=> c8(gen_c7:c814_20(x))


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

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

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

(65) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MINUS(gen_0':s12_20(n1161_20), gen_0':s12_20(n1161_20)) -> gen_c7:c814_20(n1161_20), rt in Omega(1 + n1161_20)

Induction Base:
MINUS(gen_0':s12_20(0), gen_0':s12_20(0)) ->_R^Omega(1)
c7

Induction Step:
MINUS(gen_0':s12_20(+(n1161_20, 1)), gen_0':s12_20(+(n1161_20, 1))) ->_R^Omega(1)
c8(MINUS(gen_0':s12_20(n1161_20), gen_0':s12_20(n1161_20))) ->_IH
c8(gen_c7:c814_20(c1162_20))

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

(66)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
ZERO(0') -> c3
ZERO(s(z0)) -> c4
ID(0') -> c5
ID(s(z0)) -> c6(ID(z0))
MINUS(z0, 0') -> c7
MINUS(s(z0), s(z1)) -> c8(MINUS(z0, z1))
MOD(z0, z1) -> c9(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z0))
MOD(z0, z1) -> c10(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z1))
MOD(z0, z1) -> c11(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), LE(z1, z0))
MOD(z0, z1) -> c12(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z0))
MOD(z0, z1) -> c13(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z1))
IF_MOD(true, z0, z1, z2, z3) -> c14
IF_MOD(false, z0, z1, z2, z3) -> c15(IF2(z0, z1, z2, z3))
IF2(true, z0, z1, z2) -> c16
IF2(false, z0, z1, z2) -> c17(IF3(z0, z1, z2))
IF3(true, z0, z1) -> c18(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF3(false, z0, z1) -> c19
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
zero(0') -> true
zero(s(z0)) -> false
id(0') -> 0'
id(s(z0)) -> s(id(z0))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
mod(z0, z1) -> if_mod(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1))
if_mod(true, z0, z1, z2, z3) -> 0'
if_mod(false, z0, z1, z2, z3) -> if2(z0, z1, z2, z3)
if2(true, z0, z1, z2) -> 0'
if2(false, z0, z1, z2) -> if3(z0, z1, z2)
if3(true, z0, z1) -> mod(minus(z0, z1), s(z1))
if3(false, z0, z1) -> z0

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
ZERO :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
ID :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
MINUS :: 0':s -> 0':s -> c7:c8
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
MOD :: 0':s -> 0':s -> c9:c10:c11:c12:c13
c9 :: c14:c15 -> c3:c4 -> c9:c10:c11:c12:c13
IF_MOD :: true:false -> true:false -> true:false -> 0':s -> 0':s -> c14:c15
zero :: 0':s -> true:false
le :: 0':s -> 0':s -> true:false
id :: 0':s -> 0':s
c10 :: c14:c15 -> c3:c4 -> c9:c10:c11:c12:c13
c11 :: c14:c15 -> c:c1:c2 -> c9:c10:c11:c12:c13
c12 :: c14:c15 -> c5:c6 -> c9:c10:c11:c12:c13
c13 :: c14:c15 -> c5:c6 -> c9:c10:c11:c12:c13
true :: true:false
c14 :: c14:c15
false :: true:false
c15 :: c16:c17 -> c14:c15
IF2 :: true:false -> true:false -> 0':s -> 0':s -> c16:c17
c16 :: c16:c17
c17 :: c18:c19 -> c16:c17
IF3 :: true:false -> 0':s -> 0':s -> c18:c19
c18 :: c9:c10:c11:c12:c13 -> c7:c8 -> c18:c19
minus :: 0':s -> 0':s -> 0':s
c19 :: c18:c19
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s
if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s
if3 :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_20 :: c:c1:c2
hole_0':s2_20 :: 0':s
hole_c3:c43_20 :: c3:c4
hole_c5:c64_20 :: c5:c6
hole_c7:c85_20 :: c7:c8
hole_c9:c10:c11:c12:c136_20 :: c9:c10:c11:c12:c13
hole_c14:c157_20 :: c14:c15
hole_true:false8_20 :: true:false
hole_c16:c179_20 :: c16:c17
hole_c18:c1910_20 :: c18:c19
gen_c:c1:c211_20 :: Nat -> c:c1:c2
gen_0':s12_20 :: Nat -> 0':s
gen_c5:c613_20 :: Nat -> c5:c6
gen_c7:c814_20 :: Nat -> c7:c8


Lemmas:
LE(gen_0':s12_20(n16_20), gen_0':s12_20(n16_20)) -> gen_c:c1:c211_20(n16_20), rt in Omega(1 + n16_20)
ID(gen_0':s12_20(n740_20)) -> gen_c5:c613_20(n740_20), rt in Omega(1 + n740_20)
MINUS(gen_0':s12_20(n1161_20), gen_0':s12_20(n1161_20)) -> gen_c7:c814_20(n1161_20), rt in Omega(1 + n1161_20)


Generator Equations:
gen_c:c1:c211_20(0) <=> c
gen_c:c1:c211_20(+(x, 1)) <=> c2(gen_c:c1:c211_20(x))
gen_0':s12_20(0) <=> 0'
gen_0':s12_20(+(x, 1)) <=> s(gen_0':s12_20(x))
gen_c5:c613_20(0) <=> c5
gen_c5:c613_20(+(x, 1)) <=> c6(gen_c5:c613_20(x))
gen_c7:c814_20(0) <=> c7
gen_c7:c814_20(+(x, 1)) <=> c8(gen_c7:c814_20(x))


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

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

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

(67) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s12_20(n1816_20), gen_0':s12_20(n1816_20)) -> true, rt in Omega(0)

Induction Base:
le(gen_0':s12_20(0), gen_0':s12_20(0)) ->_R^Omega(0)
true

Induction Step:
le(gen_0':s12_20(+(n1816_20, 1)), gen_0':s12_20(+(n1816_20, 1))) ->_R^Omega(0)
le(gen_0':s12_20(n1816_20), gen_0':s12_20(n1816_20)) ->_IH
true

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

(68)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
ZERO(0') -> c3
ZERO(s(z0)) -> c4
ID(0') -> c5
ID(s(z0)) -> c6(ID(z0))
MINUS(z0, 0') -> c7
MINUS(s(z0), s(z1)) -> c8(MINUS(z0, z1))
MOD(z0, z1) -> c9(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z0))
MOD(z0, z1) -> c10(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z1))
MOD(z0, z1) -> c11(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), LE(z1, z0))
MOD(z0, z1) -> c12(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z0))
MOD(z0, z1) -> c13(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z1))
IF_MOD(true, z0, z1, z2, z3) -> c14
IF_MOD(false, z0, z1, z2, z3) -> c15(IF2(z0, z1, z2, z3))
IF2(true, z0, z1, z2) -> c16
IF2(false, z0, z1, z2) -> c17(IF3(z0, z1, z2))
IF3(true, z0, z1) -> c18(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF3(false, z0, z1) -> c19
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
zero(0') -> true
zero(s(z0)) -> false
id(0') -> 0'
id(s(z0)) -> s(id(z0))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
mod(z0, z1) -> if_mod(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1))
if_mod(true, z0, z1, z2, z3) -> 0'
if_mod(false, z0, z1, z2, z3) -> if2(z0, z1, z2, z3)
if2(true, z0, z1, z2) -> 0'
if2(false, z0, z1, z2) -> if3(z0, z1, z2)
if3(true, z0, z1) -> mod(minus(z0, z1), s(z1))
if3(false, z0, z1) -> z0

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
ZERO :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
ID :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
MINUS :: 0':s -> 0':s -> c7:c8
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
MOD :: 0':s -> 0':s -> c9:c10:c11:c12:c13
c9 :: c14:c15 -> c3:c4 -> c9:c10:c11:c12:c13
IF_MOD :: true:false -> true:false -> true:false -> 0':s -> 0':s -> c14:c15
zero :: 0':s -> true:false
le :: 0':s -> 0':s -> true:false
id :: 0':s -> 0':s
c10 :: c14:c15 -> c3:c4 -> c9:c10:c11:c12:c13
c11 :: c14:c15 -> c:c1:c2 -> c9:c10:c11:c12:c13
c12 :: c14:c15 -> c5:c6 -> c9:c10:c11:c12:c13
c13 :: c14:c15 -> c5:c6 -> c9:c10:c11:c12:c13
true :: true:false
c14 :: c14:c15
false :: true:false
c15 :: c16:c17 -> c14:c15
IF2 :: true:false -> true:false -> 0':s -> 0':s -> c16:c17
c16 :: c16:c17
c17 :: c18:c19 -> c16:c17
IF3 :: true:false -> 0':s -> 0':s -> c18:c19
c18 :: c9:c10:c11:c12:c13 -> c7:c8 -> c18:c19
minus :: 0':s -> 0':s -> 0':s
c19 :: c18:c19
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s
if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s
if3 :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_20 :: c:c1:c2
hole_0':s2_20 :: 0':s
hole_c3:c43_20 :: c3:c4
hole_c5:c64_20 :: c5:c6
hole_c7:c85_20 :: c7:c8
hole_c9:c10:c11:c12:c136_20 :: c9:c10:c11:c12:c13
hole_c14:c157_20 :: c14:c15
hole_true:false8_20 :: true:false
hole_c16:c179_20 :: c16:c17
hole_c18:c1910_20 :: c18:c19
gen_c:c1:c211_20 :: Nat -> c:c1:c2
gen_0':s12_20 :: Nat -> 0':s
gen_c5:c613_20 :: Nat -> c5:c6
gen_c7:c814_20 :: Nat -> c7:c8


Lemmas:
LE(gen_0':s12_20(n16_20), gen_0':s12_20(n16_20)) -> gen_c:c1:c211_20(n16_20), rt in Omega(1 + n16_20)
ID(gen_0':s12_20(n740_20)) -> gen_c5:c613_20(n740_20), rt in Omega(1 + n740_20)
MINUS(gen_0':s12_20(n1161_20), gen_0':s12_20(n1161_20)) -> gen_c7:c814_20(n1161_20), rt in Omega(1 + n1161_20)
le(gen_0':s12_20(n1816_20), gen_0':s12_20(n1816_20)) -> true, rt in Omega(0)


Generator Equations:
gen_c:c1:c211_20(0) <=> c
gen_c:c1:c211_20(+(x, 1)) <=> c2(gen_c:c1:c211_20(x))
gen_0':s12_20(0) <=> 0'
gen_0':s12_20(+(x, 1)) <=> s(gen_0':s12_20(x))
gen_c5:c613_20(0) <=> c5
gen_c5:c613_20(+(x, 1)) <=> c6(gen_c5:c613_20(x))
gen_c7:c814_20(0) <=> c7
gen_c7:c814_20(+(x, 1)) <=> c8(gen_c7:c814_20(x))


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

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

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

(69) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
id(gen_0':s12_20(n2315_20)) -> gen_0':s12_20(n2315_20), rt in Omega(0)

Induction Base:
id(gen_0':s12_20(0)) ->_R^Omega(0)
0'

Induction Step:
id(gen_0':s12_20(+(n2315_20, 1))) ->_R^Omega(0)
s(id(gen_0':s12_20(n2315_20))) ->_IH
s(gen_0':s12_20(c2316_20))

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

(70)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
ZERO(0') -> c3
ZERO(s(z0)) -> c4
ID(0') -> c5
ID(s(z0)) -> c6(ID(z0))
MINUS(z0, 0') -> c7
MINUS(s(z0), s(z1)) -> c8(MINUS(z0, z1))
MOD(z0, z1) -> c9(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z0))
MOD(z0, z1) -> c10(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z1))
MOD(z0, z1) -> c11(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), LE(z1, z0))
MOD(z0, z1) -> c12(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z0))
MOD(z0, z1) -> c13(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z1))
IF_MOD(true, z0, z1, z2, z3) -> c14
IF_MOD(false, z0, z1, z2, z3) -> c15(IF2(z0, z1, z2, z3))
IF2(true, z0, z1, z2) -> c16
IF2(false, z0, z1, z2) -> c17(IF3(z0, z1, z2))
IF3(true, z0, z1) -> c18(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF3(false, z0, z1) -> c19
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
zero(0') -> true
zero(s(z0)) -> false
id(0') -> 0'
id(s(z0)) -> s(id(z0))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
mod(z0, z1) -> if_mod(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1))
if_mod(true, z0, z1, z2, z3) -> 0'
if_mod(false, z0, z1, z2, z3) -> if2(z0, z1, z2, z3)
if2(true, z0, z1, z2) -> 0'
if2(false, z0, z1, z2) -> if3(z0, z1, z2)
if3(true, z0, z1) -> mod(minus(z0, z1), s(z1))
if3(false, z0, z1) -> z0

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
ZERO :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
ID :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
MINUS :: 0':s -> 0':s -> c7:c8
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
MOD :: 0':s -> 0':s -> c9:c10:c11:c12:c13
c9 :: c14:c15 -> c3:c4 -> c9:c10:c11:c12:c13
IF_MOD :: true:false -> true:false -> true:false -> 0':s -> 0':s -> c14:c15
zero :: 0':s -> true:false
le :: 0':s -> 0':s -> true:false
id :: 0':s -> 0':s
c10 :: c14:c15 -> c3:c4 -> c9:c10:c11:c12:c13
c11 :: c14:c15 -> c:c1:c2 -> c9:c10:c11:c12:c13
c12 :: c14:c15 -> c5:c6 -> c9:c10:c11:c12:c13
c13 :: c14:c15 -> c5:c6 -> c9:c10:c11:c12:c13
true :: true:false
c14 :: c14:c15
false :: true:false
c15 :: c16:c17 -> c14:c15
IF2 :: true:false -> true:false -> 0':s -> 0':s -> c16:c17
c16 :: c16:c17
c17 :: c18:c19 -> c16:c17
IF3 :: true:false -> 0':s -> 0':s -> c18:c19
c18 :: c9:c10:c11:c12:c13 -> c7:c8 -> c18:c19
minus :: 0':s -> 0':s -> 0':s
c19 :: c18:c19
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s
if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s
if3 :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_20 :: c:c1:c2
hole_0':s2_20 :: 0':s
hole_c3:c43_20 :: c3:c4
hole_c5:c64_20 :: c5:c6
hole_c7:c85_20 :: c7:c8
hole_c9:c10:c11:c12:c136_20 :: c9:c10:c11:c12:c13
hole_c14:c157_20 :: c14:c15
hole_true:false8_20 :: true:false
hole_c16:c179_20 :: c16:c17
hole_c18:c1910_20 :: c18:c19
gen_c:c1:c211_20 :: Nat -> c:c1:c2
gen_0':s12_20 :: Nat -> 0':s
gen_c5:c613_20 :: Nat -> c5:c6
gen_c7:c814_20 :: Nat -> c7:c8


Lemmas:
LE(gen_0':s12_20(n16_20), gen_0':s12_20(n16_20)) -> gen_c:c1:c211_20(n16_20), rt in Omega(1 + n16_20)
ID(gen_0':s12_20(n740_20)) -> gen_c5:c613_20(n740_20), rt in Omega(1 + n740_20)
MINUS(gen_0':s12_20(n1161_20), gen_0':s12_20(n1161_20)) -> gen_c7:c814_20(n1161_20), rt in Omega(1 + n1161_20)
le(gen_0':s12_20(n1816_20), gen_0':s12_20(n1816_20)) -> true, rt in Omega(0)
id(gen_0':s12_20(n2315_20)) -> gen_0':s12_20(n2315_20), rt in Omega(0)


Generator Equations:
gen_c:c1:c211_20(0) <=> c
gen_c:c1:c211_20(+(x, 1)) <=> c2(gen_c:c1:c211_20(x))
gen_0':s12_20(0) <=> 0'
gen_0':s12_20(+(x, 1)) <=> s(gen_0':s12_20(x))
gen_c5:c613_20(0) <=> c5
gen_c5:c613_20(+(x, 1)) <=> c6(gen_c5:c613_20(x))
gen_c7:c814_20(0) <=> c7
gen_c7:c814_20(+(x, 1)) <=> c8(gen_c7:c814_20(x))


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

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

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

(71) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s12_20(n2723_20), gen_0':s12_20(n2723_20)) -> gen_0':s12_20(0), rt in Omega(0)

Induction Base:
minus(gen_0':s12_20(0), gen_0':s12_20(0)) ->_R^Omega(0)
gen_0':s12_20(0)

Induction Step:
minus(gen_0':s12_20(+(n2723_20, 1)), gen_0':s12_20(+(n2723_20, 1))) ->_R^Omega(0)
minus(gen_0':s12_20(n2723_20), gen_0':s12_20(n2723_20)) ->_IH
gen_0':s12_20(0)

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

(72)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
ZERO(0') -> c3
ZERO(s(z0)) -> c4
ID(0') -> c5
ID(s(z0)) -> c6(ID(z0))
MINUS(z0, 0') -> c7
MINUS(s(z0), s(z1)) -> c8(MINUS(z0, z1))
MOD(z0, z1) -> c9(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z0))
MOD(z0, z1) -> c10(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ZERO(z1))
MOD(z0, z1) -> c11(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), LE(z1, z0))
MOD(z0, z1) -> c12(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z0))
MOD(z0, z1) -> c13(IF_MOD(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1)), ID(z1))
IF_MOD(true, z0, z1, z2, z3) -> c14
IF_MOD(false, z0, z1, z2, z3) -> c15(IF2(z0, z1, z2, z3))
IF2(true, z0, z1, z2) -> c16
IF2(false, z0, z1, z2) -> c17(IF3(z0, z1, z2))
IF3(true, z0, z1) -> c18(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF3(false, z0, z1) -> c19
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
zero(0') -> true
zero(s(z0)) -> false
id(0') -> 0'
id(s(z0)) -> s(id(z0))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
mod(z0, z1) -> if_mod(zero(z0), zero(z1), le(z1, z0), id(z0), id(z1))
if_mod(true, z0, z1, z2, z3) -> 0'
if_mod(false, z0, z1, z2, z3) -> if2(z0, z1, z2, z3)
if2(true, z0, z1, z2) -> 0'
if2(false, z0, z1, z2) -> if3(z0, z1, z2)
if3(true, z0, z1) -> mod(minus(z0, z1), s(z1))
if3(false, z0, z1) -> z0

Types:
LE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
ZERO :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
ID :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
MINUS :: 0':s -> 0':s -> c7:c8
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
MOD :: 0':s -> 0':s -> c9:c10:c11:c12:c13
c9 :: c14:c15 -> c3:c4 -> c9:c10:c11:c12:c13
IF_MOD :: true:false -> true:false -> true:false -> 0':s -> 0':s -> c14:c15
zero :: 0':s -> true:false
le :: 0':s -> 0':s -> true:false
id :: 0':s -> 0':s
c10 :: c14:c15 -> c3:c4 -> c9:c10:c11:c12:c13
c11 :: c14:c15 -> c:c1:c2 -> c9:c10:c11:c12:c13
c12 :: c14:c15 -> c5:c6 -> c9:c10:c11:c12:c13
c13 :: c14:c15 -> c5:c6 -> c9:c10:c11:c12:c13
true :: true:false
c14 :: c14:c15
false :: true:false
c15 :: c16:c17 -> c14:c15
IF2 :: true:false -> true:false -> 0':s -> 0':s -> c16:c17
c16 :: c16:c17
c17 :: c18:c19 -> c16:c17
IF3 :: true:false -> 0':s -> 0':s -> c18:c19
c18 :: c9:c10:c11:c12:c13 -> c7:c8 -> c18:c19
minus :: 0':s -> 0':s -> 0':s
c19 :: c18:c19
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s
if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s
if3 :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_20 :: c:c1:c2
hole_0':s2_20 :: 0':s
hole_c3:c43_20 :: c3:c4
hole_c5:c64_20 :: c5:c6
hole_c7:c85_20 :: c7:c8
hole_c9:c10:c11:c12:c136_20 :: c9:c10:c11:c12:c13
hole_c14:c157_20 :: c14:c15
hole_true:false8_20 :: true:false
hole_c16:c179_20 :: c16:c17
hole_c18:c1910_20 :: c18:c19
gen_c:c1:c211_20 :: Nat -> c:c1:c2
gen_0':s12_20 :: Nat -> 0':s
gen_c5:c613_20 :: Nat -> c5:c6
gen_c7:c814_20 :: Nat -> c7:c8


Lemmas:
LE(gen_0':s12_20(n16_20), gen_0':s12_20(n16_20)) -> gen_c:c1:c211_20(n16_20), rt in Omega(1 + n16_20)
ID(gen_0':s12_20(n740_20)) -> gen_c5:c613_20(n740_20), rt in Omega(1 + n740_20)
MINUS(gen_0':s12_20(n1161_20), gen_0':s12_20(n1161_20)) -> gen_c7:c814_20(n1161_20), rt in Omega(1 + n1161_20)
le(gen_0':s12_20(n1816_20), gen_0':s12_20(n1816_20)) -> true, rt in Omega(0)
id(gen_0':s12_20(n2315_20)) -> gen_0':s12_20(n2315_20), rt in Omega(0)
minus(gen_0':s12_20(n2723_20), gen_0':s12_20(n2723_20)) -> gen_0':s12_20(0), rt in Omega(0)


Generator Equations:
gen_c:c1:c211_20(0) <=> c
gen_c:c1:c211_20(+(x, 1)) <=> c2(gen_c:c1:c211_20(x))
gen_0':s12_20(0) <=> 0'
gen_0':s12_20(+(x, 1)) <=> s(gen_0':s12_20(x))
gen_c5:c613_20(0) <=> c5
gen_c5:c613_20(+(x, 1)) <=> c6(gen_c5:c613_20(x))
gen_c7:c814_20(0) <=> c7
gen_c7:c814_20(+(x, 1)) <=> c8(gen_c7:c814_20(x))


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