WORST_CASE(Omega(n^1),O(n^1))
proof of input_c9a9TSj6a5.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxWeightedTrs
    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxTypedWeightedTrs
    (15) CompletionProof [UPPER BOUND(ID), 0 ms]
    (16) CpxTypedWeightedCompleteTrs
    (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) CompleteCoflocoProof [FINISHED, 807 ms]
    (20) BOUNDS(1, n^1)
(21) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) CdtProblem
    (23) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxRelTRS
    (25) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CpxRelTRS
    (27) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) typed CpxTrs
    (29) OrderProof [LOWER BOUND(ID), 11 ms]
    (30) typed CpxTrs
    (31) RewriteLemmaProof [LOWER BOUND(ID), 419 ms]
    (32) BEST
        (33) proven lower bound
            (34) LowerBoundPropagationProof [FINISHED, 0 ms]
            (35) BOUNDS(n^1, INF)
        (36) typed CpxTrs
            (37) RewriteLemmaProof [LOWER BOUND(ID), 955 ms]
            (38) typed CpxTrs
            (39) RewriteLemmaProof [LOWER BOUND(ID), 877 ms]
            (40) typed CpxTrs


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

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


The TRS R consists of the following rules:

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

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

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

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   pred(s(z0)) -> z0
   minus(z0, 0) -> z0
   minus(z0, s(z1)) -> pred(minus(z0, z1))
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
   if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
   if_mod(false, s(z0), s(z1)) -> s(z0)
Tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PRED(s(z0)) -> c3
   MINUS(z0, 0) -> c4
   MINUS(z0, s(z1)) -> c5(PRED(minus(z0, z1)), MINUS(z0, z1))
   MOD(0, z0) -> c6
   MOD(s(z0), 0) -> c7
   MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF_MOD(false, s(z0), s(z1)) -> c10
S tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PRED(s(z0)) -> c3
   MINUS(z0, 0) -> c4
   MINUS(z0, s(z1)) -> c5(PRED(minus(z0, z1)), MINUS(z0, z1))
   MOD(0, z0) -> c6
   MOD(s(z0), 0) -> c7
   MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF_MOD(false, s(z0), s(z1)) -> c10
K tuples:none
Defined Rule Symbols:   le_2, pred_1, minus_2, mod_2, if_mod_3

Defined Pair Symbols:   LE_2, PRED_1, MINUS_2, MOD_2, IF_MOD_3

Compound Symbols:   c, c1, c2_1, c3, c4, c5_2, c6, c7, c8_2, c9_2, c10


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 7 trailing nodes:
   LE(s(z0), 0) -> c1
   IF_MOD(false, s(z0), s(z1)) -> c10
   MOD(s(z0), 0) -> c7
   PRED(s(z0)) -> c3
   MOD(0, z0) -> c6
   MINUS(z0, 0) -> c4
   LE(0, z0) -> c

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   pred(s(z0)) -> z0
   minus(z0, 0) -> z0
   minus(z0, s(z1)) -> pred(minus(z0, z1))
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
   if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
   if_mod(false, s(z0), s(z1)) -> s(z0)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(z0, s(z1)) -> c5(PRED(minus(z0, z1)), MINUS(z0, z1))
   MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(z0, s(z1)) -> c5(PRED(minus(z0, z1)), MINUS(z0, z1))
   MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:   le_2, pred_1, minus_2, mod_2, if_mod_3

Defined Pair Symbols:   LE_2, MINUS_2, MOD_2, IF_MOD_3

Compound Symbols:   c2_1, c5_2, c8_2, c9_2


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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   pred(s(z0)) -> z0
   minus(z0, 0) -> z0
   minus(z0, s(z1)) -> pred(minus(z0, z1))
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
   if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
   if_mod(false, s(z0), s(z1)) -> s(z0)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   MINUS(z0, s(z1)) -> c5(MINUS(z0, z1))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   MINUS(z0, s(z1)) -> c5(MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:   le_2, pred_1, minus_2, mod_2, if_mod_3

Defined Pair Symbols:   LE_2, MOD_2, IF_MOD_3, MINUS_2

Compound Symbols:   c2_1, c8_2, c9_2, c5_1


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
   if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
   if_mod(false, s(z0), s(z1)) -> s(z0)

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(z0, 0) -> z0
   minus(z0, s(z1)) -> pred(minus(z0, z1))
   pred(s(z0)) -> z0
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   MINUS(z0, s(z1)) -> c5(MINUS(z0, z1))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   MINUS(z0, s(z1)) -> c5(MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:   le_2, minus_2, pred_1

Defined Pair Symbols:   LE_2, MOD_2, IF_MOD_3, MINUS_2

Compound Symbols:   c2_1, c8_2, c9_2, c5_1


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

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

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


The TRS R consists of the following rules:

   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   MINUS(z0, s(z1)) -> c5(MINUS(z0, z1))

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)
   minus(z0, 0) -> z0
   minus(z0, s(z1)) -> pred(minus(z0, z1))
   pred(s(z0)) -> z0

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

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

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   LE(s(z0), s(z1)) -> c2(LE(z0, z1)) [1]
   MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) [1]
   IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) [1]
   MINUS(z0, s(z1)) -> c5(MINUS(z0, z1)) [1]
   le(0, z0) -> true [0]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]
   minus(z0, 0) -> z0 [0]
   minus(z0, s(z1)) -> pred(minus(z0, z1)) [0]
   pred(s(z0)) -> z0 [0]

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

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

(14)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   LE(s(z0), s(z1)) -> c2(LE(z0, z1)) [1]
   MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) [1]
   IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) [1]
   MINUS(z0, s(z1)) -> c5(MINUS(z0, z1)) [1]
   le(0, z0) -> true [0]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]
   minus(z0, 0) -> z0 [0]
   minus(z0, s(z1)) -> pred(minus(z0, z1)) [0]
   pred(s(z0)) -> z0 [0]

The TRS has the following type information:
LE :: s:0 -> s:0 -> c2
s :: s:0 -> s:0
c2 :: c2 -> c2
MOD :: s:0 -> s:0 -> c8
c8 :: c9 -> c2 -> c8
IF_MOD :: true:false -> s:0 -> s:0 -> c9
le :: s:0 -> s:0 -> true:false
true :: true:false
c9 :: c8 -> c5 -> c9
minus :: s:0 -> s:0 -> s:0
MINUS :: s:0 -> s:0 -> c5
c5 :: c5 -> c5
0 :: s:0
false :: true:false
pred :: s:0 -> s:0

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

(15) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   le(v0, v1) -> null_le [0]
   minus(v0, v1) -> null_minus [0]
   pred(v0) -> null_pred [0]
   LE(v0, v1) -> null_LE [0]
   MOD(v0, v1) -> null_MOD [0]
   IF_MOD(v0, v1, v2) -> null_IF_MOD [0]
   MINUS(v0, v1) -> null_MINUS [0]

And the following fresh constants:    null_le, null_minus, null_pred, null_LE, null_MOD, null_IF_MOD, null_MINUS

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

(16)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   LE(s(z0), s(z1)) -> c2(LE(z0, z1)) [1]
   MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) [1]
   IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) [1]
   MINUS(z0, s(z1)) -> c5(MINUS(z0, z1)) [1]
   le(0, z0) -> true [0]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]
   minus(z0, 0) -> z0 [0]
   minus(z0, s(z1)) -> pred(minus(z0, z1)) [0]
   pred(s(z0)) -> z0 [0]
   le(v0, v1) -> null_le [0]
   minus(v0, v1) -> null_minus [0]
   pred(v0) -> null_pred [0]
   LE(v0, v1) -> null_LE [0]
   MOD(v0, v1) -> null_MOD [0]
   IF_MOD(v0, v1, v2) -> null_IF_MOD [0]
   MINUS(v0, v1) -> null_MINUS [0]

The TRS has the following type information:
LE :: s:0:null_minus:null_pred -> s:0:null_minus:null_pred -> c2:null_LE
s :: s:0:null_minus:null_pred -> s:0:null_minus:null_pred
c2 :: c2:null_LE -> c2:null_LE
MOD :: s:0:null_minus:null_pred -> s:0:null_minus:null_pred -> c8:null_MOD
c8 :: c9:null_IF_MOD -> c2:null_LE -> c8:null_MOD
IF_MOD :: true:false:null_le -> s:0:null_minus:null_pred -> s:0:null_minus:null_pred -> c9:null_IF_MOD
le :: s:0:null_minus:null_pred -> s:0:null_minus:null_pred -> true:false:null_le
true :: true:false:null_le
c9 :: c8:null_MOD -> c5:null_MINUS -> c9:null_IF_MOD
minus :: s:0:null_minus:null_pred -> s:0:null_minus:null_pred -> s:0:null_minus:null_pred
MINUS :: s:0:null_minus:null_pred -> s:0:null_minus:null_pred -> c5:null_MINUS
c5 :: c5:null_MINUS -> c5:null_MINUS
0 :: s:0:null_minus:null_pred
false :: true:false:null_le
pred :: s:0:null_minus:null_pred -> s:0:null_minus:null_pred
null_le :: true:false:null_le
null_minus :: s:0:null_minus:null_pred
null_pred :: s:0:null_minus:null_pred
null_LE :: c2:null_LE
null_MOD :: c8:null_MOD
null_IF_MOD :: c9:null_IF_MOD
null_MINUS :: c5:null_MINUS

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

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

   true => 2
   0 => 0
   false => 1
   null_le => 0
   null_minus => 0
   null_pred => 0
   null_LE => 0
   null_MOD => 0
   null_IF_MOD => 0
   null_MINUS => 0

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

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

   IF_MOD(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   IF_MOD(z, z', z'') -{ 1 }-> 1 + MOD(minus(z0, z1), 1 + z1) + MINUS(z0, z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = 1 + z0, z'' = 1 + z1
   LE(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   LE(z, z') -{ 1 }-> 1 + LE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MINUS(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   MINUS(z, z') -{ 1 }-> 1 + MINUS(z0, z1) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   MOD(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   MOD(z, z') -{ 1 }-> 1 + IF_MOD(le(z1, z0), 1 + z0, 1 + z1) + LE(z1, z0) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   le(z, z') -{ 0 }-> le(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   le(z, z') -{ 0 }-> 2 :|: z0 >= 0, z = 0, z' = z0
   le(z, z') -{ 0 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   minus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   minus(z, z') -{ 0 }-> pred(minus(z0, z1)) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   pred(z) -{ 0 }-> z0 :|: z = 1 + z0, z0 >= 0
   pred(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

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

(19) CompleteCoflocoProof (FINISHED)
Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V1, V, V6),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V6),0,[fun1(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V6),0,[fun2(V1, V, V6, Out)],[V1 >= 0,V >= 0,V6 >= 0]).
eq(start(V1, V, V6),0,[fun3(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V6),0,[le(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V6),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V6),0,[pred(V1, Out)],[V1 >= 0]).
eq(fun(V1, V, Out),1,[fun(V3, V2, Ret1)],[Out = 1 + Ret1,V2 >= 0,V1 = 1 + V3,V3 >= 0,V = 1 + V2]).
eq(fun1(V1, V, Out),1,[le(V4, V5, Ret010),fun2(Ret010, 1 + V5, 1 + V4, Ret01),fun(V4, V5, Ret11)],[Out = 1 + Ret01 + Ret11,V4 >= 0,V1 = 1 + V5,V5 >= 0,V = 1 + V4]).
eq(fun2(V1, V, V6, Out),1,[minus(V8, V7, Ret0101),fun1(Ret0101, 1 + V7, Ret011),fun3(V8, V7, Ret12)],[Out = 1 + Ret011 + Ret12,V1 = 2,V7 >= 0,V8 >= 0,V = 1 + V8,V6 = 1 + V7]).
eq(fun3(V1, V, Out),1,[fun3(V9, V10, Ret13)],[Out = 1 + Ret13,V1 = V9,V10 >= 0,V9 >= 0,V = 1 + V10]).
eq(le(V1, V, Out),0,[],[Out = 2,V11 >= 0,V1 = 0,V = V11]).
eq(le(V1, V, Out),0,[],[Out = 1,V1 = 1 + V12,V12 >= 0,V = 0]).
eq(le(V1, V, Out),0,[le(V14, V13, Ret)],[Out = Ret,V13 >= 0,V1 = 1 + V14,V14 >= 0,V = 1 + V13]).
eq(minus(V1, V, Out),0,[],[Out = V15,V1 = V15,V15 >= 0,V = 0]).
eq(minus(V1, V, Out),0,[minus(V17, V16, Ret0),pred(Ret0, Ret2)],[Out = Ret2,V1 = V17,V16 >= 0,V17 >= 0,V = 1 + V16]).
eq(pred(V1, Out),0,[],[Out = V18,V1 = 1 + V18,V18 >= 0]).
eq(le(V1, V, Out),0,[],[Out = 0,V20 >= 0,V19 >= 0,V1 = V20,V = V19]).
eq(minus(V1, V, Out),0,[],[Out = 0,V22 >= 0,V21 >= 0,V1 = V22,V = V21]).
eq(pred(V1, Out),0,[],[Out = 0,V23 >= 0,V1 = V23]).
eq(fun(V1, V, Out),0,[],[Out = 0,V24 >= 0,V25 >= 0,V1 = V24,V = V25]).
eq(fun1(V1, V, Out),0,[],[Out = 0,V26 >= 0,V27 >= 0,V1 = V26,V = V27]).
eq(fun2(V1, V, V6, Out),0,[],[Out = 0,V29 >= 0,V6 = V30,V28 >= 0,V1 = V29,V = V28,V30 >= 0]).
eq(fun3(V1, V, Out),0,[],[Out = 0,V32 >= 0,V31 >= 0,V1 = V32,V = V31]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun2(V1,V,V6,Out),[V1,V,V6],[Out]).
input_output_vars(fun3(V1,V,Out),[V1,V],[Out]).
input_output_vars(le(V1,V,Out),[V1,V],[Out]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(pred(V1,Out),[V1],[Out]).


CoFloCo proof output:
Preprocessing Cost Relations
=====================================

#### Computed strongly connected components 
0. recursive  : [fun/3]
1. recursive  : [fun3/3]
2. non_recursive  : [pred/2]
3. recursive [non_tail] : [minus/3]
4. recursive  : [le/3]
5. recursive [non_tail] : [fun1/3,fun2/4]
6. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into fun/3
1. SCC is partially evaluated into fun3/3
2. SCC is partially evaluated into pred/2
3. SCC is partially evaluated into minus/3
4. SCC is partially evaluated into le/3
5. SCC is partially evaluated into fun1/3
6. SCC is partially evaluated into start/3

Control-Flow Refinement of Cost Relations
=====================================

### Specialization of cost equations fun/3 
* CE 18 is refined into CE [25] 
* CE 17 is refined into CE [26] 


### Cost equations --> "Loop" of fun/3 
* CEs [26] --> Loop 18 
* CEs [25] --> Loop 19 

### Ranking functions of CR fun(V1,V,Out) 
* RF of phase [18]: [V,V1]

#### Partial ranking functions of CR fun(V1,V,Out) 
* Partial RF of phase [18]:
  - RF of loop [18:1]:
    V
    V1


### Specialization of cost equations fun3/3 
* CE 16 is refined into CE [27] 
* CE 15 is refined into CE [28] 


### Cost equations --> "Loop" of fun3/3 
* CEs [28] --> Loop 20 
* CEs [27] --> Loop 21 

### Ranking functions of CR fun3(V1,V,Out) 
* RF of phase [20]: [V]

#### Partial ranking functions of CR fun3(V1,V,Out) 
* Partial RF of phase [20]:
  - RF of loop [20:1]:
    V


### Specialization of cost equations pred/2 
* CE 23 is refined into CE [29] 
* CE 24 is refined into CE [30] 


### Cost equations --> "Loop" of pred/2 
* CEs [29] --> Loop 22 
* CEs [30] --> Loop 23 

### Ranking functions of CR pred(V1,Out) 

#### Partial ranking functions of CR pred(V1,Out) 


### Specialization of cost equations minus/3 
* CE 11 is refined into CE [31] 
* CE 9 is refined into CE [32] 
* CE 10 is refined into CE [33,34] 


### Cost equations --> "Loop" of minus/3 
* CEs [34] --> Loop 24 
* CEs [33] --> Loop 25 
* CEs [31] --> Loop 26 
* CEs [32] --> Loop 27 

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [24]: [V]
* RF of phase [25]: [V]

#### Partial ranking functions of CR minus(V1,V,Out) 
* Partial RF of phase [24]:
  - RF of loop [24:1]:
    V
* Partial RF of phase [25]:
  - RF of loop [25:1]:
    V


### Specialization of cost equations le/3 
* CE 22 is refined into CE [35] 
* CE 20 is refined into CE [36] 
* CE 19 is refined into CE [37] 
* CE 21 is refined into CE [38] 


### Cost equations --> "Loop" of le/3 
* CEs [38] --> Loop 28 
* CEs [35] --> Loop 29 
* CEs [36] --> Loop 30 
* CEs [37] --> Loop 31 

### Ranking functions of CR le(V1,V,Out) 
* RF of phase [28]: [V,V1]

#### Partial ranking functions of CR le(V1,V,Out) 
* Partial RF of phase [28]:
  - RF of loop [28:1]:
    V
    V1


### Specialization of cost equations fun1/3 
* CE 12 is refined into CE [39,40,41,42,43,44,45,46] 
* CE 14 is refined into CE [47] 
* CE 13 is refined into CE [48,49,50,51,52,53,54,55,56,57] 


### Cost equations --> "Loop" of fun1/3 
* CEs [55,56] --> Loop 32 
* CEs [57] --> Loop 33 
* CEs [54] --> Loop 34 
* CEs [53] --> Loop 35 
* CEs [51,52] --> Loop 36 
* CEs [50] --> Loop 37 
* CEs [48] --> Loop 38 
* CEs [49] --> Loop 39 
* CEs [46] --> Loop 40 
* CEs [42,44] --> Loop 41 
* CEs [47] --> Loop 42 
* CEs [39] --> Loop 43 
* CEs [40,41,43,45] --> Loop 44 

### Ranking functions of CR fun1(V1,V,Out) 
* RF of phase [32,33,34]: [V1-1,V1-V+1]
* RF of phase [38]: [V1]

#### Partial ranking functions of CR fun1(V1,V,Out) 
* Partial RF of phase [32,33,34]:
  - RF of loop [32:1,33:1,34:1]:
    V1-1
    V1-V+1
* Partial RF of phase [38]:
  - RF of loop [38:1]:
    V1


### Specialization of cost equations start/3 
* CE 1 is refined into CE [58] 
* CE 2 is refined into CE [59,60,61,62,63,64,65,66,67,68,69,70,71] 
* CE 3 is refined into CE [72,73] 
* CE 4 is refined into CE [74,75,76,77,78] 
* CE 5 is refined into CE [79,80] 
* CE 6 is refined into CE [81,82,83,84,85] 
* CE 7 is refined into CE [86,87,88] 
* CE 8 is refined into CE [89,90] 


### Cost equations --> "Loop" of start/3 
* CEs [75] --> Loop 45 
* CEs [82,86] --> Loop 46 
* CEs [59,60,61,62,63,64,65,66,67,68,69,70,71] --> Loop 47 
* CEs [58,72,73,74,76,77,78,79,80,81,83,84,85,87,88,89,90] --> Loop 48 

### Ranking functions of CR start(V1,V,V6) 

#### Partial ranking functions of CR start(V1,V,V6) 


Computing Bounds
=====================================

#### Cost of chains of fun(V1,V,Out):
* Chain [[18],19]: 1*it(18)+0
  Such that:it(18) =< Out

  with precondition: [Out>=1,V1>=Out,V>=Out] 

* Chain [19]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of fun3(V1,V,Out):
* Chain [[20],21]: 1*it(20)+0
  Such that:it(20) =< Out

  with precondition: [V1>=0,Out>=1,V>=Out] 

* Chain [21]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of pred(V1,Out):
* Chain [23]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [22]: 0
  with precondition: [V1=Out+1,V1>=1] 


#### Cost of chains of minus(V1,V,Out):
* Chain [[25],[24],27]: 0
  with precondition: [Out=0,V1>=1,V>=2] 

* Chain [[25],27]: 0
  with precondition: [Out=0,V1>=0,V>=1] 

* Chain [[25],26]: 0
  with precondition: [Out=0,V1>=0,V>=1] 

* Chain [[24],27]: 0
  with precondition: [V1=Out+V,V>=1,V1>=V] 

* Chain [27]: 0
  with precondition: [V=0,V1=Out,V1>=0] 

* Chain [26]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of le(V1,V,Out):
* Chain [[28],31]: 0
  with precondition: [Out=2,V1>=1,V>=V1] 

* Chain [[28],30]: 0
  with precondition: [Out=1,V>=1,V1>=V+1] 

* Chain [[28],29]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

* Chain [31]: 0
  with precondition: [V1=0,Out=2,V>=0] 

* Chain [30]: 0
  with precondition: [V=0,Out=1,V1>=1] 

* Chain [29]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of fun1(V1,V,Out):
* Chain [[38],44]: 2*it(38)+1
  Such that:it(38) =< Out/2

  with precondition: [V=1,Out>=3,2*V1>=Out+1] 

* Chain [[38],43]: 2*it(38)+1
  Such that:it(38) =< Out/2

  with precondition: [V=1,Out>=3,2*V1>=Out+1] 

* Chain [[38],42]: 2*it(38)+0
  Such that:it(38) =< Out/2

  with precondition: [V=1,Out>=2,2*V1>=Out] 

* Chain [[38],39,42]: 2*it(38)+2
  Such that:it(38) =< Out/2

  with precondition: [V=1,Out>=4,2*V1>=Out] 

* Chain [[32,33,34],44]: 6*it(32)+4*s(9)+1
  Such that:aux(4) =< V1-V+1
aux(7) =< V1
it(32) =< aux(7)
it(32) =< aux(4)
s(9) =< aux(7)

  with precondition: [V>=2,Out>=3,V1>=V+1,2*V1>=Out+1] 

* Chain [[32,33,34],42]: 6*it(32)+4*s(9)+0
  Such that:aux(4) =< V1-V+1
aux(8) =< V1
it(32) =< aux(8)
it(32) =< aux(4)
s(9) =< aux(8)

  with precondition: [V>=2,Out>=2,V1>=V,2*V1>=Out] 

* Chain [[32,33,34],41]: 6*it(32)+5*s(9)+1*s(13)+1
  Such that:aux(4) =< V1-V+1
s(13) =< V
aux(9) =< V1
s(9) =< aux(9)
it(32) =< aux(9)
it(32) =< aux(4)

  with precondition: [V>=2,Out>=4,V1>=V+2,2*V1>=Out+2] 

* Chain [[32,33,34],40]: 6*it(32)+4*s(9)+1*s(15)+1
  Such that:aux(3) =< V1
aux(4) =< V1-V+1
s(15) =< V
aux(10) =< V1-V
it(32) =< aux(3)
s(10) =< aux(3)
it(32) =< aux(4)
it(32) =< aux(10)
s(10) =< aux(10)
s(9) =< s(10)

  with precondition: [V>=2,Out>=4,V1>=2*V,2*V1>=Out+V] 

* Chain [[32,33,34],37,42]: 6*it(32)+4*s(9)+2
  Such that:aux(3) =< V1
aux(4) =< V1-V+1
aux(11) =< V1-V
it(32) =< aux(3)
s(10) =< aux(3)
it(32) =< aux(4)
it(32) =< aux(11)
s(10) =< aux(11)
s(9) =< s(10)

  with precondition: [V>=2,Out>=4,V1>=2*V,2*V1+2>=2*V+Out] 

* Chain [[32,33,34],36,42]: 6*it(32)+4*s(9)+2*s(16)+2
  Such that:aux(3) =< V1
aux(4) =< V1-V+1
aux(12) =< V
aux(13) =< V1-V
s(16) =< aux(12)
it(32) =< aux(3)
s(10) =< aux(3)
it(32) =< aux(4)
it(32) =< aux(13)
s(10) =< aux(13)
s(9) =< s(10)

  with precondition: [V>=2,Out>=5,V1>=2*V,2*V1+1>=Out+V] 

* Chain [[32,33,34],35,42]: 6*it(32)+4*s(9)+2*s(18)+2
  Such that:aux(3) =< V1
aux(4) =< V1-V+1
aux(14) =< V
aux(15) =< V1-V
s(18) =< aux(14)
it(32) =< aux(3)
s(10) =< aux(3)
it(32) =< aux(4)
it(32) =< aux(15)
s(10) =< aux(15)
s(9) =< s(10)

  with precondition: [V>=2,Out>=6,V1>=2*V,2*V1>=Out] 

* Chain [44]: 1
  with precondition: [Out=1,V1>=1,V>=1] 

* Chain [43]: 1
  with precondition: [V=1,Out=1,V1>=1] 

* Chain [42]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [41]: 1*s(13)+1*s(14)+1
  Such that:s(14) =< V1
s(13) =< V

  with precondition: [Out>=2,V1>=Out,V>=Out] 

* Chain [40]: 1*s(15)+1
  Such that:s(15) =< V

  with precondition: [Out>=2,V1>=V,V>=Out] 

* Chain [39,42]: 2
  with precondition: [V=1,Out=2,V1>=1] 

* Chain [37,42]: 2
  with precondition: [Out=2,V>=2,V1>=V] 

* Chain [36,42]: 2*s(16)+2
  Such that:aux(12) =< V
s(16) =< aux(12)

  with precondition: [Out>=3,V1>=V,V+1>=Out] 

* Chain [35,42]: 2*s(18)+2
  Such that:aux(14) =< V
s(18) =< aux(14)

  with precondition: [Out>=4,V1>=V,2*V>=Out] 


#### Cost of chains of start(V1,V,V6):
* Chain [48]: 14*s(71)+24*s(77)+16*s(79)+14*s(80)+18*s(81)+2
  Such that:s(73) =< V1-V
s(74) =< V1-V+1
aux(21) =< V1
aux(22) =< V
s(80) =< aux(21)
s(71) =< aux(22)
s(77) =< aux(21)
s(78) =< aux(21)
s(77) =< s(74)
s(77) =< s(73)
s(78) =< s(73)
s(79) =< s(78)
s(81) =< aux(21)
s(81) =< s(74)

  with precondition: [V1>=0] 

* Chain [47]: 8*s(86)+29*s(87)+48*s(95)+32*s(97)+28*s(98)+36*s(99)+3
  Such that:s(85) =< V
aux(25) =< V-2*V6
aux(26) =< V-2*V6+1
aux(27) =< V-V6
aux(28) =< V6
s(98) =< aux(27)
s(87) =< aux(28)
s(86) =< s(85)
s(95) =< aux(27)
s(96) =< aux(27)
s(95) =< aux(26)
s(95) =< aux(25)
s(96) =< aux(25)
s(97) =< s(96)
s(99) =< aux(27)
s(99) =< aux(26)

  with precondition: [V1=2,V>=1,V6>=1] 

* Chain [46]: 0
  with precondition: [V=0,V1>=0] 

* Chain [45]: 8*s(117)+2
  Such that:s(116) =< V1
s(117) =< s(116)

  with precondition: [V=1,V1>=1] 


Closed-form bounds of start(V1,V,V6): 
-------------------------------------
* Chain [48] with precondition: [V1>=0] 
    - Upper bound: 72*V1+2+nat(V)*14 
    - Complexity: n 
* Chain [47] with precondition: [V1=2,V>=1,V6>=1] 
    - Upper bound: 8*V+29*V6+3+nat(V-V6)*144 
    - Complexity: n 
* Chain [46] with precondition: [V=0,V1>=0] 
    - Upper bound: 0 
    - Complexity: constant 
* Chain [45] with precondition: [V=1,V1>=1] 
    - Upper bound: 8*V1+2 
    - Complexity: n 

### Maximum cost of start(V1,V,V6): max([nat(V)*14+64*V1+(8*V1+2),nat(V)*8+3+nat(V6)*29+nat(V-V6)*144]) 
Asymptotic class: n 
* Total analysis performed in 720 ms.


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

(20)
BOUNDS(1, n^1)

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

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   pred(s(z0)) -> z0
   minus(z0, 0) -> z0
   minus(z0, s(z1)) -> pred(minus(z0, z1))
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
   if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
   if_mod(false, s(z0), s(z1)) -> s(z0)
Tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PRED(s(z0)) -> c3
   MINUS(z0, 0) -> c4
   MINUS(z0, s(z1)) -> c5(PRED(minus(z0, z1)), MINUS(z0, z1))
   MOD(0, z0) -> c6
   MOD(s(z0), 0) -> c7
   MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF_MOD(false, s(z0), s(z1)) -> c10
S tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PRED(s(z0)) -> c3
   MINUS(z0, 0) -> c4
   MINUS(z0, s(z1)) -> c5(PRED(minus(z0, z1)), MINUS(z0, z1))
   MOD(0, z0) -> c6
   MOD(s(z0), 0) -> c7
   MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF_MOD(false, s(z0), s(z1)) -> c10
K tuples:none
Defined Rule Symbols:   le_2, pred_1, minus_2, mod_2, if_mod_3

Defined Pair Symbols:   LE_2, PRED_1, MINUS_2, MOD_2, IF_MOD_3

Compound Symbols:   c, c1, c2_1, c3, c4, c5_2, c6, c7, c8_2, c9_2, c10


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

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

(24)
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))
   PRED(s(z0)) -> c3
   MINUS(z0, 0) -> c4
   MINUS(z0, s(z1)) -> c5(PRED(minus(z0, z1)), MINUS(z0, z1))
   MOD(0, z0) -> c6
   MOD(s(z0), 0) -> c7
   MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF_MOD(false, s(z0), s(z1)) -> c10

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)
   pred(s(z0)) -> z0
   minus(z0, 0) -> z0
   minus(z0, s(z1)) -> pred(minus(z0, z1))
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
   if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
   if_mod(false, s(z0), s(z1)) -> s(z0)

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

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

(26)
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))
   PRED(s(z0)) -> c3
   MINUS(z0, 0') -> c4
   MINUS(z0, s(z1)) -> c5(PRED(minus(z0, z1)), MINUS(z0, z1))
   MOD(0', z0) -> c6
   MOD(s(z0), 0') -> c7
   MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
   IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
   IF_MOD(false, s(z0), s(z1)) -> c10

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)
   pred(s(z0)) -> z0
   minus(z0, 0') -> z0
   minus(z0, s(z1)) -> pred(minus(z0, z1))
   mod(0', z0) -> 0'
   mod(s(z0), 0') -> 0'
   mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
   if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
   if_mod(false, s(z0), s(z1)) -> s(z0)

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

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

(28)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
PRED(s(z0)) -> c3
MINUS(z0, 0') -> c4
MINUS(z0, s(z1)) -> c5(PRED(minus(z0, z1)), MINUS(z0, z1))
MOD(0', z0) -> c6
MOD(s(z0), 0') -> c7
MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_MOD(false, s(z0), s(z1)) -> c10
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
pred(s(z0)) -> z0
minus(z0, 0') -> z0
minus(z0, s(z1)) -> pred(minus(z0, z1))
mod(0', z0) -> 0'
mod(s(z0), 0') -> 0'
mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
if_mod(false, s(z0), s(z1)) -> s(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
PRED :: 0':s -> c3
c3 :: c3
MINUS :: 0':s -> 0':s -> c4:c5
c4 :: c4:c5
c5 :: c3 -> c4:c5 -> c4:c5
minus :: 0':s -> 0':s -> 0':s
MOD :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c9:c10 -> c:c1:c2 -> c6:c7:c8
IF_MOD :: true:false -> 0':s -> 0':s -> c9:c10
le :: 0':s -> 0':s -> true:false
true :: true:false
c9 :: c6:c7:c8 -> c4:c5 -> c9:c10
false :: true:false
c10 :: c9:c10
pred :: 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c33_11 :: c3
hole_c4:c54_11 :: c4:c5
hole_c6:c7:c85_11 :: c6:c7:c8
hole_c9:c106_11 :: c9:c10
hole_true:false7_11 :: true:false
gen_c:c1:c28_11 :: Nat -> c:c1:c2
gen_0':s9_11 :: Nat -> 0':s
gen_c4:c510_11 :: Nat -> c4:c5

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

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

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

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

(30)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
PRED(s(z0)) -> c3
MINUS(z0, 0') -> c4
MINUS(z0, s(z1)) -> c5(PRED(minus(z0, z1)), MINUS(z0, z1))
MOD(0', z0) -> c6
MOD(s(z0), 0') -> c7
MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_MOD(false, s(z0), s(z1)) -> c10
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
pred(s(z0)) -> z0
minus(z0, 0') -> z0
minus(z0, s(z1)) -> pred(minus(z0, z1))
mod(0', z0) -> 0'
mod(s(z0), 0') -> 0'
mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
if_mod(false, s(z0), s(z1)) -> s(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
PRED :: 0':s -> c3
c3 :: c3
MINUS :: 0':s -> 0':s -> c4:c5
c4 :: c4:c5
c5 :: c3 -> c4:c5 -> c4:c5
minus :: 0':s -> 0':s -> 0':s
MOD :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c9:c10 -> c:c1:c2 -> c6:c7:c8
IF_MOD :: true:false -> 0':s -> 0':s -> c9:c10
le :: 0':s -> 0':s -> true:false
true :: true:false
c9 :: c6:c7:c8 -> c4:c5 -> c9:c10
false :: true:false
c10 :: c9:c10
pred :: 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c33_11 :: c3
hole_c4:c54_11 :: c4:c5
hole_c6:c7:c85_11 :: c6:c7:c8
hole_c9:c106_11 :: c9:c10
hole_true:false7_11 :: true:false
gen_c:c1:c28_11 :: Nat -> c:c1:c2
gen_0':s9_11 :: Nat -> 0':s
gen_c4:c510_11 :: Nat -> c4:c5


Generator Equations:
gen_c:c1:c28_11(0) <=> c
gen_c:c1:c28_11(+(x, 1)) <=> c2(gen_c:c1:c28_11(x))
gen_0':s9_11(0) <=> 0'
gen_0':s9_11(+(x, 1)) <=> s(gen_0':s9_11(x))
gen_c4:c510_11(0) <=> c4
gen_c4:c510_11(+(x, 1)) <=> c5(c3, gen_c4:c510_11(x))


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

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

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LE(gen_0':s9_11(n12_11), gen_0':s9_11(n12_11)) -> gen_c:c1:c28_11(n12_11), rt in Omega(1 + n12_11)

Induction Base:
LE(gen_0':s9_11(0), gen_0':s9_11(0)) ->_R^Omega(1)
c

Induction Step:
LE(gen_0':s9_11(+(n12_11, 1)), gen_0':s9_11(+(n12_11, 1))) ->_R^Omega(1)
c2(LE(gen_0':s9_11(n12_11), gen_0':s9_11(n12_11))) ->_IH
c2(gen_c:c1:c28_11(c13_11))

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

(32)
Complex Obligation (BEST)

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

(33)
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))
PRED(s(z0)) -> c3
MINUS(z0, 0') -> c4
MINUS(z0, s(z1)) -> c5(PRED(minus(z0, z1)), MINUS(z0, z1))
MOD(0', z0) -> c6
MOD(s(z0), 0') -> c7
MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_MOD(false, s(z0), s(z1)) -> c10
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
pred(s(z0)) -> z0
minus(z0, 0') -> z0
minus(z0, s(z1)) -> pred(minus(z0, z1))
mod(0', z0) -> 0'
mod(s(z0), 0') -> 0'
mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
if_mod(false, s(z0), s(z1)) -> s(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
PRED :: 0':s -> c3
c3 :: c3
MINUS :: 0':s -> 0':s -> c4:c5
c4 :: c4:c5
c5 :: c3 -> c4:c5 -> c4:c5
minus :: 0':s -> 0':s -> 0':s
MOD :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c9:c10 -> c:c1:c2 -> c6:c7:c8
IF_MOD :: true:false -> 0':s -> 0':s -> c9:c10
le :: 0':s -> 0':s -> true:false
true :: true:false
c9 :: c6:c7:c8 -> c4:c5 -> c9:c10
false :: true:false
c10 :: c9:c10
pred :: 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c33_11 :: c3
hole_c4:c54_11 :: c4:c5
hole_c6:c7:c85_11 :: c6:c7:c8
hole_c9:c106_11 :: c9:c10
hole_true:false7_11 :: true:false
gen_c:c1:c28_11 :: Nat -> c:c1:c2
gen_0':s9_11 :: Nat -> 0':s
gen_c4:c510_11 :: Nat -> c4:c5


Generator Equations:
gen_c:c1:c28_11(0) <=> c
gen_c:c1:c28_11(+(x, 1)) <=> c2(gen_c:c1:c28_11(x))
gen_0':s9_11(0) <=> 0'
gen_0':s9_11(+(x, 1)) <=> s(gen_0':s9_11(x))
gen_c4:c510_11(0) <=> c4
gen_c4:c510_11(+(x, 1)) <=> c5(c3, gen_c4:c510_11(x))


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

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

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

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

(35)
BOUNDS(n^1, INF)

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

(36)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
PRED(s(z0)) -> c3
MINUS(z0, 0') -> c4
MINUS(z0, s(z1)) -> c5(PRED(minus(z0, z1)), MINUS(z0, z1))
MOD(0', z0) -> c6
MOD(s(z0), 0') -> c7
MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_MOD(false, s(z0), s(z1)) -> c10
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
pred(s(z0)) -> z0
minus(z0, 0') -> z0
minus(z0, s(z1)) -> pred(minus(z0, z1))
mod(0', z0) -> 0'
mod(s(z0), 0') -> 0'
mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
if_mod(false, s(z0), s(z1)) -> s(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
PRED :: 0':s -> c3
c3 :: c3
MINUS :: 0':s -> 0':s -> c4:c5
c4 :: c4:c5
c5 :: c3 -> c4:c5 -> c4:c5
minus :: 0':s -> 0':s -> 0':s
MOD :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c9:c10 -> c:c1:c2 -> c6:c7:c8
IF_MOD :: true:false -> 0':s -> 0':s -> c9:c10
le :: 0':s -> 0':s -> true:false
true :: true:false
c9 :: c6:c7:c8 -> c4:c5 -> c9:c10
false :: true:false
c10 :: c9:c10
pred :: 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c33_11 :: c3
hole_c4:c54_11 :: c4:c5
hole_c6:c7:c85_11 :: c6:c7:c8
hole_c9:c106_11 :: c9:c10
hole_true:false7_11 :: true:false
gen_c:c1:c28_11 :: Nat -> c:c1:c2
gen_0':s9_11 :: Nat -> 0':s
gen_c4:c510_11 :: Nat -> c4:c5


Lemmas:
LE(gen_0':s9_11(n12_11), gen_0':s9_11(n12_11)) -> gen_c:c1:c28_11(n12_11), rt in Omega(1 + n12_11)


Generator Equations:
gen_c:c1:c28_11(0) <=> c
gen_c:c1:c28_11(+(x, 1)) <=> c2(gen_c:c1:c28_11(x))
gen_0':s9_11(0) <=> 0'
gen_0':s9_11(+(x, 1)) <=> s(gen_0':s9_11(x))
gen_c4:c510_11(0) <=> c4
gen_c4:c510_11(+(x, 1)) <=> c5(c3, gen_c4:c510_11(x))


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

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

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

(37) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s9_11(a), gen_0':s9_11(+(1, n610_11))) -> *11_11, rt in Omega(0)

Induction Base:
minus(gen_0':s9_11(a), gen_0':s9_11(+(1, 0)))

Induction Step:
minus(gen_0':s9_11(a), gen_0':s9_11(+(1, +(n610_11, 1)))) ->_R^Omega(0)
pred(minus(gen_0':s9_11(a), gen_0':s9_11(+(1, n610_11)))) ->_IH
pred(*11_11)

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

(38)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
PRED(s(z0)) -> c3
MINUS(z0, 0') -> c4
MINUS(z0, s(z1)) -> c5(PRED(minus(z0, z1)), MINUS(z0, z1))
MOD(0', z0) -> c6
MOD(s(z0), 0') -> c7
MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_MOD(false, s(z0), s(z1)) -> c10
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
pred(s(z0)) -> z0
minus(z0, 0') -> z0
minus(z0, s(z1)) -> pred(minus(z0, z1))
mod(0', z0) -> 0'
mod(s(z0), 0') -> 0'
mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
if_mod(false, s(z0), s(z1)) -> s(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
PRED :: 0':s -> c3
c3 :: c3
MINUS :: 0':s -> 0':s -> c4:c5
c4 :: c4:c5
c5 :: c3 -> c4:c5 -> c4:c5
minus :: 0':s -> 0':s -> 0':s
MOD :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c9:c10 -> c:c1:c2 -> c6:c7:c8
IF_MOD :: true:false -> 0':s -> 0':s -> c9:c10
le :: 0':s -> 0':s -> true:false
true :: true:false
c9 :: c6:c7:c8 -> c4:c5 -> c9:c10
false :: true:false
c10 :: c9:c10
pred :: 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c33_11 :: c3
hole_c4:c54_11 :: c4:c5
hole_c6:c7:c85_11 :: c6:c7:c8
hole_c9:c106_11 :: c9:c10
hole_true:false7_11 :: true:false
gen_c:c1:c28_11 :: Nat -> c:c1:c2
gen_0':s9_11 :: Nat -> 0':s
gen_c4:c510_11 :: Nat -> c4:c5


Lemmas:
LE(gen_0':s9_11(n12_11), gen_0':s9_11(n12_11)) -> gen_c:c1:c28_11(n12_11), rt in Omega(1 + n12_11)
minus(gen_0':s9_11(a), gen_0':s9_11(+(1, n610_11))) -> *11_11, rt in Omega(0)


Generator Equations:
gen_c:c1:c28_11(0) <=> c
gen_c:c1:c28_11(+(x, 1)) <=> c2(gen_c:c1:c28_11(x))
gen_0':s9_11(0) <=> 0'
gen_0':s9_11(+(x, 1)) <=> s(gen_0':s9_11(x))
gen_c4:c510_11(0) <=> c4
gen_c4:c510_11(+(x, 1)) <=> c5(c3, gen_c4:c510_11(x))


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

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

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

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s9_11(n18559_11), gen_0':s9_11(n18559_11)) -> true, rt in Omega(0)

Induction Base:
le(gen_0':s9_11(0), gen_0':s9_11(0)) ->_R^Omega(0)
true

Induction Step:
le(gen_0':s9_11(+(n18559_11, 1)), gen_0':s9_11(+(n18559_11, 1))) ->_R^Omega(0)
le(gen_0':s9_11(n18559_11), gen_0':s9_11(n18559_11)) ->_IH
true

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

(40)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
PRED(s(z0)) -> c3
MINUS(z0, 0') -> c4
MINUS(z0, s(z1)) -> c5(PRED(minus(z0, z1)), MINUS(z0, z1))
MOD(0', z0) -> c6
MOD(s(z0), 0') -> c7
MOD(s(z0), s(z1)) -> c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) -> c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_MOD(false, s(z0), s(z1)) -> c10
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
pred(s(z0)) -> z0
minus(z0, 0') -> z0
minus(z0, s(z1)) -> pred(minus(z0, z1))
mod(0', z0) -> 0'
mod(s(z0), 0') -> 0'
mod(s(z0), s(z1)) -> if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, s(z0), s(z1)) -> mod(minus(z0, z1), s(z1))
if_mod(false, s(z0), s(z1)) -> s(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
PRED :: 0':s -> c3
c3 :: c3
MINUS :: 0':s -> 0':s -> c4:c5
c4 :: c4:c5
c5 :: c3 -> c4:c5 -> c4:c5
minus :: 0':s -> 0':s -> 0':s
MOD :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c9:c10 -> c:c1:c2 -> c6:c7:c8
IF_MOD :: true:false -> 0':s -> 0':s -> c9:c10
le :: 0':s -> 0':s -> true:false
true :: true:false
c9 :: c6:c7:c8 -> c4:c5 -> c9:c10
false :: true:false
c10 :: c9:c10
pred :: 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c33_11 :: c3
hole_c4:c54_11 :: c4:c5
hole_c6:c7:c85_11 :: c6:c7:c8
hole_c9:c106_11 :: c9:c10
hole_true:false7_11 :: true:false
gen_c:c1:c28_11 :: Nat -> c:c1:c2
gen_0':s9_11 :: Nat -> 0':s
gen_c4:c510_11 :: Nat -> c4:c5


Lemmas:
LE(gen_0':s9_11(n12_11), gen_0':s9_11(n12_11)) -> gen_c:c1:c28_11(n12_11), rt in Omega(1 + n12_11)
minus(gen_0':s9_11(a), gen_0':s9_11(+(1, n610_11))) -> *11_11, rt in Omega(0)
le(gen_0':s9_11(n18559_11), gen_0':s9_11(n18559_11)) -> true, rt in Omega(0)


Generator Equations:
gen_c:c1:c28_11(0) <=> c
gen_c:c1:c28_11(+(x, 1)) <=> c2(gen_c:c1:c28_11(x))
gen_0':s9_11(0) <=> 0'
gen_0':s9_11(+(x, 1)) <=> s(gen_0':s9_11(x))
gen_c4:c510_11(0) <=> c4
gen_c4:c510_11(+(x, 1)) <=> c5(c3, gen_c4:c510_11(x))


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