WORST_CASE(Omega(n^1),O(n^2))
proof of input_KjJxjbxoWX.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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^2).

(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), 1 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, 1718 ms]
    (20) BOUNDS(1, n^2)
(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), 7 ms]
    (30) typed CpxTrs
    (31) RewriteLemmaProof [LOWER BOUND(ID), 327 ms]
    (32) typed CpxTrs
    (33) RewriteLemmaProof [LOWER BOUND(ID), 152 ms]
    (34) BEST
        (35) proven lower bound
            (36) LowerBoundPropagationProof [FINISHED, 0 ms]
            (37) BOUNDS(n^1, INF)
        (38) 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:

   cond1(true, x, y) -> cond2(gr(x, y), x, y)
   cond2(true, x, y) -> cond1(gr0(x), y, y)
   cond2(false, x, y) -> cond1(gr0(x), p(x), y)
   gr(0, x) -> false
   gr(s(x), 0) -> true
   gr(s(x), s(y)) -> gr(x, y)
   gr0(0) -> false
   gr0(s(x)) -> true
   p(0) -> 0
   p(s(x)) -> 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:
   cond1(true, z0, z1) -> cond2(gr(z0, z1), z0, z1)
   cond2(true, z0, z1) -> cond1(gr0(z0), z1, z1)
   cond2(false, z0, z1) -> cond1(gr0(z0), p(z0), z1)
   gr(0, z0) -> false
   gr(s(z0), 0) -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   gr0(0) -> false
   gr0(s(z0)) -> true
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
   COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1), GR0(z0))
   COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1), GR0(z0))
   COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1), P(z0))
   GR(0, z0) -> c4
   GR(s(z0), 0) -> c5
   GR(s(z0), s(z1)) -> c6(GR(z0, z1))
   GR0(0) -> c7
   GR0(s(z0)) -> c8
   P(0) -> c9
   P(s(z0)) -> c10
S tuples:
   COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
   COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1), GR0(z0))
   COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1), GR0(z0))
   COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1), P(z0))
   GR(0, z0) -> c4
   GR(s(z0), 0) -> c5
   GR(s(z0), s(z1)) -> c6(GR(z0, z1))
   GR0(0) -> c7
   GR0(s(z0)) -> c8
   P(0) -> c9
   P(s(z0)) -> c10
K tuples:none
Defined Rule Symbols:   cond1_3, cond2_3, gr_2, gr0_1, p_1

Defined Pair Symbols:   COND1_3, COND2_3, GR_2, GR0_1, P_1

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c4, c5, c6_1, c7, c8, c9, c10


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   GR0(s(z0)) -> c8
   GR(s(z0), 0) -> c5
   GR(0, z0) -> c4
   P(s(z0)) -> c10
   GR0(0) -> c7
   P(0) -> c9

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   cond1(true, z0, z1) -> cond2(gr(z0, z1), z0, z1)
   cond2(true, z0, z1) -> cond1(gr0(z0), z1, z1)
   cond2(false, z0, z1) -> cond1(gr0(z0), p(z0), z1)
   gr(0, z0) -> false
   gr(s(z0), 0) -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   gr0(0) -> false
   gr0(s(z0)) -> true
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
   COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1), GR0(z0))
   COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1), GR0(z0))
   COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1), P(z0))
   GR(s(z0), s(z1)) -> c6(GR(z0, z1))
S tuples:
   COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
   COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1), GR0(z0))
   COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1), GR0(z0))
   COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1), P(z0))
   GR(s(z0), s(z1)) -> c6(GR(z0, z1))
K tuples:none
Defined Rule Symbols:   cond1_3, cond2_3, gr_2, gr0_1, p_1

Defined Pair Symbols:   COND1_3, COND2_3, GR_2

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c6_1


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

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   cond1(true, z0, z1) -> cond2(gr(z0, z1), z0, z1)
   cond2(true, z0, z1) -> cond1(gr0(z0), z1, z1)
   cond2(false, z0, z1) -> cond1(gr0(z0), p(z0), z1)
   gr(0, z0) -> false
   gr(s(z0), 0) -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   gr0(0) -> false
   gr0(s(z0)) -> true
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
   GR(s(z0), s(z1)) -> c6(GR(z0, z1))
   COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1))
   COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1))
   COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1))
S tuples:
   COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
   GR(s(z0), s(z1)) -> c6(GR(z0, z1))
   COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1))
   COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1))
   COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1))
K tuples:none
Defined Rule Symbols:   cond1_3, cond2_3, gr_2, gr0_1, p_1

Defined Pair Symbols:   COND1_3, GR_2, COND2_3

Compound Symbols:   c_2, c6_1, c1_1, c2_1, c3_1


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   cond1(true, z0, z1) -> cond2(gr(z0, z1), z0, z1)
   cond2(true, z0, z1) -> cond1(gr0(z0), z1, z1)
   cond2(false, z0, z1) -> cond1(gr0(z0), p(z0), z1)

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   gr(0, z0) -> false
   gr(s(z0), 0) -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   gr0(0) -> false
   gr0(s(z0)) -> true
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
   GR(s(z0), s(z1)) -> c6(GR(z0, z1))
   COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1))
   COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1))
   COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1))
S tuples:
   COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
   GR(s(z0), s(z1)) -> c6(GR(z0, z1))
   COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1))
   COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1))
   COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1))
K tuples:none
Defined Rule Symbols:   gr_2, gr0_1, p_1

Defined Pair Symbols:   COND1_3, GR_2, COND2_3

Compound Symbols:   c_2, c6_1, c1_1, c2_1, c3_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^2).


The TRS R consists of the following rules:

   COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
   GR(s(z0), s(z1)) -> c6(GR(z0, z1))
   COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1))
   COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1))
   COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1))

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

   gr(0, z0) -> false
   gr(s(z0), 0) -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   gr0(0) -> false
   gr0(s(z0)) -> true
   p(0) -> 0
   p(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^2).


The TRS R consists of the following rules:

   COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1)) [1]
   GR(s(z0), s(z1)) -> c6(GR(z0, z1)) [1]
   COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1)) [1]
   COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1)) [1]
   COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1)) [1]
   gr(0, z0) -> false [0]
   gr(s(z0), 0) -> true [0]
   gr(s(z0), s(z1)) -> gr(z0, z1) [0]
   gr0(0) -> false [0]
   gr0(s(z0)) -> true [0]
   p(0) -> 0 [0]
   p(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:

   COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1)) [1]
   GR(s(z0), s(z1)) -> c6(GR(z0, z1)) [1]
   COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1)) [1]
   COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1)) [1]
   COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1)) [1]
   gr(0, z0) -> false [0]
   gr(s(z0), 0) -> true [0]
   gr(s(z0), s(z1)) -> gr(z0, z1) [0]
   gr0(0) -> false [0]
   gr0(s(z0)) -> true [0]
   p(0) -> 0 [0]
   p(s(z0)) -> z0 [0]

The TRS has the following type information:
COND1 :: true:false -> s:0 -> s:0 -> c
true :: true:false
c :: c1:c2:c3 -> c6 -> c
COND2 :: true:false -> s:0 -> s:0 -> c1:c2:c3
gr :: s:0 -> s:0 -> true:false
GR :: s:0 -> s:0 -> c6
s :: s:0 -> s:0
c6 :: c6 -> c6
c1 :: c -> c1:c2:c3
gr0 :: s:0 -> true:false
false :: true:false
c2 :: c -> c1:c2:c3
p :: s:0 -> s:0
c3 :: c -> c1:c2:c3
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:

   gr(v0, v1) -> null_gr [0]
   gr0(v0) -> null_gr0 [0]
   p(v0) -> null_p [0]
   COND1(v0, v1, v2) -> null_COND1 [0]
   GR(v0, v1) -> null_GR [0]
   COND2(v0, v1, v2) -> null_COND2 [0]

And the following fresh constants:    null_gr, null_gr0, null_p, null_COND1, null_GR, null_COND2

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

(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:

   COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1)) [1]
   GR(s(z0), s(z1)) -> c6(GR(z0, z1)) [1]
   COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1)) [1]
   COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1)) [1]
   COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1)) [1]
   gr(0, z0) -> false [0]
   gr(s(z0), 0) -> true [0]
   gr(s(z0), s(z1)) -> gr(z0, z1) [0]
   gr0(0) -> false [0]
   gr0(s(z0)) -> true [0]
   p(0) -> 0 [0]
   p(s(z0)) -> z0 [0]
   gr(v0, v1) -> null_gr [0]
   gr0(v0) -> null_gr0 [0]
   p(v0) -> null_p [0]
   COND1(v0, v1, v2) -> null_COND1 [0]
   GR(v0, v1) -> null_GR [0]
   COND2(v0, v1, v2) -> null_COND2 [0]

The TRS has the following type information:
COND1 :: true:false:null_gr:null_gr0 -> s:0:null_p -> s:0:null_p -> c:null_COND1
true :: true:false:null_gr:null_gr0
c :: c1:c2:c3:null_COND2 -> c6:null_GR -> c:null_COND1
COND2 :: true:false:null_gr:null_gr0 -> s:0:null_p -> s:0:null_p -> c1:c2:c3:null_COND2
gr :: s:0:null_p -> s:0:null_p -> true:false:null_gr:null_gr0
GR :: s:0:null_p -> s:0:null_p -> c6:null_GR
s :: s:0:null_p -> s:0:null_p
c6 :: c6:null_GR -> c6:null_GR
c1 :: c:null_COND1 -> c1:c2:c3:null_COND2
gr0 :: s:0:null_p -> true:false:null_gr:null_gr0
false :: true:false:null_gr:null_gr0
c2 :: c:null_COND1 -> c1:c2:c3:null_COND2
p :: s:0:null_p -> s:0:null_p
c3 :: c:null_COND1 -> c1:c2:c3:null_COND2
0 :: s:0:null_p
null_gr :: true:false:null_gr:null_gr0
null_gr0 :: true:false:null_gr:null_gr0
null_p :: s:0:null_p
null_COND1 :: c:null_COND1
null_GR :: c6:null_GR
null_COND2 :: c1:c2:c3:null_COND2

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
   false => 1
   0 => 0
   null_gr => 0
   null_gr0 => 0
   null_p => 0
   null_COND1 => 0
   null_GR => 0
   null_COND2 => 0

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

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

   COND1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   COND1(z, z', z'') -{ 1 }-> 1 + COND2(gr(z0, z1), z0, z1) + GR(z0, z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   COND2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   COND2(z, z', z'') -{ 1 }-> 1 + COND1(gr0(z0), z1, z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   COND2(z, z', z'') -{ 1 }-> 1 + COND1(gr0(z0), p(z0), z1) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1
   GR(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   GR(z, z') -{ 1 }-> 1 + GR(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   gr(z, z') -{ 0 }-> gr(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   gr(z, z') -{ 0 }-> 2 :|: z = 1 + z0, z0 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z0 >= 0, z = 0, z' = z0
   gr(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   gr0(z) -{ 0 }-> 2 :|: z = 1 + z0, z0 >= 0
   gr0(z) -{ 0 }-> 1 :|: z = 0
   gr0(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   p(z) -{ 0 }-> z0 :|: z = 1 + z0, z0 >= 0
   p(z) -{ 0 }-> 0 :|: z = 0
   p(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, V2),0,[fun(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[fun2(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V2),0,[fun1(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[gr(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V2),0,[gr0(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V2),0,[p(V1, Out)],[V1 >= 0]).
eq(fun(V1, V, V2, Out),1,[gr(V4, V3, Ret010),fun1(Ret010, V4, V3, Ret01),fun2(V4, V3, Ret1)],[Out = 1 + Ret01 + Ret1,V1 = 2,V3 >= 0,V4 >= 0,V = V4,V2 = V3]).
eq(fun2(V1, V, Out),1,[fun2(V6, V5, Ret11)],[Out = 1 + Ret11,V5 >= 0,V1 = 1 + V6,V6 >= 0,V = 1 + V5]).
eq(fun1(V1, V, V2, Out),1,[gr0(V8, Ret10),fun(Ret10, V7, V7, Ret12)],[Out = 1 + Ret12,V1 = 2,V7 >= 0,V8 >= 0,V = V8,V2 = V7]).
eq(fun1(V1, V, V2, Out),1,[gr0(V9, Ret101),p(V9, Ret111),fun(Ret101, Ret111, V10, Ret13)],[Out = 1 + Ret13,V10 >= 0,V1 = 1,V9 >= 0,V = V9,V2 = V10]).
eq(gr(V1, V, Out),0,[],[Out = 1,V11 >= 0,V1 = 0,V = V11]).
eq(gr(V1, V, Out),0,[],[Out = 2,V1 = 1 + V12,V12 >= 0,V = 0]).
eq(gr(V1, V, Out),0,[gr(V14, V13, Ret)],[Out = Ret,V13 >= 0,V1 = 1 + V14,V14 >= 0,V = 1 + V13]).
eq(gr0(V1, Out),0,[],[Out = 1,V1 = 0]).
eq(gr0(V1, Out),0,[],[Out = 2,V1 = 1 + V15,V15 >= 0]).
eq(p(V1, Out),0,[],[Out = 0,V1 = 0]).
eq(p(V1, Out),0,[],[Out = V16,V1 = 1 + V16,V16 >= 0]).
eq(gr(V1, V, Out),0,[],[Out = 0,V18 >= 0,V17 >= 0,V1 = V18,V = V17]).
eq(gr0(V1, Out),0,[],[Out = 0,V19 >= 0,V1 = V19]).
eq(p(V1, Out),0,[],[Out = 0,V20 >= 0,V1 = V20]).
eq(fun(V1, V, V2, Out),0,[],[Out = 0,V21 >= 0,V2 = V23,V22 >= 0,V1 = V21,V = V22,V23 >= 0]).
eq(fun2(V1, V, Out),0,[],[Out = 0,V24 >= 0,V25 >= 0,V1 = V24,V = V25]).
eq(fun1(V1, V, V2, Out),0,[],[Out = 0,V26 >= 0,V2 = V27,V28 >= 0,V1 = V26,V = V28,V27 >= 0]).
input_output_vars(fun(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(fun2(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(gr(V1,V,Out),[V1,V],[Out]).
input_output_vars(gr0(V1,Out),[V1],[Out]).
input_output_vars(p(V1,Out),[V1],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [gr0/2]
1. non_recursive  : [p/2]
2. recursive  : [fun2/3]
3. recursive  : [gr/3]
4. recursive [non_tail] : [fun/4,fun1/4]
5. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into gr0/2
1. SCC is partially evaluated into p/2
2. SCC is partially evaluated into fun2/3
3. SCC is partially evaluated into gr/3
4. SCC is partially evaluated into fun1/4
5. SCC is partially evaluated into start/3

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

### Specialization of cost equations gr0/2 
* CE 20 is refined into CE [24] 
* CE 21 is refined into CE [25] 
* CE 19 is refined into CE [26] 


### Cost equations --> "Loop" of gr0/2 
* CEs [24] --> Loop 18 
* CEs [25] --> Loop 19 
* CEs [26] --> Loop 20 

### Ranking functions of CR gr0(V1,Out) 

#### Partial ranking functions of CR gr0(V1,Out) 


### Specialization of cost equations p/2 
* CE 23 is refined into CE [27] 
* CE 22 is refined into CE [28] 


### Cost equations --> "Loop" of p/2 
* CEs [27] --> Loop 21 
* CEs [28] --> Loop 22 

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

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


### Specialization of cost equations fun2/3 
* CE 18 is refined into CE [29] 
* CE 17 is refined into CE [30] 


### Cost equations --> "Loop" of fun2/3 
* CEs [30] --> Loop 23 
* CEs [29] --> Loop 24 

### Ranking functions of CR fun2(V1,V,Out) 
* RF of phase [23]: [V,V1]

#### Partial ranking functions of CR fun2(V1,V,Out) 
* Partial RF of phase [23]:
  - RF of loop [23:1]:
    V
    V1


### Specialization of cost equations gr/3 
* CE 11 is refined into CE [31] 
* CE 9 is refined into CE [32] 
* CE 8 is refined into CE [33] 
* CE 10 is refined into CE [34] 


### Cost equations --> "Loop" of gr/3 
* CEs [34] --> Loop 25 
* CEs [31] --> Loop 26 
* CEs [32] --> Loop 27 
* CEs [33] --> Loop 28 

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

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


### Specialization of cost equations fun1/4 
* CE 16 is refined into CE [35] 
* CE 13 is refined into CE [36,37,38] 
* CE 12 is refined into CE [39,40,41,42,43] 
* CE 15 is refined into CE [44,45,46,47,48] 
* CE 14 is refined into CE [49,50,51,52,53,54,55,56,57,58] 


### Cost equations --> "Loop" of fun1/4 
* CEs [48] --> Loop 29 
* CEs [47] --> Loop 30 
* CEs [46] --> Loop 31 
* CEs [45] --> Loop 32 
* CEs [44] --> Loop 33 
* CEs [58] --> Loop 34 
* CEs [57] --> Loop 35 
* CEs [56] --> Loop 36 
* CEs [55] --> Loop 37 
* CEs [54] --> Loop 38 
* CEs [53] --> Loop 39 
* CEs [50] --> Loop 40 
* CEs [52] --> Loop 41 
* CEs [49,51] --> Loop 42 
* CEs [35] --> Loop 43 
* CEs [36,37,38] --> Loop 44 
* CEs [39,40,41,42,43] --> Loop 45 

### Ranking functions of CR fun1(V1,V,V2,Out) 
* RF of phase [36,37]: [V-1]

#### Partial ranking functions of CR fun1(V1,V,V2,Out) 
* Partial RF of phase [36,37]:
  - RF of loop [36:1,37:1]:
    V-1


### Specialization of cost equations start/3 
* CE 1 is refined into CE [59] 
* CE 2 is refined into CE [60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95] 
* CE 3 is refined into CE [96,97] 
* CE 4 is refined into CE [98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120] 
* CE 5 is refined into CE [121,122,123,124,125] 
* CE 6 is refined into CE [126,127,128] 
* CE 7 is refined into CE [129,130] 


### Cost equations --> "Loop" of start/3 
* CEs [122] --> Loop 46 
* CEs [62,63,64,65,113] --> Loop 47 
* CEs [60,61,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,112,114,115,116,117,118,119] --> Loop 48 
* CEs [98,99,100,101,102,103,104,105,106,107,108,109,110,111] --> Loop 49 
* CEs [59,96,97,120,121,123,124,125,126,127,128,129,130] --> Loop 50 

### Ranking functions of CR start(V1,V,V2) 

#### Partial ranking functions of CR start(V1,V,V2) 


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

#### Cost of chains of gr0(V1,Out):
* Chain [20]: 0
  with precondition: [V1=0,Out=1] 

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

* Chain [18]: 0
  with precondition: [Out=2,V1>=1] 


#### Cost of chains of p(V1,Out):
* Chain [22]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [21]: 0
  with precondition: [V1=Out+1,V1>=1] 


#### Cost of chains of fun2(V1,V,Out):
* Chain [[23],24]: 1*it(23)+0
  Such that:it(23) =< Out

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

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


#### Cost of chains of gr(V1,V,Out):
* Chain [[25],28]: 0
  with precondition: [Out=1,V1>=1,V>=V1] 

* Chain [[25],27]: 0
  with precondition: [Out=2,V>=1,V1>=V+1] 

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

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

* Chain [27]: 0
  with precondition: [V=0,Out=2,V1>=1] 

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


#### Cost of chains of fun1(V1,V,V2,Out):
* Chain [[36,37],45]: 4*it(36)+1*s(3)+1
  Such that:aux(4) =< V
it(36) =< aux(4)
s(3) =< it(36)*aux(4)

  with precondition: [V1=1,V>=2,Out>=3,V2+1>=V] 

* Chain [[36,37],43]: 4*it(36)+1*s(3)+0
  Such that:aux(5) =< V
it(36) =< aux(5)
s(3) =< it(36)*aux(5)

  with precondition: [V1=1,V>=2,Out>=2,V2+1>=V] 

* Chain [[36,37],42,45]: 4*it(36)+1*s(3)+3
  Such that:aux(6) =< V
it(36) =< aux(6)
s(3) =< it(36)*aux(6)

  with precondition: [V1=1,V>=2,Out>=5,V2+1>=V] 

* Chain [[36,37],42,43]: 4*it(36)+1*s(3)+2
  Such that:aux(7) =< V
it(36) =< aux(7)
s(3) =< it(36)*aux(7)

  with precondition: [V1=1,V>=2,Out>=4,V2+1>=V] 

* Chain [[36,37],40,43]: 4*it(36)+1*s(3)+2
  Such that:aux(8) =< V
it(36) =< aux(8)
s(3) =< it(36)*aux(8)

  with precondition: [V1=1,V>=2,Out>=4,V2+1>=V] 

* Chain [[36,37],39,43]: 4*it(36)+1*s(3)+2
  Such that:aux(9) =< V
it(36) =< aux(9)
s(3) =< it(36)*aux(9)

  with precondition: [V1=1,V>=2,Out>=4,V2+1>=V] 

* Chain [[36,37],38,43]: 4*it(36)+1*s(3)+1*s(4)+2
  Such that:s(4) =< V2
aux(10) =< V
it(36) =< aux(10)
s(3) =< it(36)*aux(10)

  with precondition: [V1=1,V>=3,Out>=5,V2+1>=V] 

* Chain [45]: 1
  with precondition: [V1=1,Out=1,V>=0,V2>=0] 

* Chain [44]: 1
  with precondition: [V1=2,Out=1,V>=0,V2>=0] 

* Chain [43]: 0
  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 

* Chain [42,45]: 3
  with precondition: [V1=1,Out=3,V>=1,V2>=0] 

* Chain [42,43]: 2
  with precondition: [V1=1,Out=2,V>=1,V2>=0] 

* Chain [41,44]: 3
  with precondition: [V1=1,V2=0,Out=3,V>=2] 

* Chain [41,43]: 2
  with precondition: [V1=1,V2=0,Out=2,V>=2] 

* Chain [41,33,45]: 5
  with precondition: [V1=1,V2=0,Out=5,V>=2] 

* Chain [41,33,43]: 4
  with precondition: [V1=1,V2=0,Out=4,V>=2] 

* Chain [41,32,43]: 4
  with precondition: [V1=1,V2=0,Out=4,V>=2] 

* Chain [40,43]: 2
  with precondition: [V1=1,Out=2,V>=1,V2>=0] 

* Chain [39,43]: 2
  with precondition: [V1=1,Out=2,V>=1,V2>=0] 

* Chain [38,43]: 1*s(4)+2
  Such that:s(4) =< V2

  with precondition: [V1=1,Out>=3,V+1>=Out,V2+2>=Out] 

* Chain [35,44]: 3
  with precondition: [V1=1,Out=3,V2>=1,V>=V2+2] 

* Chain [35,43]: 2
  with precondition: [V1=1,Out=2,V2>=1,V>=V2+2] 

* Chain [35,32,43]: 4
  with precondition: [V1=1,Out=4,V2>=1,V>=V2+2] 

* Chain [35,31,43]: 1*s(5)+4
  Such that:s(5) =< V2

  with precondition: [V1=1,Out>=5,V>=V2+2,V2+4>=Out] 

* Chain [35,30,[36,37],45]: 4*it(36)+1*s(3)+5
  Such that:aux(4) =< V2
it(36) =< aux(4)
s(3) =< it(36)*aux(4)

  with precondition: [V1=1,V2>=2,Out>=7,V>=V2+2] 

* Chain [35,30,[36,37],43]: 4*it(36)+1*s(3)+4
  Such that:aux(5) =< V2
it(36) =< aux(5)
s(3) =< it(36)*aux(5)

  with precondition: [V1=1,V2>=2,Out>=6,V>=V2+2] 

* Chain [35,30,[36,37],42,45]: 4*it(36)+1*s(3)+7
  Such that:aux(6) =< V2
it(36) =< aux(6)
s(3) =< it(36)*aux(6)

  with precondition: [V1=1,V2>=2,Out>=9,V>=V2+2] 

* Chain [35,30,[36,37],42,43]: 4*it(36)+1*s(3)+6
  Such that:aux(7) =< V2
it(36) =< aux(7)
s(3) =< it(36)*aux(7)

  with precondition: [V1=1,V2>=2,Out>=8,V>=V2+2] 

* Chain [35,30,[36,37],40,43]: 4*it(36)+1*s(3)+6
  Such that:aux(8) =< V2
it(36) =< aux(8)
s(3) =< it(36)*aux(8)

  with precondition: [V1=1,V2>=2,Out>=8,V>=V2+2] 

* Chain [35,30,[36,37],39,43]: 4*it(36)+1*s(3)+6
  Such that:aux(9) =< V2
it(36) =< aux(9)
s(3) =< it(36)*aux(9)

  with precondition: [V1=1,V2>=2,Out>=8,V>=V2+2] 

* Chain [35,30,[36,37],38,43]: 5*it(36)+1*s(3)+6
  Such that:aux(11) =< V2
it(36) =< aux(11)
s(3) =< it(36)*aux(11)

  with precondition: [V1=1,V2>=3,Out>=9,V>=V2+2] 

* Chain [35,30,45]: 5
  with precondition: [V1=1,Out=5,V2>=1,V>=V2+2] 

* Chain [35,30,43]: 4
  with precondition: [V1=1,Out=4,V2>=1,V>=V2+2] 

* Chain [35,30,42,45]: 7
  with precondition: [V1=1,Out=7,V2>=1,V>=V2+2] 

* Chain [35,30,42,43]: 6
  with precondition: [V1=1,Out=6,V2>=1,V>=V2+2] 

* Chain [35,30,40,43]: 6
  with precondition: [V1=1,Out=6,V2>=1,V>=V2+2] 

* Chain [35,30,39,43]: 6
  with precondition: [V1=1,Out=6,V2>=1,V>=V2+2] 

* Chain [35,30,38,43]: 1*s(4)+6
  Such that:s(4) =< V2

  with precondition: [V1=1,Out>=7,V>=V2+2,V2+5>=Out] 

* Chain [35,29,[36,37],45]: 5*it(36)+1*s(3)+5
  Such that:aux(12) =< V2
it(36) =< aux(12)
s(3) =< it(36)*aux(12)

  with precondition: [V1=1,V2>=2,Out>=8,V>=V2+2] 

* Chain [35,29,[36,37],43]: 5*it(36)+1*s(3)+4
  Such that:aux(13) =< V2
it(36) =< aux(13)
s(3) =< it(36)*aux(13)

  with precondition: [V1=1,V2>=2,Out>=7,V>=V2+2] 

* Chain [35,29,[36,37],42,45]: 5*it(36)+1*s(3)+7
  Such that:aux(14) =< V2
it(36) =< aux(14)
s(3) =< it(36)*aux(14)

  with precondition: [V1=1,V2>=2,Out>=10,V>=V2+2] 

* Chain [35,29,[36,37],42,43]: 5*it(36)+1*s(3)+6
  Such that:aux(15) =< V2
it(36) =< aux(15)
s(3) =< it(36)*aux(15)

  with precondition: [V1=1,V2>=2,Out>=9,V>=V2+2] 

* Chain [35,29,[36,37],40,43]: 5*it(36)+1*s(3)+6
  Such that:aux(16) =< V2
it(36) =< aux(16)
s(3) =< it(36)*aux(16)

  with precondition: [V1=1,V2>=2,Out>=9,V>=V2+2] 

* Chain [35,29,[36,37],39,43]: 5*it(36)+1*s(3)+6
  Such that:aux(17) =< V2
it(36) =< aux(17)
s(3) =< it(36)*aux(17)

  with precondition: [V1=1,V2>=2,Out>=9,V>=V2+2] 

* Chain [35,29,[36,37],38,43]: 6*it(36)+1*s(3)+6
  Such that:aux(18) =< V2
it(36) =< aux(18)
s(3) =< it(36)*aux(18)

  with precondition: [V1=1,V2>=3,Out>=10,V>=V2+2] 

* Chain [35,29,45]: 1*s(6)+5
  Such that:s(6) =< V2

  with precondition: [V1=1,Out>=6,V>=V2+2,V2+5>=Out] 

* Chain [35,29,43]: 1*s(6)+4
  Such that:s(6) =< V2

  with precondition: [V1=1,Out>=5,V>=V2+2,V2+4>=Out] 

* Chain [35,29,42,45]: 1*s(6)+7
  Such that:s(6) =< V2

  with precondition: [V1=1,Out>=8,V>=V2+2,V2+7>=Out] 

* Chain [35,29,42,43]: 1*s(6)+6
  Such that:s(6) =< V2

  with precondition: [V1=1,Out>=7,V>=V2+2,V2+6>=Out] 

* Chain [35,29,40,43]: 1*s(6)+6
  Such that:s(6) =< V2

  with precondition: [V1=1,Out>=7,V>=V2+2,V2+6>=Out] 

* Chain [35,29,39,43]: 1*s(6)+6
  Such that:s(6) =< V2

  with precondition: [V1=1,Out>=7,V>=V2+2,V2+6>=Out] 

* Chain [35,29,38,43]: 2*s(4)+6
  Such that:aux(19) =< V2
s(4) =< aux(19)

  with precondition: [V1=1,V2>=2,Out>=8,V>=V2+2,2*V2+5>=Out] 

* Chain [34,44]: 1*s(7)+3
  Such that:s(7) =< V2

  with precondition: [V1=1,Out>=4,V>=V2+2,V2+3>=Out] 

* Chain [34,43]: 1*s(7)+2
  Such that:s(7) =< V2

  with precondition: [V1=1,Out>=3,V>=V2+2,V2+2>=Out] 

* Chain [34,32,43]: 1*s(7)+4
  Such that:s(7) =< V2

  with precondition: [V1=1,Out>=5,V>=V2+2,V2+4>=Out] 

* Chain [34,31,43]: 2*s(5)+4
  Such that:aux(20) =< V2
s(5) =< aux(20)

  with precondition: [V1=1,Out>=6,V>=V2+2,2*V2+4>=Out] 

* Chain [34,30,[36,37],45]: 5*it(36)+1*s(3)+5
  Such that:aux(21) =< V2
it(36) =< aux(21)
s(3) =< it(36)*aux(21)

  with precondition: [V1=1,V2>=2,Out>=8,V>=V2+2] 

* Chain [34,30,[36,37],43]: 5*it(36)+1*s(3)+4
  Such that:aux(22) =< V2
it(36) =< aux(22)
s(3) =< it(36)*aux(22)

  with precondition: [V1=1,V2>=2,Out>=7,V>=V2+2] 

* Chain [34,30,[36,37],42,45]: 5*it(36)+1*s(3)+7
  Such that:aux(23) =< V2
it(36) =< aux(23)
s(3) =< it(36)*aux(23)

  with precondition: [V1=1,V2>=2,Out>=10,V>=V2+2] 

* Chain [34,30,[36,37],42,43]: 5*it(36)+1*s(3)+6
  Such that:aux(24) =< V2
it(36) =< aux(24)
s(3) =< it(36)*aux(24)

  with precondition: [V1=1,V2>=2,Out>=9,V>=V2+2] 

* Chain [34,30,[36,37],40,43]: 5*it(36)+1*s(3)+6
  Such that:aux(25) =< V2
it(36) =< aux(25)
s(3) =< it(36)*aux(25)

  with precondition: [V1=1,V2>=2,Out>=9,V>=V2+2] 

* Chain [34,30,[36,37],39,43]: 5*it(36)+1*s(3)+6
  Such that:aux(26) =< V2
it(36) =< aux(26)
s(3) =< it(36)*aux(26)

  with precondition: [V1=1,V2>=2,Out>=9,V>=V2+2] 

* Chain [34,30,[36,37],38,43]: 6*it(36)+1*s(3)+6
  Such that:aux(27) =< V2
it(36) =< aux(27)
s(3) =< it(36)*aux(27)

  with precondition: [V1=1,V2>=3,Out>=10,V>=V2+2] 

* Chain [34,30,45]: 1*s(7)+5
  Such that:s(7) =< V2

  with precondition: [V1=1,Out>=6,V>=V2+2,V2+5>=Out] 

* Chain [34,30,43]: 1*s(7)+4
  Such that:s(7) =< V2

  with precondition: [V1=1,Out>=5,V>=V2+2,V2+4>=Out] 

* Chain [34,30,42,45]: 1*s(7)+7
  Such that:s(7) =< V2

  with precondition: [V1=1,Out>=8,V>=V2+2,V2+7>=Out] 

* Chain [34,30,42,43]: 1*s(7)+6
  Such that:s(7) =< V2

  with precondition: [V1=1,Out>=7,V>=V2+2,V2+6>=Out] 

* Chain [34,30,40,43]: 1*s(7)+6
  Such that:s(7) =< V2

  with precondition: [V1=1,Out>=7,V>=V2+2,V2+6>=Out] 

* Chain [34,30,39,43]: 1*s(7)+6
  Such that:s(7) =< V2

  with precondition: [V1=1,Out>=7,V>=V2+2,V2+6>=Out] 

* Chain [34,30,38,43]: 2*s(4)+6
  Such that:aux(28) =< V2
s(4) =< aux(28)

  with precondition: [V1=1,V2>=2,Out>=8,V>=V2+2,2*V2+5>=Out] 

* Chain [34,29,[36,37],45]: 6*it(36)+1*s(3)+5
  Such that:aux(29) =< V2
it(36) =< aux(29)
s(3) =< it(36)*aux(29)

  with precondition: [V1=1,V2>=2,Out>=9,V>=V2+2] 

* Chain [34,29,[36,37],43]: 6*it(36)+1*s(3)+4
  Such that:aux(30) =< V2
it(36) =< aux(30)
s(3) =< it(36)*aux(30)

  with precondition: [V1=1,V2>=2,Out>=8,V>=V2+2] 

* Chain [34,29,[36,37],42,45]: 6*it(36)+1*s(3)+7
  Such that:aux(31) =< V2
it(36) =< aux(31)
s(3) =< it(36)*aux(31)

  with precondition: [V1=1,V2>=2,Out>=11,V>=V2+2] 

* Chain [34,29,[36,37],42,43]: 6*it(36)+1*s(3)+6
  Such that:aux(32) =< V2
it(36) =< aux(32)
s(3) =< it(36)*aux(32)

  with precondition: [V1=1,V2>=2,Out>=10,V>=V2+2] 

* Chain [34,29,[36,37],40,43]: 6*it(36)+1*s(3)+6
  Such that:aux(33) =< V2
it(36) =< aux(33)
s(3) =< it(36)*aux(33)

  with precondition: [V1=1,V2>=2,Out>=10,V>=V2+2] 

* Chain [34,29,[36,37],39,43]: 6*it(36)+1*s(3)+6
  Such that:aux(34) =< V2
it(36) =< aux(34)
s(3) =< it(36)*aux(34)

  with precondition: [V1=1,V2>=2,Out>=10,V>=V2+2] 

* Chain [34,29,[36,37],38,43]: 7*it(36)+1*s(3)+6
  Such that:aux(35) =< V2
it(36) =< aux(35)
s(3) =< it(36)*aux(35)

  with precondition: [V1=1,V2>=3,Out>=11,V>=V2+2] 

* Chain [34,29,45]: 2*s(6)+5
  Such that:aux(36) =< V2
s(6) =< aux(36)

  with precondition: [V1=1,Out>=7,V>=V2+2,2*V2+5>=Out] 

* Chain [34,29,43]: 2*s(6)+4
  Such that:aux(37) =< V2
s(6) =< aux(37)

  with precondition: [V1=1,Out>=6,V>=V2+2,2*V2+4>=Out] 

* Chain [34,29,42,45]: 2*s(6)+7
  Such that:aux(38) =< V2
s(6) =< aux(38)

  with precondition: [V1=1,Out>=9,V>=V2+2,2*V2+7>=Out] 

* Chain [34,29,42,43]: 2*s(6)+6
  Such that:aux(39) =< V2
s(6) =< aux(39)

  with precondition: [V1=1,Out>=8,V>=V2+2,2*V2+6>=Out] 

* Chain [34,29,40,43]: 2*s(6)+6
  Such that:aux(40) =< V2
s(6) =< aux(40)

  with precondition: [V1=1,Out>=8,V>=V2+2,2*V2+6>=Out] 

* Chain [34,29,39,43]: 2*s(6)+6
  Such that:aux(41) =< V2
s(6) =< aux(41)

  with precondition: [V1=1,Out>=8,V>=V2+2,2*V2+6>=Out] 

* Chain [34,29,38,43]: 3*s(4)+6
  Such that:aux(42) =< V2
s(4) =< aux(42)

  with precondition: [V1=1,V2>=2,Out>=9,V>=V2+2,3*V2+5>=Out] 

* Chain [33,45]: 3
  with precondition: [V1=2,V2=0,Out=3,V>=1] 

* Chain [33,43]: 2
  with precondition: [V1=2,V2=0,Out=2,V>=1] 

* Chain [32,43]: 2
  with precondition: [V1=2,Out=2,V>=1,V2>=0] 

* Chain [31,43]: 1*s(5)+2
  Such that:s(5) =< V2

  with precondition: [V1=2,V>=1,Out>=3,V2+2>=Out] 

* Chain [30,[36,37],45]: 4*it(36)+1*s(3)+3
  Such that:aux(4) =< V2
it(36) =< aux(4)
s(3) =< it(36)*aux(4)

  with precondition: [V1=2,V>=1,V2>=2,Out>=5] 

* Chain [30,[36,37],43]: 4*it(36)+1*s(3)+2
  Such that:aux(5) =< V2
it(36) =< aux(5)
s(3) =< it(36)*aux(5)

  with precondition: [V1=2,V>=1,V2>=2,Out>=4] 

* Chain [30,[36,37],42,45]: 4*it(36)+1*s(3)+5
  Such that:aux(6) =< V2
it(36) =< aux(6)
s(3) =< it(36)*aux(6)

  with precondition: [V1=2,V>=1,V2>=2,Out>=7] 

* Chain [30,[36,37],42,43]: 4*it(36)+1*s(3)+4
  Such that:aux(7) =< V2
it(36) =< aux(7)
s(3) =< it(36)*aux(7)

  with precondition: [V1=2,V>=1,V2>=2,Out>=6] 

* Chain [30,[36,37],40,43]: 4*it(36)+1*s(3)+4
  Such that:aux(8) =< V2
it(36) =< aux(8)
s(3) =< it(36)*aux(8)

  with precondition: [V1=2,V>=1,V2>=2,Out>=6] 

* Chain [30,[36,37],39,43]: 4*it(36)+1*s(3)+4
  Such that:aux(9) =< V2
it(36) =< aux(9)
s(3) =< it(36)*aux(9)

  with precondition: [V1=2,V>=1,V2>=2,Out>=6] 

* Chain [30,[36,37],38,43]: 5*it(36)+1*s(3)+4
  Such that:aux(11) =< V2
it(36) =< aux(11)
s(3) =< it(36)*aux(11)

  with precondition: [V1=2,V>=1,V2>=3,Out>=7] 

* Chain [30,45]: 3
  with precondition: [V1=2,Out=3,V>=1,V2>=1] 

* Chain [30,43]: 2
  with precondition: [V1=2,Out=2,V>=1,V2>=1] 

* Chain [30,42,45]: 5
  with precondition: [V1=2,Out=5,V>=1,V2>=1] 

* Chain [30,42,43]: 4
  with precondition: [V1=2,Out=4,V>=1,V2>=1] 

* Chain [30,40,43]: 4
  with precondition: [V1=2,Out=4,V>=1,V2>=1] 

* Chain [30,39,43]: 4
  with precondition: [V1=2,Out=4,V>=1,V2>=1] 

* Chain [30,38,43]: 1*s(4)+4
  Such that:s(4) =< V2

  with precondition: [V1=2,V>=1,Out>=5,V2+3>=Out] 

* Chain [29,[36,37],45]: 5*it(36)+1*s(3)+3
  Such that:aux(12) =< V2
it(36) =< aux(12)
s(3) =< it(36)*aux(12)

  with precondition: [V1=2,V>=1,V2>=2,Out>=6] 

* Chain [29,[36,37],43]: 5*it(36)+1*s(3)+2
  Such that:aux(13) =< V2
it(36) =< aux(13)
s(3) =< it(36)*aux(13)

  with precondition: [V1=2,V>=1,V2>=2,Out>=5] 

* Chain [29,[36,37],42,45]: 5*it(36)+1*s(3)+5
  Such that:aux(14) =< V2
it(36) =< aux(14)
s(3) =< it(36)*aux(14)

  with precondition: [V1=2,V>=1,V2>=2,Out>=8] 

* Chain [29,[36,37],42,43]: 5*it(36)+1*s(3)+4
  Such that:aux(15) =< V2
it(36) =< aux(15)
s(3) =< it(36)*aux(15)

  with precondition: [V1=2,V>=1,V2>=2,Out>=7] 

* Chain [29,[36,37],40,43]: 5*it(36)+1*s(3)+4
  Such that:aux(16) =< V2
it(36) =< aux(16)
s(3) =< it(36)*aux(16)

  with precondition: [V1=2,V>=1,V2>=2,Out>=7] 

* Chain [29,[36,37],39,43]: 5*it(36)+1*s(3)+4
  Such that:aux(17) =< V2
it(36) =< aux(17)
s(3) =< it(36)*aux(17)

  with precondition: [V1=2,V>=1,V2>=2,Out>=7] 

* Chain [29,[36,37],38,43]: 6*it(36)+1*s(3)+4
  Such that:aux(18) =< V2
it(36) =< aux(18)
s(3) =< it(36)*aux(18)

  with precondition: [V1=2,V>=1,V2>=3,Out>=8] 

* Chain [29,45]: 1*s(6)+3
  Such that:s(6) =< V2

  with precondition: [V1=2,V>=1,Out>=4,V2+3>=Out] 

* Chain [29,43]: 1*s(6)+2
  Such that:s(6) =< V2

  with precondition: [V1=2,V>=1,Out>=3,V2+2>=Out] 

* Chain [29,42,45]: 1*s(6)+5
  Such that:s(6) =< V2

  with precondition: [V1=2,V>=1,Out>=6,V2+5>=Out] 

* Chain [29,42,43]: 1*s(6)+4
  Such that:s(6) =< V2

  with precondition: [V1=2,V>=1,Out>=5,V2+4>=Out] 

* Chain [29,40,43]: 1*s(6)+4
  Such that:s(6) =< V2

  with precondition: [V1=2,V>=1,Out>=5,V2+4>=Out] 

* Chain [29,39,43]: 1*s(6)+4
  Such that:s(6) =< V2

  with precondition: [V1=2,V>=1,Out>=5,V2+4>=Out] 

* Chain [29,38,43]: 2*s(4)+4
  Such that:aux(19) =< V2
s(4) =< aux(19)

  with precondition: [V1=2,V>=1,V2>=2,Out>=6,2*V2+3>=Out] 


#### Cost of chains of start(V1,V,V2):
* Chain [50]: 1*s(201)+0
  Such that:s(201) =< V

  with precondition: [V1>=0] 

* Chain [49]: 184*s(202)+28*s(211)+7*s(212)+28*s(215)+7
  Such that:s(210) =< V
aux(54) =< V2
s(202) =< aux(54)
s(211) =< s(210)
s(212) =< s(211)*s(210)
s(215) =< s(202)*aux(54)

  with precondition: [V1=1,V>=0,V2>=0] 

* Chain [48]: 240*s(222)+62*s(223)+14*s(229)+42*s(260)+6
  Such that:aux(61) =< V
aux(62) =< V2
s(223) =< aux(61)
s(222) =< aux(62)
s(229) =< s(223)*aux(61)
s(260) =< s(222)*aux(62)

  with precondition: [V1=2,V>=0,V2>=0] 

* Chain [47]: 4
  with precondition: [V1=2,V2=0,V>=1] 

* Chain [46]: 0
  with precondition: [V=0,V1>=1] 


Closed-form bounds of start(V1,V,V2): 
-------------------------------------
* Chain [50] with precondition: [V1>=0] 
    - Upper bound: nat(V) 
    - Complexity: n 
* Chain [49] with precondition: [V1=1,V>=0,V2>=0] 
    - Upper bound: 28*V+7+7*V*V+184*V2+28*V2*V2 
    - Complexity: n^2 
* Chain [48] with precondition: [V1=2,V>=0,V2>=0] 
    - Upper bound: 62*V+6+14*V*V+240*V2+42*V2*V2 
    - Complexity: n^2 
* Chain [47] with precondition: [V1=2,V2=0,V>=1] 
    - Upper bound: 4 
    - Complexity: constant 
* Chain [46] with precondition: [V=0,V1>=1] 
    - Upper bound: 0 
    - Complexity: constant 

### Maximum cost of start(V1,V,V2): max([4,nat(V)*27+6+nat(V)*7*nat(V)+nat(V2)*184+nat(V2)*28*nat(V2)+max([1,nat(V)*7*nat(V)+nat(V)*34+nat(V2)*56+nat(V2)*14*nat(V2)])+nat(V)]) 
Asymptotic class: n^2 
* Total analysis performed in 1636 ms.


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

(20)
BOUNDS(1, n^2)

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

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   cond1(true, z0, z1) -> cond2(gr(z0, z1), z0, z1)
   cond2(true, z0, z1) -> cond1(gr0(z0), z1, z1)
   cond2(false, z0, z1) -> cond1(gr0(z0), p(z0), z1)
   gr(0, z0) -> false
   gr(s(z0), 0) -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   gr0(0) -> false
   gr0(s(z0)) -> true
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
   COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1), GR0(z0))
   COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1), GR0(z0))
   COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1), P(z0))
   GR(0, z0) -> c4
   GR(s(z0), 0) -> c5
   GR(s(z0), s(z1)) -> c6(GR(z0, z1))
   GR0(0) -> c7
   GR0(s(z0)) -> c8
   P(0) -> c9
   P(s(z0)) -> c10
S tuples:
   COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
   COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1), GR0(z0))
   COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1), GR0(z0))
   COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1), P(z0))
   GR(0, z0) -> c4
   GR(s(z0), 0) -> c5
   GR(s(z0), s(z1)) -> c6(GR(z0, z1))
   GR0(0) -> c7
   GR0(s(z0)) -> c8
   P(0) -> c9
   P(s(z0)) -> c10
K tuples:none
Defined Rule Symbols:   cond1_3, cond2_3, gr_2, gr0_1, p_1

Defined Pair Symbols:   COND1_3, COND2_3, GR_2, GR0_1, P_1

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c4, c5, c6_1, c7, c8, c9, 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:

   COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
   COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1), GR0(z0))
   COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1), GR0(z0))
   COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1), P(z0))
   GR(0, z0) -> c4
   GR(s(z0), 0) -> c5
   GR(s(z0), s(z1)) -> c6(GR(z0, z1))
   GR0(0) -> c7
   GR0(s(z0)) -> c8
   P(0) -> c9
   P(s(z0)) -> c10

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

   cond1(true, z0, z1) -> cond2(gr(z0, z1), z0, z1)
   cond2(true, z0, z1) -> cond1(gr0(z0), z1, z1)
   cond2(false, z0, z1) -> cond1(gr0(z0), p(z0), z1)
   gr(0, z0) -> false
   gr(s(z0), 0) -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   gr0(0) -> false
   gr0(s(z0)) -> true
   p(0) -> 0
   p(s(z0)) -> 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:

   COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
   COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1), GR0(z0))
   COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1), GR0(z0))
   COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1), P(z0))
   GR(0', z0) -> c4
   GR(s(z0), 0') -> c5
   GR(s(z0), s(z1)) -> c6(GR(z0, z1))
   GR0(0') -> c7
   GR0(s(z0)) -> c8
   P(0') -> c9
   P(s(z0)) -> c10

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

   cond1(true, z0, z1) -> cond2(gr(z0, z1), z0, z1)
   cond2(true, z0, z1) -> cond1(gr0(z0), z1, z1)
   cond2(false, z0, z1) -> cond1(gr0(z0), p(z0), z1)
   gr(0', z0) -> false
   gr(s(z0), 0') -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   gr0(0') -> false
   gr0(s(z0)) -> true
   p(0') -> 0'
   p(s(z0)) -> z0

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

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

(28)
Obligation:
Innermost TRS:
Rules:
COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1), GR0(z0))
COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1), GR0(z0))
COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1), P(z0))
GR(0', z0) -> c4
GR(s(z0), 0') -> c5
GR(s(z0), s(z1)) -> c6(GR(z0, z1))
GR0(0') -> c7
GR0(s(z0)) -> c8
P(0') -> c9
P(s(z0)) -> c10
cond1(true, z0, z1) -> cond2(gr(z0, z1), z0, z1)
cond2(true, z0, z1) -> cond1(gr0(z0), z1, z1)
cond2(false, z0, z1) -> cond1(gr0(z0), p(z0), z1)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
gr0(0') -> false
gr0(s(z0)) -> true
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: true:false -> 0':s -> 0':s -> c
true :: true:false
c :: c1:c2:c3 -> c4:c5:c6 -> c
COND2 :: true:false -> 0':s -> 0':s -> c1:c2:c3
gr :: 0':s -> 0':s -> true:false
GR :: 0':s -> 0':s -> c4:c5:c6
c1 :: c -> c7:c8 -> c1:c2:c3
gr0 :: 0':s -> true:false
GR0 :: 0':s -> c7:c8
false :: true:false
c2 :: c -> c7:c8 -> c1:c2:c3
p :: 0':s -> 0':s
c3 :: c -> c9:c10 -> c1:c2:c3
P :: 0':s -> c9:c10
0' :: 0':s
c4 :: c4:c5:c6
s :: 0':s -> 0':s
c5 :: c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
c7 :: c7:c8
c8 :: c7:c8
c9 :: c9:c10
c10 :: c9:c10
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
hole_c1_11 :: c
hole_true:false2_11 :: true:false
hole_0':s3_11 :: 0':s
hole_c1:c2:c34_11 :: c1:c2:c3
hole_c4:c5:c65_11 :: c4:c5:c6
hole_c7:c86_11 :: c7:c8
hole_c9:c107_11 :: c9:c10
hole_cond1:cond28_11 :: cond1:cond2
gen_0':s9_11 :: Nat -> 0':s
gen_c4:c5:c610_11 :: Nat -> c4:c5:c6

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

(29) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
COND1, COND2, gr, GR, cond1, cond2

They will be analysed ascendingly in the following order:
COND1 = COND2
gr < COND1
GR < COND1
gr < cond1
cond1 = cond2

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

(30)
Obligation:
Innermost TRS:
Rules:
COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1), GR0(z0))
COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1), GR0(z0))
COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1), P(z0))
GR(0', z0) -> c4
GR(s(z0), 0') -> c5
GR(s(z0), s(z1)) -> c6(GR(z0, z1))
GR0(0') -> c7
GR0(s(z0)) -> c8
P(0') -> c9
P(s(z0)) -> c10
cond1(true, z0, z1) -> cond2(gr(z0, z1), z0, z1)
cond2(true, z0, z1) -> cond1(gr0(z0), z1, z1)
cond2(false, z0, z1) -> cond1(gr0(z0), p(z0), z1)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
gr0(0') -> false
gr0(s(z0)) -> true
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: true:false -> 0':s -> 0':s -> c
true :: true:false
c :: c1:c2:c3 -> c4:c5:c6 -> c
COND2 :: true:false -> 0':s -> 0':s -> c1:c2:c3
gr :: 0':s -> 0':s -> true:false
GR :: 0':s -> 0':s -> c4:c5:c6
c1 :: c -> c7:c8 -> c1:c2:c3
gr0 :: 0':s -> true:false
GR0 :: 0':s -> c7:c8
false :: true:false
c2 :: c -> c7:c8 -> c1:c2:c3
p :: 0':s -> 0':s
c3 :: c -> c9:c10 -> c1:c2:c3
P :: 0':s -> c9:c10
0' :: 0':s
c4 :: c4:c5:c6
s :: 0':s -> 0':s
c5 :: c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
c7 :: c7:c8
c8 :: c7:c8
c9 :: c9:c10
c10 :: c9:c10
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
hole_c1_11 :: c
hole_true:false2_11 :: true:false
hole_0':s3_11 :: 0':s
hole_c1:c2:c34_11 :: c1:c2:c3
hole_c4:c5:c65_11 :: c4:c5:c6
hole_c7:c86_11 :: c7:c8
hole_c9:c107_11 :: c9:c10
hole_cond1:cond28_11 :: cond1:cond2
gen_0':s9_11 :: Nat -> 0':s
gen_c4:c5:c610_11 :: Nat -> c4:c5:c6


Generator Equations:
gen_0':s9_11(0) <=> 0'
gen_0':s9_11(+(x, 1)) <=> s(gen_0':s9_11(x))
gen_c4:c5:c610_11(0) <=> c4
gen_c4:c5:c610_11(+(x, 1)) <=> c6(gen_c4:c5:c610_11(x))


The following defined symbols remain to be analysed:
gr, COND1, COND2, GR, cond1, cond2

They will be analysed ascendingly in the following order:
COND1 = COND2
gr < COND1
GR < COND1
gr < cond1
cond1 = cond2

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gr(gen_0':s9_11(n12_11), gen_0':s9_11(n12_11)) -> false, rt in Omega(0)

Induction Base:
gr(gen_0':s9_11(0), gen_0':s9_11(0)) ->_R^Omega(0)
false

Induction Step:
gr(gen_0':s9_11(+(n12_11, 1)), gen_0':s9_11(+(n12_11, 1))) ->_R^Omega(0)
gr(gen_0':s9_11(n12_11), gen_0':s9_11(n12_11)) ->_IH
false

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

(32)
Obligation:
Innermost TRS:
Rules:
COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1), GR0(z0))
COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1), GR0(z0))
COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1), P(z0))
GR(0', z0) -> c4
GR(s(z0), 0') -> c5
GR(s(z0), s(z1)) -> c6(GR(z0, z1))
GR0(0') -> c7
GR0(s(z0)) -> c8
P(0') -> c9
P(s(z0)) -> c10
cond1(true, z0, z1) -> cond2(gr(z0, z1), z0, z1)
cond2(true, z0, z1) -> cond1(gr0(z0), z1, z1)
cond2(false, z0, z1) -> cond1(gr0(z0), p(z0), z1)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
gr0(0') -> false
gr0(s(z0)) -> true
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: true:false -> 0':s -> 0':s -> c
true :: true:false
c :: c1:c2:c3 -> c4:c5:c6 -> c
COND2 :: true:false -> 0':s -> 0':s -> c1:c2:c3
gr :: 0':s -> 0':s -> true:false
GR :: 0':s -> 0':s -> c4:c5:c6
c1 :: c -> c7:c8 -> c1:c2:c3
gr0 :: 0':s -> true:false
GR0 :: 0':s -> c7:c8
false :: true:false
c2 :: c -> c7:c8 -> c1:c2:c3
p :: 0':s -> 0':s
c3 :: c -> c9:c10 -> c1:c2:c3
P :: 0':s -> c9:c10
0' :: 0':s
c4 :: c4:c5:c6
s :: 0':s -> 0':s
c5 :: c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
c7 :: c7:c8
c8 :: c7:c8
c9 :: c9:c10
c10 :: c9:c10
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
hole_c1_11 :: c
hole_true:false2_11 :: true:false
hole_0':s3_11 :: 0':s
hole_c1:c2:c34_11 :: c1:c2:c3
hole_c4:c5:c65_11 :: c4:c5:c6
hole_c7:c86_11 :: c7:c8
hole_c9:c107_11 :: c9:c10
hole_cond1:cond28_11 :: cond1:cond2
gen_0':s9_11 :: Nat -> 0':s
gen_c4:c5:c610_11 :: Nat -> c4:c5:c6


Lemmas:
gr(gen_0':s9_11(n12_11), gen_0':s9_11(n12_11)) -> false, rt in Omega(0)


Generator Equations:
gen_0':s9_11(0) <=> 0'
gen_0':s9_11(+(x, 1)) <=> s(gen_0':s9_11(x))
gen_c4:c5:c610_11(0) <=> c4
gen_c4:c5:c610_11(+(x, 1)) <=> c6(gen_c4:c5:c610_11(x))


The following defined symbols remain to be analysed:
GR, COND1, COND2, cond1, cond2

They will be analysed ascendingly in the following order:
COND1 = COND2
GR < COND1
cond1 = cond2

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

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
GR(gen_0':s9_11(n341_11), gen_0':s9_11(n341_11)) -> gen_c4:c5:c610_11(n341_11), rt in Omega(1 + n341_11)

Induction Base:
GR(gen_0':s9_11(0), gen_0':s9_11(0)) ->_R^Omega(1)
c4

Induction Step:
GR(gen_0':s9_11(+(n341_11, 1)), gen_0':s9_11(+(n341_11, 1))) ->_R^Omega(1)
c6(GR(gen_0':s9_11(n341_11), gen_0':s9_11(n341_11))) ->_IH
c6(gen_c4:c5:c610_11(c342_11))

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

(34)
Complex Obligation (BEST)

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

(35)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1), GR0(z0))
COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1), GR0(z0))
COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1), P(z0))
GR(0', z0) -> c4
GR(s(z0), 0') -> c5
GR(s(z0), s(z1)) -> c6(GR(z0, z1))
GR0(0') -> c7
GR0(s(z0)) -> c8
P(0') -> c9
P(s(z0)) -> c10
cond1(true, z0, z1) -> cond2(gr(z0, z1), z0, z1)
cond2(true, z0, z1) -> cond1(gr0(z0), z1, z1)
cond2(false, z0, z1) -> cond1(gr0(z0), p(z0), z1)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
gr0(0') -> false
gr0(s(z0)) -> true
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: true:false -> 0':s -> 0':s -> c
true :: true:false
c :: c1:c2:c3 -> c4:c5:c6 -> c
COND2 :: true:false -> 0':s -> 0':s -> c1:c2:c3
gr :: 0':s -> 0':s -> true:false
GR :: 0':s -> 0':s -> c4:c5:c6
c1 :: c -> c7:c8 -> c1:c2:c3
gr0 :: 0':s -> true:false
GR0 :: 0':s -> c7:c8
false :: true:false
c2 :: c -> c7:c8 -> c1:c2:c3
p :: 0':s -> 0':s
c3 :: c -> c9:c10 -> c1:c2:c3
P :: 0':s -> c9:c10
0' :: 0':s
c4 :: c4:c5:c6
s :: 0':s -> 0':s
c5 :: c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
c7 :: c7:c8
c8 :: c7:c8
c9 :: c9:c10
c10 :: c9:c10
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
hole_c1_11 :: c
hole_true:false2_11 :: true:false
hole_0':s3_11 :: 0':s
hole_c1:c2:c34_11 :: c1:c2:c3
hole_c4:c5:c65_11 :: c4:c5:c6
hole_c7:c86_11 :: c7:c8
hole_c9:c107_11 :: c9:c10
hole_cond1:cond28_11 :: cond1:cond2
gen_0':s9_11 :: Nat -> 0':s
gen_c4:c5:c610_11 :: Nat -> c4:c5:c6


Lemmas:
gr(gen_0':s9_11(n12_11), gen_0':s9_11(n12_11)) -> false, rt in Omega(0)


Generator Equations:
gen_0':s9_11(0) <=> 0'
gen_0':s9_11(+(x, 1)) <=> s(gen_0':s9_11(x))
gen_c4:c5:c610_11(0) <=> c4
gen_c4:c5:c610_11(+(x, 1)) <=> c6(gen_c4:c5:c610_11(x))


The following defined symbols remain to be analysed:
GR, COND1, COND2, cond1, cond2

They will be analysed ascendingly in the following order:
COND1 = COND2
GR < COND1
cond1 = cond2

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

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

(37)
BOUNDS(n^1, INF)

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

(38)
Obligation:
Innermost TRS:
Rules:
COND1(true, z0, z1) -> c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
COND2(true, z0, z1) -> c1(COND1(gr0(z0), z1, z1), GR0(z0))
COND2(false, z0, z1) -> c2(COND1(gr0(z0), p(z0), z1), GR0(z0))
COND2(false, z0, z1) -> c3(COND1(gr0(z0), p(z0), z1), P(z0))
GR(0', z0) -> c4
GR(s(z0), 0') -> c5
GR(s(z0), s(z1)) -> c6(GR(z0, z1))
GR0(0') -> c7
GR0(s(z0)) -> c8
P(0') -> c9
P(s(z0)) -> c10
cond1(true, z0, z1) -> cond2(gr(z0, z1), z0, z1)
cond2(true, z0, z1) -> cond1(gr0(z0), z1, z1)
cond2(false, z0, z1) -> cond1(gr0(z0), p(z0), z1)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
gr0(0') -> false
gr0(s(z0)) -> true
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: true:false -> 0':s -> 0':s -> c
true :: true:false
c :: c1:c2:c3 -> c4:c5:c6 -> c
COND2 :: true:false -> 0':s -> 0':s -> c1:c2:c3
gr :: 0':s -> 0':s -> true:false
GR :: 0':s -> 0':s -> c4:c5:c6
c1 :: c -> c7:c8 -> c1:c2:c3
gr0 :: 0':s -> true:false
GR0 :: 0':s -> c7:c8
false :: true:false
c2 :: c -> c7:c8 -> c1:c2:c3
p :: 0':s -> 0':s
c3 :: c -> c9:c10 -> c1:c2:c3
P :: 0':s -> c9:c10
0' :: 0':s
c4 :: c4:c5:c6
s :: 0':s -> 0':s
c5 :: c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
c7 :: c7:c8
c8 :: c7:c8
c9 :: c9:c10
c10 :: c9:c10
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
hole_c1_11 :: c
hole_true:false2_11 :: true:false
hole_0':s3_11 :: 0':s
hole_c1:c2:c34_11 :: c1:c2:c3
hole_c4:c5:c65_11 :: c4:c5:c6
hole_c7:c86_11 :: c7:c8
hole_c9:c107_11 :: c9:c10
hole_cond1:cond28_11 :: cond1:cond2
gen_0':s9_11 :: Nat -> 0':s
gen_c4:c5:c610_11 :: Nat -> c4:c5:c6


Lemmas:
gr(gen_0':s9_11(n12_11), gen_0':s9_11(n12_11)) -> false, rt in Omega(0)
GR(gen_0':s9_11(n341_11), gen_0':s9_11(n341_11)) -> gen_c4:c5:c610_11(n341_11), rt in Omega(1 + n341_11)


Generator Equations:
gen_0':s9_11(0) <=> 0'
gen_0':s9_11(+(x, 1)) <=> s(gen_0':s9_11(x))
gen_c4:c5:c610_11(0) <=> c4
gen_c4:c5:c610_11(+(x, 1)) <=> c6(gen_c4:c5:c610_11(x))


The following defined symbols remain to be analysed:
cond2, COND1, COND2, cond1

They will be analysed ascendingly in the following order:
COND1 = COND2
cond1 = cond2