WORST_CASE(Omega(n^1),O(n^2))
proof of input_tASkpqDJtf.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 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, 577 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), 18 ms]
    (30) typed CpxTrs
    (31) RewriteLemmaProof [LOWER BOUND(ID), 317 ms]
    (32) typed CpxTrs
    (33) RewriteLemmaProof [LOWER BOUND(ID), 114 ms]
    (34) BEST
        (35) proven lower bound
            (36) LowerBoundPropagationProof [FINISHED, 0 ms]
            (37) BOUNDS(n^1, INF)
        (38) typed CpxTrs
            (39) RewriteLemmaProof [LOWER BOUND(ID), 1080 ms]
            (40) typed CpxTrs
            (41) RewriteLemmaProof [LOWER BOUND(ID), 1939 ms]
            (42) BOUNDS(1, INF)


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

(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(s(x), y) -> cond2(gr(s(x), y), s(x), y)
   cond2(true, x, y) -> cond1(y, y)
   cond2(false, x, y) -> cond1(p(x), y)
   gr(0, x) -> false
   gr(s(x), 0) -> true
   gr(s(x), s(y)) -> gr(x, y)
   neq(0, 0) -> false
   neq(0, s(x)) -> true
   neq(s(x), 0) -> true
   neq(s(x), s(y)) -> neq(x, y)
   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(s(z0), z1) -> cond2(gr(s(z0), z1), s(z0), z1)
   cond2(true, z0, z1) -> cond1(z1, z1)
   cond2(false, z0, z1) -> cond1(p(z0), z1)
   gr(0, z0) -> false
   gr(s(z0), 0) -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   neq(0, 0) -> false
   neq(0, s(z0)) -> true
   neq(s(z0), 0) -> true
   neq(s(z0), s(z1)) -> neq(z0, z1)
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   COND1(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1))
   COND2(true, z0, z1) -> c1(COND1(z1, z1))
   COND2(false, z0, z1) -> c2(COND1(p(z0), z1), P(z0))
   GR(0, z0) -> c3
   GR(s(z0), 0) -> c4
   GR(s(z0), s(z1)) -> c5(GR(z0, z1))
   NEQ(0, 0) -> c6
   NEQ(0, s(z0)) -> c7
   NEQ(s(z0), 0) -> c8
   NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1))
   P(0) -> c10
   P(s(z0)) -> c11
S tuples:
   COND1(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1))
   COND2(true, z0, z1) -> c1(COND1(z1, z1))
   COND2(false, z0, z1) -> c2(COND1(p(z0), z1), P(z0))
   GR(0, z0) -> c3
   GR(s(z0), 0) -> c4
   GR(s(z0), s(z1)) -> c5(GR(z0, z1))
   NEQ(0, 0) -> c6
   NEQ(0, s(z0)) -> c7
   NEQ(s(z0), 0) -> c8
   NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1))
   P(0) -> c10
   P(s(z0)) -> c11
K tuples:none
Defined Rule Symbols:   cond1_2, cond2_3, gr_2, neq_2, p_1

Defined Pair Symbols:   COND1_2, COND2_3, GR_2, NEQ_2, P_1

Compound Symbols:   c_2, c1_1, c2_2, c3, c4, c5_1, c6, c7, c8, c9_1, c10, c11


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 7 trailing nodes:
   NEQ(0, s(z0)) -> c7
   GR(0, z0) -> c3
   P(s(z0)) -> c11
   NEQ(s(z0), 0) -> c8
   P(0) -> c10
   GR(s(z0), 0) -> c4
   NEQ(0, 0) -> c6

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   cond1(s(z0), z1) -> cond2(gr(s(z0), z1), s(z0), z1)
   cond2(true, z0, z1) -> cond1(z1, z1)
   cond2(false, z0, z1) -> cond1(p(z0), z1)
   gr(0, z0) -> false
   gr(s(z0), 0) -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   neq(0, 0) -> false
   neq(0, s(z0)) -> true
   neq(s(z0), 0) -> true
   neq(s(z0), s(z1)) -> neq(z0, z1)
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   COND1(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1))
   COND2(true, z0, z1) -> c1(COND1(z1, z1))
   COND2(false, z0, z1) -> c2(COND1(p(z0), z1), P(z0))
   GR(s(z0), s(z1)) -> c5(GR(z0, z1))
   NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1))
S tuples:
   COND1(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1))
   COND2(true, z0, z1) -> c1(COND1(z1, z1))
   COND2(false, z0, z1) -> c2(COND1(p(z0), z1), P(z0))
   GR(s(z0), s(z1)) -> c5(GR(z0, z1))
   NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1))
K tuples:none
Defined Rule Symbols:   cond1_2, cond2_3, gr_2, neq_2, p_1

Defined Pair Symbols:   COND1_2, COND2_3, GR_2, NEQ_2

Compound Symbols:   c_2, c1_1, c2_2, c5_1, c9_1


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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   cond1(s(z0), z1) -> cond2(gr(s(z0), z1), s(z0), z1)
   cond2(true, z0, z1) -> cond1(z1, z1)
   cond2(false, z0, z1) -> cond1(p(z0), z1)
   gr(0, z0) -> false
   gr(s(z0), 0) -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   neq(0, 0) -> false
   neq(0, s(z0)) -> true
   neq(s(z0), 0) -> true
   neq(s(z0), s(z1)) -> neq(z0, z1)
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   COND1(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1))
   COND2(true, z0, z1) -> c1(COND1(z1, z1))
   GR(s(z0), s(z1)) -> c5(GR(z0, z1))
   NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1))
   COND2(false, z0, z1) -> c2(COND1(p(z0), z1))
S tuples:
   COND1(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1))
   COND2(true, z0, z1) -> c1(COND1(z1, z1))
   GR(s(z0), s(z1)) -> c5(GR(z0, z1))
   NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1))
   COND2(false, z0, z1) -> c2(COND1(p(z0), z1))
K tuples:none
Defined Rule Symbols:   cond1_2, cond2_3, gr_2, neq_2, p_1

Defined Pair Symbols:   COND1_2, COND2_3, GR_2, NEQ_2

Compound Symbols:   c_2, c1_1, c5_1, c9_1, c2_1


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   cond1(s(z0), z1) -> cond2(gr(s(z0), z1), s(z0), z1)
   cond2(true, z0, z1) -> cond1(z1, z1)
   cond2(false, z0, z1) -> cond1(p(z0), z1)
   neq(0, 0) -> false
   neq(0, s(z0)) -> true
   neq(s(z0), 0) -> true
   neq(s(z0), s(z1)) -> neq(z0, z1)

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   gr(s(z0), 0) -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   gr(0, z0) -> false
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   COND1(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1))
   COND2(true, z0, z1) -> c1(COND1(z1, z1))
   GR(s(z0), s(z1)) -> c5(GR(z0, z1))
   NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1))
   COND2(false, z0, z1) -> c2(COND1(p(z0), z1))
S tuples:
   COND1(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1))
   COND2(true, z0, z1) -> c1(COND1(z1, z1))
   GR(s(z0), s(z1)) -> c5(GR(z0, z1))
   NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1))
   COND2(false, z0, z1) -> c2(COND1(p(z0), z1))
K tuples:none
Defined Rule Symbols:   gr_2, p_1

Defined Pair Symbols:   COND1_2, COND2_3, GR_2, NEQ_2

Compound Symbols:   c_2, c1_1, c5_1, c9_1, c2_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(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1))
   COND2(true, z0, z1) -> c1(COND1(z1, z1))
   GR(s(z0), s(z1)) -> c5(GR(z0, z1))
   NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1))
   COND2(false, z0, z1) -> c2(COND1(p(z0), z1))

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

   gr(s(z0), 0) -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   gr(0, z0) -> false
   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(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1)) [1]
   COND2(true, z0, z1) -> c1(COND1(z1, z1)) [1]
   GR(s(z0), s(z1)) -> c5(GR(z0, z1)) [1]
   NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1)) [1]
   COND2(false, z0, z1) -> c2(COND1(p(z0), z1)) [1]
   gr(s(z0), 0) -> true [0]
   gr(s(z0), s(z1)) -> gr(z0, z1) [0]
   gr(0, z0) -> false [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(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1)) [1]
   COND2(true, z0, z1) -> c1(COND1(z1, z1)) [1]
   GR(s(z0), s(z1)) -> c5(GR(z0, z1)) [1]
   NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1)) [1]
   COND2(false, z0, z1) -> c2(COND1(p(z0), z1)) [1]
   gr(s(z0), 0) -> true [0]
   gr(s(z0), s(z1)) -> gr(z0, z1) [0]
   gr(0, z0) -> false [0]
   p(0) -> 0 [0]
   p(s(z0)) -> z0 [0]

The TRS has the following type information:
COND1 :: s:0 -> s:0 -> c
s :: s:0 -> s:0
c :: c1:c2 -> c5 -> c
COND2 :: true:false -> s:0 -> s:0 -> c1:c2
gr :: s:0 -> s:0 -> true:false
GR :: s:0 -> s:0 -> c5
true :: true:false
c1 :: c -> c1:c2
c5 :: c5 -> c5
NEQ :: s:0 -> s:0 -> c9
c9 :: c9 -> c9
false :: true:false
c2 :: c -> c1:c2
p :: s:0 -> s:0
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]
   p(v0) -> null_p [0]
   COND1(v0, v1) -> null_COND1 [0]
   COND2(v0, v1, v2) -> null_COND2 [0]
   GR(v0, v1) -> null_GR [0]
   NEQ(v0, v1) -> null_NEQ [0]

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

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

(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(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1)) [1]
   COND2(true, z0, z1) -> c1(COND1(z1, z1)) [1]
   GR(s(z0), s(z1)) -> c5(GR(z0, z1)) [1]
   NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1)) [1]
   COND2(false, z0, z1) -> c2(COND1(p(z0), z1)) [1]
   gr(s(z0), 0) -> true [0]
   gr(s(z0), s(z1)) -> gr(z0, z1) [0]
   gr(0, z0) -> false [0]
   p(0) -> 0 [0]
   p(s(z0)) -> z0 [0]
   gr(v0, v1) -> null_gr [0]
   p(v0) -> null_p [0]
   COND1(v0, v1) -> null_COND1 [0]
   COND2(v0, v1, v2) -> null_COND2 [0]
   GR(v0, v1) -> null_GR [0]
   NEQ(v0, v1) -> null_NEQ [0]

The TRS has the following type information:
COND1 :: s:0:null_p -> s:0:null_p -> c:null_COND1
s :: s:0:null_p -> s:0:null_p
c :: c1:c2:null_COND2 -> c5:null_GR -> c:null_COND1
COND2 :: true:false:null_gr -> s:0:null_p -> s:0:null_p -> c1:c2:null_COND2
gr :: s:0:null_p -> s:0:null_p -> true:false:null_gr
GR :: s:0:null_p -> s:0:null_p -> c5:null_GR
true :: true:false:null_gr
c1 :: c:null_COND1 -> c1:c2:null_COND2
c5 :: c5:null_GR -> c5:null_GR
NEQ :: s:0:null_p -> s:0:null_p -> c9:null_NEQ
c9 :: c9:null_NEQ -> c9:null_NEQ
false :: true:false:null_gr
c2 :: c:null_COND1 -> c1:c2:null_COND2
p :: s:0:null_p -> s:0:null_p
0 :: s:0:null_p
null_gr :: true:false:null_gr
null_p :: s:0:null_p
null_COND1 :: c:null_COND1
null_COND2 :: c1:c2:null_COND2
null_GR :: c5:null_GR
null_NEQ :: c9:null_NEQ

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_p => 0
   null_COND1 => 0
   null_COND2 => 0
   null_GR => 0
   null_NEQ => 0

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

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

   COND1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   COND1(z, z') -{ 1 }-> 1 + COND2(gr(1 + z0, z1), 1 + z0, z1) + GR(1 + z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   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(z1, z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   COND2(z, z', z'') -{ 1 }-> 1 + COND1(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
   NEQ(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   NEQ(z, z') -{ 1 }-> 1 + NEQ(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
   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, V4),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V4),0,[fun1(V1, V, V4, Out)],[V1 >= 0,V >= 0,V4 >= 0]).
eq(start(V1, V, V4),0,[fun2(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V4),0,[fun3(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V4),0,[gr(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V4),0,[p(V1, Out)],[V1 >= 0]).
eq(fun(V1, V, Out),1,[gr(1 + V3, V2, Ret010),fun1(Ret010, 1 + V3, V2, Ret01),fun2(1 + V3, V2, Ret1)],[Out = 1 + Ret01 + Ret1,V2 >= 0,V1 = 1 + V3,V = V2,V3 >= 0]).
eq(fun1(V1, V, V4, Out),1,[fun(V5, V5, Ret11)],[Out = 1 + Ret11,V1 = 2,V5 >= 0,V6 >= 0,V = V6,V4 = V5]).
eq(fun2(V1, V, Out),1,[fun2(V8, V7, Ret12)],[Out = 1 + Ret12,V7 >= 0,V1 = 1 + V8,V8 >= 0,V = 1 + V7]).
eq(fun3(V1, V, Out),1,[fun3(V9, V10, Ret13)],[Out = 1 + Ret13,V10 >= 0,V1 = 1 + V9,V9 >= 0,V = 1 + V10]).
eq(fun1(V1, V, V4, Out),1,[p(V12, Ret10),fun(Ret10, V11, Ret14)],[Out = 1 + Ret14,V11 >= 0,V1 = 1,V12 >= 0,V = V12,V4 = V11]).
eq(gr(V1, V, Out),0,[],[Out = 2,V1 = 1 + V13,V13 >= 0,V = 0]).
eq(gr(V1, V, Out),0,[gr(V15, V14, Ret)],[Out = Ret,V14 >= 0,V1 = 1 + V15,V15 >= 0,V = 1 + V14]).
eq(gr(V1, V, Out),0,[],[Out = 1,V16 >= 0,V1 = 0,V = V16]).
eq(p(V1, Out),0,[],[Out = 0,V1 = 0]).
eq(p(V1, Out),0,[],[Out = V17,V1 = 1 + V17,V17 >= 0]).
eq(gr(V1, V, Out),0,[],[Out = 0,V19 >= 0,V18 >= 0,V1 = V19,V = V18]).
eq(p(V1, Out),0,[],[Out = 0,V20 >= 0,V1 = V20]).
eq(fun(V1, V, Out),0,[],[Out = 0,V22 >= 0,V21 >= 0,V1 = V22,V = V21]).
eq(fun1(V1, V, V4, Out),0,[],[Out = 0,V23 >= 0,V4 = V25,V24 >= 0,V1 = V23,V = V24,V25 >= 0]).
eq(fun2(V1, V, Out),0,[],[Out = 0,V26 >= 0,V27 >= 0,V1 = V26,V = V27]).
eq(fun3(V1, V, Out),0,[],[Out = 0,V29 >= 0,V28 >= 0,V1 = V29,V = V28]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,V4,Out),[V1,V,V4],[Out]).
input_output_vars(fun2(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun3(V1,V,Out),[V1,V],[Out]).
input_output_vars(gr(V1,V,Out),[V1,V],[Out]).
input_output_vars(p(V1,Out),[V1],[Out]).


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

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

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into p/2
1. SCC is partially evaluated into fun2/3
2. SCC is partially evaluated into gr/3
3. SCC is partially evaluated into fun/3
4. SCC is partially evaluated into fun3/3
5. SCC is partially evaluated into start/3

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

### Specialization of cost equations p/2 
* CE 10 is refined into CE [23] 
* CE 9 is refined into CE [24] 


### Cost equations --> "Loop" of p/2 
* CEs [23] --> Loop 16 
* CEs [24] --> Loop 17 

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

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


### Specialization of cost equations fun2/3 
* CE 16 is refined into CE [25] 
* CE 15 is refined into CE [26] 


### Cost equations --> "Loop" of fun2/3 
* CEs [26] --> Loop 18 
* CEs [25] --> Loop 19 

### Ranking functions of CR fun2(V1,V,Out) 
* RF of phase [18]: [V,V1]

#### Partial ranking functions of CR fun2(V1,V,Out) 
* Partial RF of phase [18]:
  - RF of loop [18:1]:
    V
    V1


### Specialization of cost equations gr/3 
* CE 22 is refined into CE [27] 
* CE 19 is refined into CE [28] 
* CE 21 is refined into CE [29] 
* CE 20 is refined into CE [30] 


### Cost equations --> "Loop" of gr/3 
* CEs [30] --> Loop 20 
* CEs [27] --> Loop 21 
* CEs [28] --> Loop 22 
* CEs [29] --> Loop 23 

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

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


### Specialization of cost equations fun/3 
* CE 11 is refined into CE [31,32,33,34,35,36,37] 
* CE 14 is refined into CE [38] 
* CE 12 is refined into CE [39,40,41,42] 
* CE 13 is refined into CE [43,44,45] 


### Cost equations --> "Loop" of fun/3 
* CEs [45] --> Loop 24 
* CEs [44] --> Loop 25 
* CEs [42] --> Loop 26 
* CEs [41] --> Loop 27 
* CEs [40] --> Loop 28 
* CEs [39] --> Loop 29 
* CEs [43] --> Loop 30 
* CEs [37] --> Loop 31 
* CEs [33,35] --> Loop 32 
* CEs [38] --> Loop 33 
* CEs [31,32,34,36] --> Loop 34 

### Ranking functions of CR fun(V1,V,Out) 
* RF of phase [26,27]: [V1]

#### Partial ranking functions of CR fun(V1,V,Out) 
* Partial RF of phase [26,27]:
  - RF of loop [26:1,27:1]:
    V1


### Specialization of cost equations fun3/3 
* CE 18 is refined into CE [46] 
* CE 17 is refined into CE [47] 


### Cost equations --> "Loop" of fun3/3 
* CEs [47] --> Loop 35 
* CEs [46] --> Loop 36 

### Ranking functions of CR fun3(V1,V,Out) 
* RF of phase [35]: [V,V1]

#### Partial ranking functions of CR fun3(V1,V,Out) 
* Partial RF of phase [35]:
  - RF of loop [35:1]:
    V
    V1


### Specialization of cost equations start/3 
* CE 3 is refined into CE [48,49,50,51] 
* CE 1 is refined into CE [52] 
* CE 2 is refined into CE [53,54,55,56,57,58,59,60] 
* CE 4 is refined into CE [61,62,63,64,65,66,67] 
* CE 5 is refined into CE [68,69] 
* CE 6 is refined into CE [70,71] 
* CE 7 is refined into CE [72,73,74,75,76] 
* CE 8 is refined into CE [77,78] 


### Cost equations --> "Loop" of start/3 
* CEs [61,73] --> Loop 37 
* CEs [48,49,50,51] --> Loop 38 
* CEs [53,54,55,56,57,58,59,60] --> Loop 39 
* CEs [52,62,63,64,65,66,67,68,69,70,71,72,74,75,76,77,78] --> Loop 40 

### Ranking functions of CR start(V1,V,V4) 

#### Partial ranking functions of CR start(V1,V,V4) 


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

#### Cost of chains of p(V1,Out):
* Chain [17]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [16]: 0
  with precondition: [V1=Out+1,V1>=1] 


#### Cost of chains of fun2(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 gr(V1,V,Out):
* Chain [[20],23]: 0
  with precondition: [Out=1,V1>=1,V>=V1] 

* Chain [[20],22]: 0
  with precondition: [Out=2,V>=1,V1>=V+1] 

* Chain [[20],21]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

* Chain [23]: 0
  with precondition: [V1=0,Out=1,V>=0] 

* Chain [22]: 0
  with precondition: [V=0,Out=2,V1>=1] 

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


#### Cost of chains of fun(V1,V,Out):
* Chain [[26,27],34]: 4*it(26)+1*s(3)+1
  Such that:aux(4) =< V1
it(26) =< aux(4)
s(3) =< it(26)*aux(4)

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

* Chain [[26,27],33]: 4*it(26)+1*s(3)+0
  Such that:aux(5) =< V1
it(26) =< aux(5)
s(3) =< it(26)*aux(5)

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

* Chain [[26,27],32]: 5*it(26)+1*s(3)+1*s(4)+1
  Such that:s(4) =< V
aux(6) =< V1
it(26) =< aux(6)
s(3) =< it(26)*aux(6)

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

* Chain [[26,27],29,33]: 4*it(26)+1*s(3)+2
  Such that:aux(7) =< V1
it(26) =< aux(7)
s(3) =< it(26)*aux(7)

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

* Chain [[26,27],28,33]: 5*it(26)+1*s(3)+2
  Such that:aux(8) =< V1
it(26) =< aux(8)
s(3) =< it(26)*aux(8)

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

* Chain [34]: 1
  with precondition: [Out=1,V1>=1,V>=0] 

* Chain [33]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [32]: 1*s(4)+1*s(5)+1
  Such that:s(5) =< V1
s(4) =< V

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

* Chain [31]: 1*s(7)+1
  Such that:s(7) =< V

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

* Chain [30,33]: 2
  with precondition: [V=0,Out=2,V1>=1] 

* Chain [29,33]: 2
  with precondition: [Out=2,V1>=1,V>=V1] 

* Chain [28,33]: 1*s(6)+2
  Such that:s(6) =< V1

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

* Chain [25,[26,27],34]: 4*it(26)+1*s(3)+3
  Such that:aux(4) =< V
it(26) =< aux(4)
s(3) =< it(26)*aux(4)

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

* Chain [25,[26,27],33]: 4*it(26)+1*s(3)+2
  Such that:aux(5) =< V
it(26) =< aux(5)
s(3) =< it(26)*aux(5)

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

* Chain [25,[26,27],32]: 6*it(26)+1*s(3)+3
  Such that:aux(9) =< V
it(26) =< aux(9)
s(3) =< it(26)*aux(9)

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

* Chain [25,[26,27],29,33]: 4*it(26)+1*s(3)+4
  Such that:aux(7) =< V
it(26) =< aux(7)
s(3) =< it(26)*aux(7)

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

* Chain [25,[26,27],28,33]: 5*it(26)+1*s(3)+4
  Such that:aux(8) =< V
it(26) =< aux(8)
s(3) =< it(26)*aux(8)

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

* Chain [25,34]: 3
  with precondition: [Out=3,V>=1,V1>=V+1] 

* Chain [25,33]: 2
  with precondition: [Out=2,V>=1,V1>=V+1] 

* Chain [25,32]: 2*s(4)+3
  Such that:aux(10) =< V
s(4) =< aux(10)

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

* Chain [25,29,33]: 4
  with precondition: [Out=4,V>=1,V1>=V+1] 

* Chain [25,28,33]: 1*s(6)+4
  Such that:s(6) =< V

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

* Chain [24,[26,27],34]: 5*it(26)+1*s(3)+3
  Such that:aux(11) =< V
it(26) =< aux(11)
s(3) =< it(26)*aux(11)

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

* Chain [24,[26,27],33]: 5*it(26)+1*s(3)+2
  Such that:aux(12) =< V
it(26) =< aux(12)
s(3) =< it(26)*aux(12)

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

* Chain [24,[26,27],32]: 7*it(26)+1*s(3)+3
  Such that:aux(13) =< V
it(26) =< aux(13)
s(3) =< it(26)*aux(13)

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

* Chain [24,[26,27],29,33]: 5*it(26)+1*s(3)+4
  Such that:aux(14) =< V
it(26) =< aux(14)
s(3) =< it(26)*aux(14)

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

* Chain [24,[26,27],28,33]: 6*it(26)+1*s(3)+4
  Such that:aux(15) =< V
it(26) =< aux(15)
s(3) =< it(26)*aux(15)

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

* Chain [24,34]: 1*s(8)+3
  Such that:s(8) =< V

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

* Chain [24,33]: 1*s(8)+2
  Such that:s(8) =< V

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

* Chain [24,32]: 3*s(4)+3
  Such that:aux(16) =< V
s(4) =< aux(16)

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

* Chain [24,29,33]: 1*s(8)+4
  Such that:s(8) =< V

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

* Chain [24,28,33]: 2*s(6)+4
  Such that:aux(17) =< V
s(6) =< aux(17)

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


#### Cost of chains of fun3(V1,V,Out):
* Chain [[35],36]: 1*it(35)+0
  Such that:it(35) =< Out

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

* Chain [36]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of start(V1,V,V4):
* Chain [40]: 67*s(69)+24*s(71)+5*s(72)+10*s(79)+4
  Such that:aux(21) =< V1
aux(22) =< V
s(71) =< aux(21)
s(69) =< aux(22)
s(72) =< s(71)*aux(21)
s(79) =< s(69)*aux(22)

  with precondition: [V1>=0] 

* Chain [39]: 65*s(82)+24*s(84)+5*s(85)+10*s(92)+5
  Such that:aux(23) =< V
aux(24) =< V4
s(84) =< aux(23)
s(82) =< aux(24)
s(85) =< s(84)*aux(23)
s(92) =< s(82)*aux(24)

  with precondition: [V1=1,V>=0,V4>=0] 

* Chain [38]: 27*s(93)+5*s(96)+3
  Such that:aux(27) =< V4
s(93) =< aux(27)
s(96) =< s(93)*aux(27)

  with precondition: [V1=2,V>=0,V4>=0] 

* Chain [37]: 2
  with precondition: [V=0,V1>=1] 


Closed-form bounds of start(V1,V,V4): 
-------------------------------------
* Chain [40] with precondition: [V1>=0] 
    - Upper bound: 24*V1+4+5*V1*V1+nat(V)*67+nat(V)*10*nat(V) 
    - Complexity: n^2 
* Chain [39] with precondition: [V1=1,V>=0,V4>=0] 
    - Upper bound: 24*V+5+5*V*V+65*V4+10*V4*V4 
    - Complexity: n^2 
* Chain [38] with precondition: [V1=2,V>=0,V4>=0] 
    - Upper bound: 27*V4+3+5*V4*V4 
    - Complexity: n^2 
* Chain [37] with precondition: [V=0,V1>=1] 
    - Upper bound: 2 
    - Complexity: constant 

### Maximum cost of start(V1,V,V4): max([nat(V4)*27+1+nat(V4)*5*nat(V4),nat(V)*24+2+nat(V)*5*nat(V)+max([nat(V4)*65+1+nat(V4)*10*nat(V4),5*V1*V1+24*V1+nat(V)*43+nat(V)*5*nat(V)])])+2 
Asymptotic class: n^2 
* Total analysis performed in 565 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(s(z0), z1) -> cond2(gr(s(z0), z1), s(z0), z1)
   cond2(true, z0, z1) -> cond1(z1, z1)
   cond2(false, z0, z1) -> cond1(p(z0), z1)
   gr(0, z0) -> false
   gr(s(z0), 0) -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   neq(0, 0) -> false
   neq(0, s(z0)) -> true
   neq(s(z0), 0) -> true
   neq(s(z0), s(z1)) -> neq(z0, z1)
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   COND1(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1))
   COND2(true, z0, z1) -> c1(COND1(z1, z1))
   COND2(false, z0, z1) -> c2(COND1(p(z0), z1), P(z0))
   GR(0, z0) -> c3
   GR(s(z0), 0) -> c4
   GR(s(z0), s(z1)) -> c5(GR(z0, z1))
   NEQ(0, 0) -> c6
   NEQ(0, s(z0)) -> c7
   NEQ(s(z0), 0) -> c8
   NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1))
   P(0) -> c10
   P(s(z0)) -> c11
S tuples:
   COND1(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1))
   COND2(true, z0, z1) -> c1(COND1(z1, z1))
   COND2(false, z0, z1) -> c2(COND1(p(z0), z1), P(z0))
   GR(0, z0) -> c3
   GR(s(z0), 0) -> c4
   GR(s(z0), s(z1)) -> c5(GR(z0, z1))
   NEQ(0, 0) -> c6
   NEQ(0, s(z0)) -> c7
   NEQ(s(z0), 0) -> c8
   NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1))
   P(0) -> c10
   P(s(z0)) -> c11
K tuples:none
Defined Rule Symbols:   cond1_2, cond2_3, gr_2, neq_2, p_1

Defined Pair Symbols:   COND1_2, COND2_3, GR_2, NEQ_2, P_1

Compound Symbols:   c_2, c1_1, c2_2, c3, c4, c5_1, c6, c7, c8, c9_1, c10, c11


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

(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(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1))
   COND2(true, z0, z1) -> c1(COND1(z1, z1))
   COND2(false, z0, z1) -> c2(COND1(p(z0), z1), P(z0))
   GR(0, z0) -> c3
   GR(s(z0), 0) -> c4
   GR(s(z0), s(z1)) -> c5(GR(z0, z1))
   NEQ(0, 0) -> c6
   NEQ(0, s(z0)) -> c7
   NEQ(s(z0), 0) -> c8
   NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1))
   P(0) -> c10
   P(s(z0)) -> c11

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

   cond1(s(z0), z1) -> cond2(gr(s(z0), z1), s(z0), z1)
   cond2(true, z0, z1) -> cond1(z1, z1)
   cond2(false, z0, z1) -> cond1(p(z0), z1)
   gr(0, z0) -> false
   gr(s(z0), 0) -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   neq(0, 0) -> false
   neq(0, s(z0)) -> true
   neq(s(z0), 0) -> true
   neq(s(z0), s(z1)) -> neq(z0, z1)
   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(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1))
   COND2(true, z0, z1) -> c1(COND1(z1, z1))
   COND2(false, z0, z1) -> c2(COND1(p(z0), z1), P(z0))
   GR(0', z0) -> c3
   GR(s(z0), 0') -> c4
   GR(s(z0), s(z1)) -> c5(GR(z0, z1))
   NEQ(0', 0') -> c6
   NEQ(0', s(z0)) -> c7
   NEQ(s(z0), 0') -> c8
   NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1))
   P(0') -> c10
   P(s(z0)) -> c11

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

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

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

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

(28)
Obligation:
Innermost TRS:
Rules:
COND1(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1))
COND2(true, z0, z1) -> c1(COND1(z1, z1))
COND2(false, z0, z1) -> c2(COND1(p(z0), z1), P(z0))
GR(0', z0) -> c3
GR(s(z0), 0') -> c4
GR(s(z0), s(z1)) -> c5(GR(z0, z1))
NEQ(0', 0') -> c6
NEQ(0', s(z0)) -> c7
NEQ(s(z0), 0') -> c8
NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1))
P(0') -> c10
P(s(z0)) -> c11
cond1(s(z0), z1) -> cond2(gr(s(z0), z1), s(z0), z1)
cond2(true, z0, z1) -> cond1(z1, z1)
cond2(false, z0, z1) -> cond1(p(z0), z1)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
neq(0', 0') -> false
neq(0', s(z0)) -> true
neq(s(z0), 0') -> true
neq(s(z0), s(z1)) -> neq(z0, z1)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: s:0' -> s:0' -> c
s :: s:0' -> s:0'
c :: c1:c2 -> c3:c4:c5 -> c
COND2 :: true:false -> s:0' -> s:0' -> c1:c2
gr :: s:0' -> s:0' -> true:false
GR :: s:0' -> s:0' -> c3:c4:c5
true :: true:false
c1 :: c -> c1:c2
false :: true:false
c2 :: c -> c10:c11 -> c1:c2
p :: s:0' -> s:0'
P :: s:0' -> c10:c11
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
NEQ :: s:0' -> s:0' -> c6:c7:c8:c9
c6 :: c6:c7:c8:c9
c7 :: c6:c7:c8:c9
c8 :: c6:c7:c8:c9
c9 :: c6:c7:c8:c9 -> c6:c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11
cond1 :: s:0' -> s:0' -> cond1:cond2
cond2 :: true:false -> s:0' -> s:0' -> cond1:cond2
neq :: s:0' -> s:0' -> true:false
hole_c1_12 :: c
hole_s:0'2_12 :: s:0'
hole_c1:c23_12 :: c1:c2
hole_c3:c4:c54_12 :: c3:c4:c5
hole_true:false5_12 :: true:false
hole_c10:c116_12 :: c10:c11
hole_c6:c7:c8:c97_12 :: c6:c7:c8:c9
hole_cond1:cond28_12 :: cond1:cond2
gen_s:0'9_12 :: Nat -> s:0'
gen_c3:c4:c510_12 :: Nat -> c3:c4:c5
gen_c6:c7:c8:c911_12 :: Nat -> c6:c7:c8:c9

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

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

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

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

(30)
Obligation:
Innermost TRS:
Rules:
COND1(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1))
COND2(true, z0, z1) -> c1(COND1(z1, z1))
COND2(false, z0, z1) -> c2(COND1(p(z0), z1), P(z0))
GR(0', z0) -> c3
GR(s(z0), 0') -> c4
GR(s(z0), s(z1)) -> c5(GR(z0, z1))
NEQ(0', 0') -> c6
NEQ(0', s(z0)) -> c7
NEQ(s(z0), 0') -> c8
NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1))
P(0') -> c10
P(s(z0)) -> c11
cond1(s(z0), z1) -> cond2(gr(s(z0), z1), s(z0), z1)
cond2(true, z0, z1) -> cond1(z1, z1)
cond2(false, z0, z1) -> cond1(p(z0), z1)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
neq(0', 0') -> false
neq(0', s(z0)) -> true
neq(s(z0), 0') -> true
neq(s(z0), s(z1)) -> neq(z0, z1)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: s:0' -> s:0' -> c
s :: s:0' -> s:0'
c :: c1:c2 -> c3:c4:c5 -> c
COND2 :: true:false -> s:0' -> s:0' -> c1:c2
gr :: s:0' -> s:0' -> true:false
GR :: s:0' -> s:0' -> c3:c4:c5
true :: true:false
c1 :: c -> c1:c2
false :: true:false
c2 :: c -> c10:c11 -> c1:c2
p :: s:0' -> s:0'
P :: s:0' -> c10:c11
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
NEQ :: s:0' -> s:0' -> c6:c7:c8:c9
c6 :: c6:c7:c8:c9
c7 :: c6:c7:c8:c9
c8 :: c6:c7:c8:c9
c9 :: c6:c7:c8:c9 -> c6:c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11
cond1 :: s:0' -> s:0' -> cond1:cond2
cond2 :: true:false -> s:0' -> s:0' -> cond1:cond2
neq :: s:0' -> s:0' -> true:false
hole_c1_12 :: c
hole_s:0'2_12 :: s:0'
hole_c1:c23_12 :: c1:c2
hole_c3:c4:c54_12 :: c3:c4:c5
hole_true:false5_12 :: true:false
hole_c10:c116_12 :: c10:c11
hole_c6:c7:c8:c97_12 :: c6:c7:c8:c9
hole_cond1:cond28_12 :: cond1:cond2
gen_s:0'9_12 :: Nat -> s:0'
gen_c3:c4:c510_12 :: Nat -> c3:c4:c5
gen_c6:c7:c8:c911_12 :: Nat -> c6:c7:c8:c9


Generator Equations:
gen_s:0'9_12(0) <=> 0'
gen_s:0'9_12(+(x, 1)) <=> s(gen_s:0'9_12(x))
gen_c3:c4:c510_12(0) <=> c3
gen_c3:c4:c510_12(+(x, 1)) <=> c5(gen_c3:c4:c510_12(x))
gen_c6:c7:c8:c911_12(0) <=> c6
gen_c6:c7:c8:c911_12(+(x, 1)) <=> c9(gen_c6:c7:c8:c911_12(x))


The following defined symbols remain to be analysed:
gr, COND1, GR, NEQ, cond1, neq

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

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gr(gen_s:0'9_12(n13_12), gen_s:0'9_12(n13_12)) -> false, rt in Omega(0)

Induction Base:
gr(gen_s:0'9_12(0), gen_s:0'9_12(0)) ->_R^Omega(0)
false

Induction Step:
gr(gen_s:0'9_12(+(n13_12, 1)), gen_s:0'9_12(+(n13_12, 1))) ->_R^Omega(0)
gr(gen_s:0'9_12(n13_12), gen_s:0'9_12(n13_12)) ->_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(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1))
COND2(true, z0, z1) -> c1(COND1(z1, z1))
COND2(false, z0, z1) -> c2(COND1(p(z0), z1), P(z0))
GR(0', z0) -> c3
GR(s(z0), 0') -> c4
GR(s(z0), s(z1)) -> c5(GR(z0, z1))
NEQ(0', 0') -> c6
NEQ(0', s(z0)) -> c7
NEQ(s(z0), 0') -> c8
NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1))
P(0') -> c10
P(s(z0)) -> c11
cond1(s(z0), z1) -> cond2(gr(s(z0), z1), s(z0), z1)
cond2(true, z0, z1) -> cond1(z1, z1)
cond2(false, z0, z1) -> cond1(p(z0), z1)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
neq(0', 0') -> false
neq(0', s(z0)) -> true
neq(s(z0), 0') -> true
neq(s(z0), s(z1)) -> neq(z0, z1)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: s:0' -> s:0' -> c
s :: s:0' -> s:0'
c :: c1:c2 -> c3:c4:c5 -> c
COND2 :: true:false -> s:0' -> s:0' -> c1:c2
gr :: s:0' -> s:0' -> true:false
GR :: s:0' -> s:0' -> c3:c4:c5
true :: true:false
c1 :: c -> c1:c2
false :: true:false
c2 :: c -> c10:c11 -> c1:c2
p :: s:0' -> s:0'
P :: s:0' -> c10:c11
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
NEQ :: s:0' -> s:0' -> c6:c7:c8:c9
c6 :: c6:c7:c8:c9
c7 :: c6:c7:c8:c9
c8 :: c6:c7:c8:c9
c9 :: c6:c7:c8:c9 -> c6:c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11
cond1 :: s:0' -> s:0' -> cond1:cond2
cond2 :: true:false -> s:0' -> s:0' -> cond1:cond2
neq :: s:0' -> s:0' -> true:false
hole_c1_12 :: c
hole_s:0'2_12 :: s:0'
hole_c1:c23_12 :: c1:c2
hole_c3:c4:c54_12 :: c3:c4:c5
hole_true:false5_12 :: true:false
hole_c10:c116_12 :: c10:c11
hole_c6:c7:c8:c97_12 :: c6:c7:c8:c9
hole_cond1:cond28_12 :: cond1:cond2
gen_s:0'9_12 :: Nat -> s:0'
gen_c3:c4:c510_12 :: Nat -> c3:c4:c5
gen_c6:c7:c8:c911_12 :: Nat -> c6:c7:c8:c9


Lemmas:
gr(gen_s:0'9_12(n13_12), gen_s:0'9_12(n13_12)) -> false, rt in Omega(0)


Generator Equations:
gen_s:0'9_12(0) <=> 0'
gen_s:0'9_12(+(x, 1)) <=> s(gen_s:0'9_12(x))
gen_c3:c4:c510_12(0) <=> c3
gen_c3:c4:c510_12(+(x, 1)) <=> c5(gen_c3:c4:c510_12(x))
gen_c6:c7:c8:c911_12(0) <=> c6
gen_c6:c7:c8:c911_12(+(x, 1)) <=> c9(gen_c6:c7:c8:c911_12(x))


The following defined symbols remain to be analysed:
GR, COND1, NEQ, cond1, neq

They will be analysed ascendingly in the following order:
GR < COND1

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

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
GR(gen_s:0'9_12(n364_12), gen_s:0'9_12(n364_12)) -> gen_c3:c4:c510_12(n364_12), rt in Omega(1 + n364_12)

Induction Base:
GR(gen_s:0'9_12(0), gen_s:0'9_12(0)) ->_R^Omega(1)
c3

Induction Step:
GR(gen_s:0'9_12(+(n364_12, 1)), gen_s:0'9_12(+(n364_12, 1))) ->_R^Omega(1)
c5(GR(gen_s:0'9_12(n364_12), gen_s:0'9_12(n364_12))) ->_IH
c5(gen_c3:c4:c510_12(c365_12))

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(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1))
COND2(true, z0, z1) -> c1(COND1(z1, z1))
COND2(false, z0, z1) -> c2(COND1(p(z0), z1), P(z0))
GR(0', z0) -> c3
GR(s(z0), 0') -> c4
GR(s(z0), s(z1)) -> c5(GR(z0, z1))
NEQ(0', 0') -> c6
NEQ(0', s(z0)) -> c7
NEQ(s(z0), 0') -> c8
NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1))
P(0') -> c10
P(s(z0)) -> c11
cond1(s(z0), z1) -> cond2(gr(s(z0), z1), s(z0), z1)
cond2(true, z0, z1) -> cond1(z1, z1)
cond2(false, z0, z1) -> cond1(p(z0), z1)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
neq(0', 0') -> false
neq(0', s(z0)) -> true
neq(s(z0), 0') -> true
neq(s(z0), s(z1)) -> neq(z0, z1)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: s:0' -> s:0' -> c
s :: s:0' -> s:0'
c :: c1:c2 -> c3:c4:c5 -> c
COND2 :: true:false -> s:0' -> s:0' -> c1:c2
gr :: s:0' -> s:0' -> true:false
GR :: s:0' -> s:0' -> c3:c4:c5
true :: true:false
c1 :: c -> c1:c2
false :: true:false
c2 :: c -> c10:c11 -> c1:c2
p :: s:0' -> s:0'
P :: s:0' -> c10:c11
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
NEQ :: s:0' -> s:0' -> c6:c7:c8:c9
c6 :: c6:c7:c8:c9
c7 :: c6:c7:c8:c9
c8 :: c6:c7:c8:c9
c9 :: c6:c7:c8:c9 -> c6:c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11
cond1 :: s:0' -> s:0' -> cond1:cond2
cond2 :: true:false -> s:0' -> s:0' -> cond1:cond2
neq :: s:0' -> s:0' -> true:false
hole_c1_12 :: c
hole_s:0'2_12 :: s:0'
hole_c1:c23_12 :: c1:c2
hole_c3:c4:c54_12 :: c3:c4:c5
hole_true:false5_12 :: true:false
hole_c10:c116_12 :: c10:c11
hole_c6:c7:c8:c97_12 :: c6:c7:c8:c9
hole_cond1:cond28_12 :: cond1:cond2
gen_s:0'9_12 :: Nat -> s:0'
gen_c3:c4:c510_12 :: Nat -> c3:c4:c5
gen_c6:c7:c8:c911_12 :: Nat -> c6:c7:c8:c9


Lemmas:
gr(gen_s:0'9_12(n13_12), gen_s:0'9_12(n13_12)) -> false, rt in Omega(0)


Generator Equations:
gen_s:0'9_12(0) <=> 0'
gen_s:0'9_12(+(x, 1)) <=> s(gen_s:0'9_12(x))
gen_c3:c4:c510_12(0) <=> c3
gen_c3:c4:c510_12(+(x, 1)) <=> c5(gen_c3:c4:c510_12(x))
gen_c6:c7:c8:c911_12(0) <=> c6
gen_c6:c7:c8:c911_12(+(x, 1)) <=> c9(gen_c6:c7:c8:c911_12(x))


The following defined symbols remain to be analysed:
GR, COND1, NEQ, cond1, neq

They will be analysed ascendingly in the following order:
GR < COND1

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

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

(37)
BOUNDS(n^1, INF)

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

(38)
Obligation:
Innermost TRS:
Rules:
COND1(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1))
COND2(true, z0, z1) -> c1(COND1(z1, z1))
COND2(false, z0, z1) -> c2(COND1(p(z0), z1), P(z0))
GR(0', z0) -> c3
GR(s(z0), 0') -> c4
GR(s(z0), s(z1)) -> c5(GR(z0, z1))
NEQ(0', 0') -> c6
NEQ(0', s(z0)) -> c7
NEQ(s(z0), 0') -> c8
NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1))
P(0') -> c10
P(s(z0)) -> c11
cond1(s(z0), z1) -> cond2(gr(s(z0), z1), s(z0), z1)
cond2(true, z0, z1) -> cond1(z1, z1)
cond2(false, z0, z1) -> cond1(p(z0), z1)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
neq(0', 0') -> false
neq(0', s(z0)) -> true
neq(s(z0), 0') -> true
neq(s(z0), s(z1)) -> neq(z0, z1)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: s:0' -> s:0' -> c
s :: s:0' -> s:0'
c :: c1:c2 -> c3:c4:c5 -> c
COND2 :: true:false -> s:0' -> s:0' -> c1:c2
gr :: s:0' -> s:0' -> true:false
GR :: s:0' -> s:0' -> c3:c4:c5
true :: true:false
c1 :: c -> c1:c2
false :: true:false
c2 :: c -> c10:c11 -> c1:c2
p :: s:0' -> s:0'
P :: s:0' -> c10:c11
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
NEQ :: s:0' -> s:0' -> c6:c7:c8:c9
c6 :: c6:c7:c8:c9
c7 :: c6:c7:c8:c9
c8 :: c6:c7:c8:c9
c9 :: c6:c7:c8:c9 -> c6:c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11
cond1 :: s:0' -> s:0' -> cond1:cond2
cond2 :: true:false -> s:0' -> s:0' -> cond1:cond2
neq :: s:0' -> s:0' -> true:false
hole_c1_12 :: c
hole_s:0'2_12 :: s:0'
hole_c1:c23_12 :: c1:c2
hole_c3:c4:c54_12 :: c3:c4:c5
hole_true:false5_12 :: true:false
hole_c10:c116_12 :: c10:c11
hole_c6:c7:c8:c97_12 :: c6:c7:c8:c9
hole_cond1:cond28_12 :: cond1:cond2
gen_s:0'9_12 :: Nat -> s:0'
gen_c3:c4:c510_12 :: Nat -> c3:c4:c5
gen_c6:c7:c8:c911_12 :: Nat -> c6:c7:c8:c9


Lemmas:
gr(gen_s:0'9_12(n13_12), gen_s:0'9_12(n13_12)) -> false, rt in Omega(0)
GR(gen_s:0'9_12(n364_12), gen_s:0'9_12(n364_12)) -> gen_c3:c4:c510_12(n364_12), rt in Omega(1 + n364_12)


Generator Equations:
gen_s:0'9_12(0) <=> 0'
gen_s:0'9_12(+(x, 1)) <=> s(gen_s:0'9_12(x))
gen_c3:c4:c510_12(0) <=> c3
gen_c3:c4:c510_12(+(x, 1)) <=> c5(gen_c3:c4:c510_12(x))
gen_c6:c7:c8:c911_12(0) <=> c6
gen_c6:c7:c8:c911_12(+(x, 1)) <=> c9(gen_c6:c7:c8:c911_12(x))


The following defined symbols remain to be analysed:
COND1, NEQ, cond1, neq
----------------------------------------

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
NEQ(gen_s:0'9_12(n9654_12), gen_s:0'9_12(n9654_12)) -> gen_c6:c7:c8:c911_12(n9654_12), rt in Omega(1 + n9654_12)

Induction Base:
NEQ(gen_s:0'9_12(0), gen_s:0'9_12(0)) ->_R^Omega(1)
c6

Induction Step:
NEQ(gen_s:0'9_12(+(n9654_12, 1)), gen_s:0'9_12(+(n9654_12, 1))) ->_R^Omega(1)
c9(NEQ(gen_s:0'9_12(n9654_12), gen_s:0'9_12(n9654_12))) ->_IH
c9(gen_c6:c7:c8:c911_12(c9655_12))

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

(40)
Obligation:
Innermost TRS:
Rules:
COND1(s(z0), z1) -> c(COND2(gr(s(z0), z1), s(z0), z1), GR(s(z0), z1))
COND2(true, z0, z1) -> c1(COND1(z1, z1))
COND2(false, z0, z1) -> c2(COND1(p(z0), z1), P(z0))
GR(0', z0) -> c3
GR(s(z0), 0') -> c4
GR(s(z0), s(z1)) -> c5(GR(z0, z1))
NEQ(0', 0') -> c6
NEQ(0', s(z0)) -> c7
NEQ(s(z0), 0') -> c8
NEQ(s(z0), s(z1)) -> c9(NEQ(z0, z1))
P(0') -> c10
P(s(z0)) -> c11
cond1(s(z0), z1) -> cond2(gr(s(z0), z1), s(z0), z1)
cond2(true, z0, z1) -> cond1(z1, z1)
cond2(false, z0, z1) -> cond1(p(z0), z1)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
neq(0', 0') -> false
neq(0', s(z0)) -> true
neq(s(z0), 0') -> true
neq(s(z0), s(z1)) -> neq(z0, z1)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: s:0' -> s:0' -> c
s :: s:0' -> s:0'
c :: c1:c2 -> c3:c4:c5 -> c
COND2 :: true:false -> s:0' -> s:0' -> c1:c2
gr :: s:0' -> s:0' -> true:false
GR :: s:0' -> s:0' -> c3:c4:c5
true :: true:false
c1 :: c -> c1:c2
false :: true:false
c2 :: c -> c10:c11 -> c1:c2
p :: s:0' -> s:0'
P :: s:0' -> c10:c11
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
NEQ :: s:0' -> s:0' -> c6:c7:c8:c9
c6 :: c6:c7:c8:c9
c7 :: c6:c7:c8:c9
c8 :: c6:c7:c8:c9
c9 :: c6:c7:c8:c9 -> c6:c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11
cond1 :: s:0' -> s:0' -> cond1:cond2
cond2 :: true:false -> s:0' -> s:0' -> cond1:cond2
neq :: s:0' -> s:0' -> true:false
hole_c1_12 :: c
hole_s:0'2_12 :: s:0'
hole_c1:c23_12 :: c1:c2
hole_c3:c4:c54_12 :: c3:c4:c5
hole_true:false5_12 :: true:false
hole_c10:c116_12 :: c10:c11
hole_c6:c7:c8:c97_12 :: c6:c7:c8:c9
hole_cond1:cond28_12 :: cond1:cond2
gen_s:0'9_12 :: Nat -> s:0'
gen_c3:c4:c510_12 :: Nat -> c3:c4:c5
gen_c6:c7:c8:c911_12 :: Nat -> c6:c7:c8:c9


Lemmas:
gr(gen_s:0'9_12(n13_12), gen_s:0'9_12(n13_12)) -> false, rt in Omega(0)
GR(gen_s:0'9_12(n364_12), gen_s:0'9_12(n364_12)) -> gen_c3:c4:c510_12(n364_12), rt in Omega(1 + n364_12)
NEQ(gen_s:0'9_12(n9654_12), gen_s:0'9_12(n9654_12)) -> gen_c6:c7:c8:c911_12(n9654_12), rt in Omega(1 + n9654_12)


Generator Equations:
gen_s:0'9_12(0) <=> 0'
gen_s:0'9_12(+(x, 1)) <=> s(gen_s:0'9_12(x))
gen_c3:c4:c510_12(0) <=> c3
gen_c3:c4:c510_12(+(x, 1)) <=> c5(gen_c3:c4:c510_12(x))
gen_c6:c7:c8:c911_12(0) <=> c6
gen_c6:c7:c8:c911_12(+(x, 1)) <=> c9(gen_c6:c7:c8:c911_12(x))


The following defined symbols remain to be analysed:
cond1, neq
----------------------------------------

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
neq(gen_s:0'9_12(n33868_12), gen_s:0'9_12(n33868_12)) -> false, rt in Omega(0)

Induction Base:
neq(gen_s:0'9_12(0), gen_s:0'9_12(0)) ->_R^Omega(0)
false

Induction Step:
neq(gen_s:0'9_12(+(n33868_12, 1)), gen_s:0'9_12(+(n33868_12, 1))) ->_R^Omega(0)
neq(gen_s:0'9_12(n33868_12), gen_s:0'9_12(n33868_12)) ->_IH
false

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

(42)
BOUNDS(1, INF)