WORST_CASE(Omega(n^1),?)
proof of input_zbtxVQ6MIH.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, INF).

(0) CpxTRS
(1) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CdtProblem
(3) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 12 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 339 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 28 ms]
        (18) typed CpxTrs


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

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


The TRS R consists of the following rules:

   p(0) -> 0
   p(s(X)) -> X
   leq(0, Y) -> true
   leq(s(X), 0) -> false
   leq(s(X), s(Y)) -> leq(X, Y)
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y)))

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

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

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> 0
   p(s(z0)) -> z0
   leq(0, z0) -> true
   leq(s(z0), 0) -> false
   leq(s(z0), s(z1)) -> leq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   diff(z0, z1) -> if(leq(z0, z1), 0, s(diff(p(z0), z1)))
Tuples:
   P(0) -> c
   P(s(z0)) -> c1
   LEQ(0, z0) -> c2
   LEQ(s(z0), 0) -> c3
   LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
   IF(true, z0, z1) -> c5
   IF(false, z0, z1) -> c6
   DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), LEQ(z0, z1))
   DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0))
S tuples:
   P(0) -> c
   P(s(z0)) -> c1
   LEQ(0, z0) -> c2
   LEQ(s(z0), 0) -> c3
   LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
   IF(true, z0, z1) -> c5
   IF(false, z0, z1) -> c6
   DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), LEQ(z0, z1))
   DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0))
K tuples:none
Defined Rule Symbols:   p_1, leq_2, if_3, diff_2

Defined Pair Symbols:   P_1, LEQ_2, IF_3, DIFF_2

Compound Symbols:   c, c1, c2, c3, c4_1, c5, c6, c7_2, c8_3


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

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

(4)
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:

   P(0) -> c
   P(s(z0)) -> c1
   LEQ(0, z0) -> c2
   LEQ(s(z0), 0) -> c3
   LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
   IF(true, z0, z1) -> c5
   IF(false, z0, z1) -> c6
   DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), LEQ(z0, z1))
   DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0))

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

   p(0) -> 0
   p(s(z0)) -> z0
   leq(0, z0) -> true
   leq(s(z0), 0) -> false
   leq(s(z0), s(z1)) -> leq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   diff(z0, z1) -> if(leq(z0, z1), 0, s(diff(p(z0), z1)))

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

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

(6)
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:

   P(0') -> c
   P(s(z0)) -> c1
   LEQ(0', z0) -> c2
   LEQ(s(z0), 0') -> c3
   LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
   IF(true, z0, z1) -> c5
   IF(false, z0, z1) -> c6
   DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), LEQ(z0, z1))
   DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0))

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

   p(0') -> 0'
   p(s(z0)) -> z0
   leq(0', z0) -> true
   leq(s(z0), 0') -> false
   leq(s(z0), s(z1)) -> leq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   diff(z0, z1) -> if(leq(z0, z1), 0', s(diff(p(z0), z1)))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
P(0') -> c
P(s(z0)) -> c1
LEQ(0', z0) -> c2
LEQ(s(z0), 0') -> c3
LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
IF(true, z0, z1) -> c5
IF(false, z0, z1) -> c6
DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), LEQ(z0, z1))
DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0))
p(0') -> 0'
p(s(z0)) -> z0
leq(0', z0) -> true
leq(s(z0), 0') -> false
leq(s(z0), s(z1)) -> leq(z0, z1)
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1
diff(z0, z1) -> if(leq(z0, z1), 0', s(diff(p(z0), z1)))

Types:
P :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
LEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
IF :: true:false -> 0':s -> 0':s -> c5:c6
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c5:c6
DIFF :: 0':s -> 0':s -> c7:c8
c7 :: c5:c6 -> c2:c3:c4 -> c7:c8
leq :: 0':s -> 0':s -> true:false
diff :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
c8 :: c5:c6 -> c7:c8 -> c:c1 -> c7:c8
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_9 :: c:c1
hole_0':s2_9 :: 0':s
hole_c2:c3:c43_9 :: c2:c3:c4
hole_c5:c64_9 :: c5:c6
hole_true:false5_9 :: true:false
hole_c7:c86_9 :: c7:c8
gen_0':s7_9 :: Nat -> 0':s
gen_c2:c3:c48_9 :: Nat -> c2:c3:c4
gen_c7:c89_9 :: Nat -> c7:c8

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
LEQ, DIFF, leq, diff

They will be analysed ascendingly in the following order:
LEQ < DIFF
leq < DIFF
diff < DIFF
leq < diff

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

(10)
Obligation:
Innermost TRS:
Rules:
P(0') -> c
P(s(z0)) -> c1
LEQ(0', z0) -> c2
LEQ(s(z0), 0') -> c3
LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
IF(true, z0, z1) -> c5
IF(false, z0, z1) -> c6
DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), LEQ(z0, z1))
DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0))
p(0') -> 0'
p(s(z0)) -> z0
leq(0', z0) -> true
leq(s(z0), 0') -> false
leq(s(z0), s(z1)) -> leq(z0, z1)
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1
diff(z0, z1) -> if(leq(z0, z1), 0', s(diff(p(z0), z1)))

Types:
P :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
LEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
IF :: true:false -> 0':s -> 0':s -> c5:c6
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c5:c6
DIFF :: 0':s -> 0':s -> c7:c8
c7 :: c5:c6 -> c2:c3:c4 -> c7:c8
leq :: 0':s -> 0':s -> true:false
diff :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
c8 :: c5:c6 -> c7:c8 -> c:c1 -> c7:c8
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_9 :: c:c1
hole_0':s2_9 :: 0':s
hole_c2:c3:c43_9 :: c2:c3:c4
hole_c5:c64_9 :: c5:c6
hole_true:false5_9 :: true:false
hole_c7:c86_9 :: c7:c8
gen_0':s7_9 :: Nat -> 0':s
gen_c2:c3:c48_9 :: Nat -> c2:c3:c4
gen_c7:c89_9 :: Nat -> c7:c8


Generator Equations:
gen_0':s7_9(0) <=> 0'
gen_0':s7_9(+(x, 1)) <=> s(gen_0':s7_9(x))
gen_c2:c3:c48_9(0) <=> c2
gen_c2:c3:c48_9(+(x, 1)) <=> c4(gen_c2:c3:c48_9(x))
gen_c7:c89_9(0) <=> c7(c5, c2)
gen_c7:c89_9(+(x, 1)) <=> c8(c5, gen_c7:c89_9(x), c)


The following defined symbols remain to be analysed:
LEQ, DIFF, leq, diff

They will be analysed ascendingly in the following order:
LEQ < DIFF
leq < DIFF
diff < DIFF
leq < diff

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LEQ(gen_0':s7_9(n11_9), gen_0':s7_9(n11_9)) -> gen_c2:c3:c48_9(n11_9), rt in Omega(1 + n11_9)

Induction Base:
LEQ(gen_0':s7_9(0), gen_0':s7_9(0)) ->_R^Omega(1)
c2

Induction Step:
LEQ(gen_0':s7_9(+(n11_9, 1)), gen_0':s7_9(+(n11_9, 1))) ->_R^Omega(1)
c4(LEQ(gen_0':s7_9(n11_9), gen_0':s7_9(n11_9))) ->_IH
c4(gen_c2:c3:c48_9(c12_9))

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

(12)
Complex Obligation (BEST)

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

(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
P(0') -> c
P(s(z0)) -> c1
LEQ(0', z0) -> c2
LEQ(s(z0), 0') -> c3
LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
IF(true, z0, z1) -> c5
IF(false, z0, z1) -> c6
DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), LEQ(z0, z1))
DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0))
p(0') -> 0'
p(s(z0)) -> z0
leq(0', z0) -> true
leq(s(z0), 0') -> false
leq(s(z0), s(z1)) -> leq(z0, z1)
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1
diff(z0, z1) -> if(leq(z0, z1), 0', s(diff(p(z0), z1)))

Types:
P :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
LEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
IF :: true:false -> 0':s -> 0':s -> c5:c6
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c5:c6
DIFF :: 0':s -> 0':s -> c7:c8
c7 :: c5:c6 -> c2:c3:c4 -> c7:c8
leq :: 0':s -> 0':s -> true:false
diff :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
c8 :: c5:c6 -> c7:c8 -> c:c1 -> c7:c8
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_9 :: c:c1
hole_0':s2_9 :: 0':s
hole_c2:c3:c43_9 :: c2:c3:c4
hole_c5:c64_9 :: c5:c6
hole_true:false5_9 :: true:false
hole_c7:c86_9 :: c7:c8
gen_0':s7_9 :: Nat -> 0':s
gen_c2:c3:c48_9 :: Nat -> c2:c3:c4
gen_c7:c89_9 :: Nat -> c7:c8


Generator Equations:
gen_0':s7_9(0) <=> 0'
gen_0':s7_9(+(x, 1)) <=> s(gen_0':s7_9(x))
gen_c2:c3:c48_9(0) <=> c2
gen_c2:c3:c48_9(+(x, 1)) <=> c4(gen_c2:c3:c48_9(x))
gen_c7:c89_9(0) <=> c7(c5, c2)
gen_c7:c89_9(+(x, 1)) <=> c8(c5, gen_c7:c89_9(x), c)


The following defined symbols remain to be analysed:
LEQ, DIFF, leq, diff

They will be analysed ascendingly in the following order:
LEQ < DIFF
leq < DIFF
diff < DIFF
leq < diff

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
P(0') -> c
P(s(z0)) -> c1
LEQ(0', z0) -> c2
LEQ(s(z0), 0') -> c3
LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
IF(true, z0, z1) -> c5
IF(false, z0, z1) -> c6
DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), LEQ(z0, z1))
DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0))
p(0') -> 0'
p(s(z0)) -> z0
leq(0', z0) -> true
leq(s(z0), 0') -> false
leq(s(z0), s(z1)) -> leq(z0, z1)
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1
diff(z0, z1) -> if(leq(z0, z1), 0', s(diff(p(z0), z1)))

Types:
P :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
LEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
IF :: true:false -> 0':s -> 0':s -> c5:c6
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c5:c6
DIFF :: 0':s -> 0':s -> c7:c8
c7 :: c5:c6 -> c2:c3:c4 -> c7:c8
leq :: 0':s -> 0':s -> true:false
diff :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
c8 :: c5:c6 -> c7:c8 -> c:c1 -> c7:c8
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_9 :: c:c1
hole_0':s2_9 :: 0':s
hole_c2:c3:c43_9 :: c2:c3:c4
hole_c5:c64_9 :: c5:c6
hole_true:false5_9 :: true:false
hole_c7:c86_9 :: c7:c8
gen_0':s7_9 :: Nat -> 0':s
gen_c2:c3:c48_9 :: Nat -> c2:c3:c4
gen_c7:c89_9 :: Nat -> c7:c8


Lemmas:
LEQ(gen_0':s7_9(n11_9), gen_0':s7_9(n11_9)) -> gen_c2:c3:c48_9(n11_9), rt in Omega(1 + n11_9)


Generator Equations:
gen_0':s7_9(0) <=> 0'
gen_0':s7_9(+(x, 1)) <=> s(gen_0':s7_9(x))
gen_c2:c3:c48_9(0) <=> c2
gen_c2:c3:c48_9(+(x, 1)) <=> c4(gen_c2:c3:c48_9(x))
gen_c7:c89_9(0) <=> c7(c5, c2)
gen_c7:c89_9(+(x, 1)) <=> c8(c5, gen_c7:c89_9(x), c)


The following defined symbols remain to be analysed:
leq, DIFF, diff

They will be analysed ascendingly in the following order:
leq < DIFF
diff < DIFF
leq < diff

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
leq(gen_0':s7_9(n590_9), gen_0':s7_9(n590_9)) -> true, rt in Omega(0)

Induction Base:
leq(gen_0':s7_9(0), gen_0':s7_9(0)) ->_R^Omega(0)
true

Induction Step:
leq(gen_0':s7_9(+(n590_9, 1)), gen_0':s7_9(+(n590_9, 1))) ->_R^Omega(0)
leq(gen_0':s7_9(n590_9), gen_0':s7_9(n590_9)) ->_IH
true

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

(18)
Obligation:
Innermost TRS:
Rules:
P(0') -> c
P(s(z0)) -> c1
LEQ(0', z0) -> c2
LEQ(s(z0), 0') -> c3
LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1))
IF(true, z0, z1) -> c5
IF(false, z0, z1) -> c6
DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), LEQ(z0, z1))
DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0', s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0))
p(0') -> 0'
p(s(z0)) -> z0
leq(0', z0) -> true
leq(s(z0), 0') -> false
leq(s(z0), s(z1)) -> leq(z0, z1)
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1
diff(z0, z1) -> if(leq(z0, z1), 0', s(diff(p(z0), z1)))

Types:
P :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
LEQ :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
IF :: true:false -> 0':s -> 0':s -> c5:c6
true :: true:false
c5 :: c5:c6
false :: true:false
c6 :: c5:c6
DIFF :: 0':s -> 0':s -> c7:c8
c7 :: c5:c6 -> c2:c3:c4 -> c7:c8
leq :: 0':s -> 0':s -> true:false
diff :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
c8 :: c5:c6 -> c7:c8 -> c:c1 -> c7:c8
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_9 :: c:c1
hole_0':s2_9 :: 0':s
hole_c2:c3:c43_9 :: c2:c3:c4
hole_c5:c64_9 :: c5:c6
hole_true:false5_9 :: true:false
hole_c7:c86_9 :: c7:c8
gen_0':s7_9 :: Nat -> 0':s
gen_c2:c3:c48_9 :: Nat -> c2:c3:c4
gen_c7:c89_9 :: Nat -> c7:c8


Lemmas:
LEQ(gen_0':s7_9(n11_9), gen_0':s7_9(n11_9)) -> gen_c2:c3:c48_9(n11_9), rt in Omega(1 + n11_9)
leq(gen_0':s7_9(n590_9), gen_0':s7_9(n590_9)) -> true, rt in Omega(0)


Generator Equations:
gen_0':s7_9(0) <=> 0'
gen_0':s7_9(+(x, 1)) <=> s(gen_0':s7_9(x))
gen_c2:c3:c48_9(0) <=> c2
gen_c2:c3:c48_9(+(x, 1)) <=> c4(gen_c2:c3:c48_9(x))
gen_c7:c89_9(0) <=> c7(c5, c2)
gen_c7:c89_9(+(x, 1)) <=> c8(c5, gen_c7:c89_9(x), c)


The following defined symbols remain to be analysed:
diff, DIFF

They will be analysed ascendingly in the following order:
diff < DIFF