*** 1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
evenodd(x,0()) -> not(evenodd(x,s(0())))
evenodd(0(),s(0())) -> false()
evenodd(s(x),s(0())) -> evenodd(x,0())
not(false()) -> true()
not(true()) -> false()
Weak DP Rules:
Weak TRS Rules:
Signature:
{evenodd/2,not/1} / {0/0,false/0,s/1,true/0}
Obligation:
Full
basic terms: {evenodd,not}/{0,false,s,true}
Applied Processor:
ToInnermost
Proof:
switch to innermost, as the system is overlay and right linear and does not contain weak rules
*** 1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
evenodd(x,0()) -> not(evenodd(x,s(0())))
evenodd(0(),s(0())) -> false()
evenodd(s(x),s(0())) -> evenodd(x,0())
not(false()) -> true()
not(true()) -> false()
Weak DP Rules:
Weak TRS Rules:
Signature:
{evenodd/2,not/1} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {evenodd,not}/{0,false,s,true}
Applied Processor:
Bounds {initialAutomaton = minimal, enrichment = match}
Proof:
The problem is match-bounded by 3.
The enriched problem is compatible with follwoing automaton.
0_0() -> 2
0_1() -> 5
0_2() -> 8
evenodd_0(2,2) -> 1
evenodd_1(2,4) -> 3
evenodd_1(2,5) -> 1
evenodd_1(2,5) -> 3
evenodd_1(2,5) -> 6
evenodd_2(2,7) -> 6
false_0() -> 2
false_1() -> 1
false_1() -> 3
false_1() -> 6
false_2() -> 1
false_3() -> 1
false_3() -> 3
false_3() -> 6
not_0(2) -> 1
not_1(3) -> 1
not_2(6) -> 1
not_2(6) -> 3
not_2(6) -> 6
s_0(2) -> 2
s_1(5) -> 4
s_2(8) -> 7
true_0() -> 2
true_1() -> 1
true_2() -> 1
true_2() -> 3
true_2() -> 6
true_3() -> 1
true_3() -> 3
true_3() -> 6
*** 1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
evenodd(x,0()) -> not(evenodd(x,s(0())))
evenodd(0(),s(0())) -> false()
evenodd(s(x),s(0())) -> evenodd(x,0())
not(false()) -> true()
not(true()) -> false()
Signature:
{evenodd/2,not/1} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {evenodd,not}/{0,false,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).