LMPO
YES(?,ELEMENTARY)
We consider the following Problem:
Strict Trs:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, mark(f(X)) -> a__f(mark(X))
, mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,ELEMENTARY)
Proof:
The input was oriented with the instance of
Lightweight Multiset Path Order () as induced by the safe mapping
safe(a__f) = {1}, safe(f) = {1}, safe(a__c) = {1}, safe(g) = {1},
safe(d) = {1}, safe(a__h) = {1}, safe(mark) = {}, safe(c) = {1},
safe(h) = {1}
and precedence
a__f > a__c, a__h > a__c, mark > a__f, mark > a__c, mark > a__h .
Following symbols are considered recursive:
{a__f, a__c, a__h, mark}
The recursion depth is 3 .
For your convenience, here are the oriented rules in predicative
notation (possibly applying argument filtering):
Strict DPs: {}
Weak DPs : {}
Strict Trs:
{ a__f(; f(; X)) -> a__c(; f(; g(; f(; X))))
, a__c(; X) -> d(; X)
, a__h(; X) -> a__c(; d(; X))
, mark(f(; X);) -> a__f(; mark(X;))
, mark(c(; X);) -> a__c(; X)
, mark(h(; X);) -> a__h(; mark(X;))
, mark(g(; X);) -> g(; X)
, mark(d(; X);) -> d(; X)
, a__f(; X) -> f(; X)
, a__c(; X) -> c(; X)
, a__h(; X) -> h(; X)}
Weak Trs : {}
Hurray, we answered YES(?,ELEMENTARY)
MPO
YES(?,PRIMREC)
We consider the following Problem:
Strict Trs:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, mark(f(X)) -> a__f(mark(X))
, mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,PRIMREC)
Proof:
The input was oriented with the instance of
'multiset path orders' as induced by the precedence
a__f > f, a__f > a__c, a__f > g, a__c > d, a__c > c, a__h > a__c,
a__h > d, a__h > h, mark > a__f, mark > a__c, mark > a__h .
Hurray, we answered YES(?,PRIMREC)
POP*
YES(?,POLY)
We consider the following Problem:
Strict Trs:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, mark(f(X)) -> a__f(mark(X))
, mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,POLY)
Proof:
The input was oriented with the instance of
Polynomial Path Order () as induced by the safe mapping
safe(a__f) = {1}, safe(f) = {1}, safe(a__c) = {1}, safe(g) = {1},
safe(d) = {1}, safe(a__h) = {1}, safe(mark) = {}, safe(c) = {1},
safe(h) = {1}
and precedence
a__f > a__c, a__h > a__c, mark > a__f, mark > a__c, mark > a__h .
Following symbols are considered recursive:
{a__f, a__c, a__h, mark}
The recursion depth is 3 .
For your convenience, here are the oriented rules in predicative
notation (possibly applying argument filtering):
Strict DPs: {}
Weak DPs : {}
Strict Trs:
{ a__f(; f(; X)) -> a__c(; f(; g(; f(; X))))
, a__c(; X) -> d(; X)
, a__h(; X) -> a__c(; d(; X))
, mark(f(; X);) -> a__f(; mark(X;))
, mark(c(; X);) -> a__c(; X)
, mark(h(; X);) -> a__h(; mark(X;))
, mark(g(; X);) -> g(; X)
, mark(d(; X);) -> d(; X)
, a__f(; X) -> f(; X)
, a__c(; X) -> c(; X)
, a__h(; X) -> h(; X)}
Weak Trs : {}
Hurray, we answered YES(?,POLY)
POP* (PS)
YES(?,POLY)
We consider the following Problem:
Strict Trs:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, mark(f(X)) -> a__f(mark(X))
, mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,POLY)
Proof:
The input was oriented with the instance of
Polynomial Path Order (PS) as induced by the safe mapping
safe(a__f) = {1}, safe(f) = {1}, safe(a__c) = {1}, safe(g) = {1},
safe(d) = {1}, safe(a__h) = {1}, safe(mark) = {}, safe(c) = {1},
safe(h) = {1}
and precedence
a__f > a__c, a__h > a__c, mark > a__f, mark > a__c, mark > a__h .
Following symbols are considered recursive:
{a__f, a__c, a__h, mark}
The recursion depth is 3 .
For your convenience, here are the oriented rules in predicative
notation (possibly applying argument filtering):
Strict DPs: {}
Weak DPs : {}
Strict Trs:
{ a__f(; f(; X)) -> a__c(; f(; g(; f(; X))))
, a__c(; X) -> d(; X)
, a__h(; X) -> a__c(; d(; X))
, mark(f(; X);) -> a__f(; mark(X;))
, mark(c(; X);) -> a__c(; X)
, mark(h(; X);) -> a__h(; mark(X;))
, mark(g(; X);) -> g(; X)
, mark(d(; X);) -> d(; X)
, a__f(; X) -> f(; X)
, a__c(; X) -> c(; X)
, a__h(; X) -> h(; X)}
Weak Trs : {}
Hurray, we answered YES(?,POLY)
Small POP*
YES(?,O(n^1))
We consider the following Problem:
Strict Trs:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, mark(f(X)) -> a__f(mark(X))
, mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The input was oriented with the instance of
Small Polynomial Path Order (WSC,
Nat 1-bounded) as induced by the safe mapping
safe(a__f) = {1}, safe(f) = {1}, safe(a__c) = {1}, safe(g) = {1},
safe(d) = {1}, safe(a__h) = {1}, safe(mark) = {}, safe(c) = {1},
safe(h) = {1}
and precedence
a__f > a__c, a__h > a__c, mark > a__f, mark > a__c, mark > a__h .
Following symbols are considered recursive:
{mark}
The recursion depth is 1 .
For your convenience, here are the oriented rules in predicative
notation (possibly applying argument filtering):
Strict DPs: {}
Weak DPs : {}
Strict Trs:
{ a__f(; f(; X)) -> a__c(; f(; g(; f(; X))))
, a__c(; X) -> d(; X)
, a__h(; X) -> a__c(; d(; X))
, mark(f(; X);) -> a__f(; mark(X;))
, mark(c(; X);) -> a__c(; X)
, mark(h(; X);) -> a__h(; mark(X;))
, mark(g(; X);) -> g(; X)
, mark(d(; X);) -> d(; X)
, a__f(; X) -> f(; X)
, a__c(; X) -> c(; X)
, a__h(; X) -> h(; X)}
Weak Trs : {}
Hurray, we answered YES(?,O(n^1))
Small POP* (PS)
YES(?,O(n^1))
We consider the following Problem:
Strict Trs:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, mark(f(X)) -> a__f(mark(X))
, mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The input was oriented with the instance of
Small Polynomial Path Order (WSC,
PS,
Nat 1-bounded) as induced by the safe mapping
safe(a__f) = {1}, safe(f) = {1}, safe(a__c) = {1}, safe(g) = {1},
safe(d) = {1}, safe(a__h) = {1}, safe(mark) = {}, safe(c) = {1},
safe(h) = {1}
and precedence
a__f > a__c, a__h > a__c, mark > a__f, mark > a__c, mark > a__h .
Following symbols are considered recursive:
{mark}
The recursion depth is 1 .
For your convenience, here are the oriented rules in predicative
notation (possibly applying argument filtering):
Strict DPs: {}
Weak DPs : {}
Strict Trs:
{ a__f(; f(; X)) -> a__c(; f(; g(; f(; X))))
, a__c(; X) -> d(; X)
, a__h(; X) -> a__c(; d(; X))
, mark(f(; X);) -> a__f(; mark(X;))
, mark(c(; X);) -> a__c(; X)
, mark(h(; X);) -> a__h(; mark(X;))
, mark(g(; X);) -> g(; X)
, mark(d(; X);) -> d(; X)
, a__f(; X) -> f(; X)
, a__c(; X) -> c(; X)
, a__h(; X) -> h(; X)}
Weak Trs : {}
Hurray, we answered YES(?,O(n^1))