(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__and(true, X) → mark(X)
a__and(false, Y) → false
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(add(X, Y))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
a__from(X) → cons(X, from(s(X)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(add(X1, X2)) → a__add(mark(X1), X2)
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(X)
mark(true) → true
mark(false) → false
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__add(X1, X2) → add(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:
[aand2, mark1, false, aif3, aadd2, 0, add2, nil, and2, if3] > true
[aand2, mark1, false, aif3, aadd2, 0, add2, nil, and2, if3] > afrom1 > s1 > [afirst2, cons2] > first2
[aand2, mark1, false, aif3, aadd2, 0, add2, nil, and2, if3] > afrom1 > from1
Status:
from1: multiset
aand2: multiset
true: multiset
mark1: multiset
aadd2: multiset
afrom1: [1]
and2: multiset
0: multiset
first2: multiset
afirst2: [2,1]
add2: multiset
cons2: [1,2]
if3: multiset
false: multiset
s1: multiset
aif3: multiset
nil: multiset
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__and(true, X) → mark(X)
a__and(false, Y) → false
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(add(X, Y))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
a__from(X) → cons(X, from(s(X)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(add(X1, X2)) → a__add(mark(X1), X2)
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(X)
mark(true) → true
mark(false) → false
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__and(X1, X2) → and(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__add(X1, X2) → add(X1, X2)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:
aand2 > and2 > if3
aif3 > if3
aadd2 > add2 > if3
Status:
add2: multiset
if3: multiset
aand2: multiset
aadd2: multiset
and2: [2,1]
aif3: multiset
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__and(X1, X2) → and(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__add(X1, X2) → add(X1, X2)
(4) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(5) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(6) TRUE