(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(concat(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = 1 + x1 + x2
POL(false) = 0
POL(leaf) = 1
POL(lessleaves(x1, x2)) = x1 + x2
POL(true) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
concat(leaf, Y) → Y
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
concat(cons(U, V), Y) → cons(U, concat(V, Y))
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:
[concat2, cons2]
Status:
cons2: [1,2]
concat2: [1,2]
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
concat(cons(U, V), Y) → cons(U, concat(V, Y))
(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
(7) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(8) TRUE
(9) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(10) TRUE