Certification Problem

Input (TPDB TRS_Standard/Applicative_05/TreeLevels)

The rewrite relation of the following TRS is considered.

app(app(map,f),nil) → nil (1)
app(app(map,f),app(app(cons,x),xs)) → app(app(cons,app(f,x)),app(app(map,f),xs)) (2)
app(app(append,xs),nil) → xs (3)
app(app(append,nil),ys) → ys (4)
app(app(append,app(app(cons,x),xs)),ys) → app(app(cons,x),app(app(append,xs),ys)) (5)
app(app(zip,nil),yss) → yss (6)
app(app(zip,xss),nil) → xss (7)
app(app(zip,app(app(cons,xs),xss)),app(app(cons,ys),yss)) → app(app(cons,app(app(append,xs),ys)),app(app(zip,xss),yss)) (8)
app(app(combine,xs),nil) → xs (9)
app(app(combine,xs),app(app(cons,ys),yss)) → app(app(combine,app(app(zip,xs),ys)),yss) (10)
app(levels,app(app(node,x),xs)) → app(app(cons,app(app(cons,x),nil)),app(app(combine,nil),app(app(map,levels),xs))) (11)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Uncurrying

We uncurry the binary symbol app in combination with the following symbol map which also determines the applicative arities of these symbols.

map is mapped to map, map1(x1), map2(x1, x2)
nil is mapped to nil
cons is mapped to cons, cons1(x1), cons2(x1, x2)
append is mapped to append, append1(x1), append2(x1, x2)
zip is mapped to zip, zip1(x1), zip2(x1, x2)
combine is mapped to combine, combine1(x1), combine2(x1, x2)
levels is mapped to levels, levels1(x1)
node is mapped to node, node1(x1), node2(x1, x2)


There are no uncurry rules.
No rules have to be added for the eta-expansion.

Uncurrying the rules and adding the uncurrying rules yields the new set of rules
map2(f,nil) → nil (25)
map2(f,cons2(x,xs)) → cons2(app(f,x),map2(f,xs)) (26)
append2(xs,nil) → xs (27)
append2(nil,ys) → ys (28)
append2(cons2(x,xs),ys) → cons2(x,append2(xs,ys)) (29)
zip2(nil,yss) → yss (30)
zip2(xss,nil) → xss (31)
zip2(cons2(xs,xss),cons2(ys,yss)) → cons2(append2(xs,ys),zip2(xss,yss)) (32)
combine2(xs,nil) → xs (33)
combine2(xs,cons2(ys,yss)) → combine2(zip2(xs,ys),yss) (34)
levels1(node2(x,xs)) → cons2(cons2(x,nil),combine2(nil,map2(levels,xs))) (35)
app(map,y1) → map1(y1) (12)
app(map1(x0),y1) → map2(x0,y1) (13)
app(cons,y1) → cons1(y1) (14)
app(cons1(x0),y1) → cons2(x0,y1) (15)
app(append,y1) → append1(y1) (16)
app(append1(x0),y1) → append2(x0,y1) (17)
app(zip,y1) → zip1(y1) (18)
app(zip1(x0),y1) → zip2(x0,y1) (19)
app(combine,y1) → combine1(y1) (20)
app(combine1(x0),y1) → combine2(x0,y1) (21)
app(levels,y1) → levels1(y1) (22)
app(node,y1) → node1(y1) (23)
app(node1(x0),y1) → node2(x0,y1) (24)

1.1 Rule Removal

Using the
prec(map2) = 10 stat(map2) = mul
prec(nil) = 1 stat(nil) = mul
prec(cons2) = 0 stat(cons2) = mul
prec(app) = 10 stat(app) = mul
prec(append2) = 2 stat(append2) = mul
prec(zip2) = 3 stat(zip2) = mul
prec(combine2) = 4 stat(combine2) = lex
prec(levels1) = 10 stat(levels1) = mul
prec(node2) = 10 stat(node2) = mul
prec(levels) = 5 stat(levels) = mul
prec(map) = 11 stat(map) = mul
prec(map1) = 10 stat(map1) = mul
prec(cons) = 12 stat(cons) = mul
prec(cons1) = 6 stat(cons1) = lex
prec(append) = 13 stat(append) = mul
prec(append1) = 7 stat(append1) = lex
prec(zip) = 14 stat(zip) = mul
prec(zip1) = 8 stat(zip1) = mul
prec(combine) = 15 stat(combine) = mul
prec(node) = 16 stat(node) = mul
prec(node1) = 9 stat(node1) = lex

π(map2) = [1,2]
π(nil) = []
π(cons2) = [1,2]
π(app) = [1,2]
π(append2) = [1,2]
π(zip2) = [1,2]
π(combine2) = [2,1]
π(levels1) = [1]
π(node2) = [1,2]
π(levels) = []
π(map) = []
π(map1) = [1]
π(cons) = []
π(cons1) = [1]
π(append) = []
π(append1) = [1]
π(zip) = []
π(zip1) = [1]
π(combine) = []
π(combine1) = 1
π(node) = []
π(node1) = [1]

all of the following rules can be deleted.
map2(f,nil) → nil (25)
map2(f,cons2(x,xs)) → cons2(app(f,x),map2(f,xs)) (26)
append2(xs,nil) → xs (27)
append2(nil,ys) → ys (28)
append2(cons2(x,xs),ys) → cons2(x,append2(xs,ys)) (29)
zip2(nil,yss) → yss (30)
zip2(xss,nil) → xss (31)
zip2(cons2(xs,xss),cons2(ys,yss)) → cons2(append2(xs,ys),zip2(xss,yss)) (32)
combine2(xs,nil) → xs (33)
combine2(xs,cons2(ys,yss)) → combine2(zip2(xs,ys),yss) (34)
levels1(node2(x,xs)) → cons2(cons2(x,nil),combine2(nil,map2(levels,xs))) (35)
app(map,y1) → map1(y1) (12)
app(map1(x0),y1) → map2(x0,y1) (13)
app(cons,y1) → cons1(y1) (14)
app(cons1(x0),y1) → cons2(x0,y1) (15)
app(append,y1) → append1(y1) (16)
app(append1(x0),y1) → append2(x0,y1) (17)
app(zip,y1) → zip1(y1) (18)
app(zip1(x0),y1) → zip2(x0,y1) (19)
app(combine,y1) → combine1(y1) (20)
app(combine1(x0),y1) → combine2(x0,y1) (21)
app(levels,y1) → levels1(y1) (22)
app(node,y1) → node1(y1) (23)
app(node1(x0),y1) → node2(x0,y1) (24)

1.1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.