Certification Problem

Input (TPDB TRS_Standard/AG01/#3.5a)

The rewrite relation of the following TRS is considered.

le(0,y) true (1)
le(s(x),0) false (2)
le(s(x),s(y)) le(x,y) (3)
pred(s(x)) x (4)
minus(x,0) x (5)
minus(x,s(y)) pred(minus(x,y)) (6)
mod(0,y) 0 (7)
mod(s(x),0) 0 (8)
mod(s(x),s(y)) if_mod(le(y,x),s(x),s(y)) (9)
if_mod(true,s(x),s(y)) mod(minus(x,y),s(y)) (10)
if_mod(false,s(x),s(y)) s(x) (11)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
minus#(x,s(y)) minus#(x,y) (12)
if_mod#(true,s(x),s(y)) minus#(x,y) (13)
le#(s(x),s(y)) le#(x,y) (14)
if_mod#(true,s(x),s(y)) mod#(minus(x,y),s(y)) (15)
mod#(s(x),s(y)) if_mod#(le(y,x),s(x),s(y)) (16)
minus#(x,s(y)) pred#(minus(x,y)) (17)
mod#(s(x),s(y)) le#(y,x) (18)

1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.