Certification Problem

Input (TPDB TRS_Standard/Applicative_05/BTreeMember)

The rewrite relation of the following TRS is considered.

app(app(lt,app(s,x)),app(s,y))	→	app(app(lt,x),y)	(1)
app(app(lt,0),app(s,y))	→	true	(2)
app(app(lt,y),0)	→	false	(3)
app(app(eq,x),x)	→	true	(4)
app(app(eq,app(s,x)),0)	→	false	(5)
app(app(eq,0),app(s,x))	→	false	(6)
app(app(member,w),null)	→	false	(7)
app(app(member,w),app(app(app(fork,x),y),z))	→	app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))	(8)

Prove or disprove termination.

Yes.

The following set of initial dependency pairs has been identified.

app^#(app(lt,app(s,x)),app(s,y))	→	app^#(lt,x)	(9)
app^#(app(lt,app(s,x)),app(s,y))	→	app^#(app(lt,x),y)	(10)
app^#(app(member,w),app(app(app(fork,x),y),z))	→	app^#(app(member,w),z)	(11)
app^#(app(member,w),app(app(app(fork,x),y),z))	→	app^#(eq,w)	(12)
app^#(app(member,w),app(app(app(fork,x),y),z))	→	app^#(app(eq,w),y)	(13)
app^#(app(member,w),app(app(app(fork,x),y),z))	→	app^#(if,app(app(eq,w),y))	(14)
app^#(app(member,w),app(app(app(fork,x),y),z))	→	app^#(app(if,app(app(eq,w),y)),true)	(15)
app^#(app(member,w),app(app(app(fork,x),y),z))	→	app^#(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z))	(16)
app^#(app(member,w),app(app(app(fork,x),y),z))	→	app^#(app(member,w),x)	(17)
app^#(app(member,w),app(app(app(fork,x),y),z))	→	app^#(lt,w)	(18)
app^#(app(member,w),app(app(app(fork,x),y),z))	→	app^#(app(lt,w),y)	(19)
app^#(app(member,w),app(app(app(fork,x),y),z))	→	app^#(if,app(app(lt,w),y))	(20)
app^#(app(member,w),app(app(app(fork,x),y),z))	→	app^#(app(if,app(app(lt,w),y)),app(app(member,w),x))	(21)
app^#(app(member,w),app(app(app(fork,x),y),z))	→	app^#(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))	(22)

The dependency pairs are split into 2 components.

The 1^st component contains the pair

app^#(app(member,w),app(app(app(fork,x),y),z))	→	app^#(app(member,w),z)	(11)
app^#(app(member,w),app(app(app(fork,x),y),z))	→	app^#(app(member,w),x)	(17)

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

app^#(app(member,w),app(app(app(fork,x),y),z))	→	app^#(app(member,w),z)	(11)

2	>	2
1	≥	1
app^#(app(member,w),app(app(app(fork,x),y),z))	→	app^#(app(member,w),x)	(17)

2	>	2
1	≥	1

As there is no critical graph in the transitive closure, there are no infinite chains.