YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
=(x:S,x:S) -> ttrue
=(x:S,y:S) -> if(x:S,y:S,if(y:S,ffalse,ttrue))
=(x:S,y:S) -> if(x:S,y:S,not(y:S))
and(x:S,y:S) -> if(x:S,y:S,ffalse)
if(ffalse,x:S,y:S) -> y:S
if(ttrue,x:S,y:S) -> x:S
if(x:S,x:S,if(x:S,ffalse,ttrue)) -> ttrue
implies(x:S,y:S) -> if(x:S,y:S,ttrue)
not(x:S) -> if(x:S,ffalse,ttrue)
or(x:S,y:S) -> if(x:S,ttrue,y:S)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 =#(x:S,y:S) -> IF(x:S,y:S,if(y:S,ffalse,ttrue))
 =#(x:S,y:S) -> IF(x:S,y:S,not(y:S))
 =#(x:S,y:S) -> IF(y:S,ffalse,ttrue)
 =#(x:S,y:S) -> NOT(y:S)
 AND(x:S,y:S) -> IF(x:S,y:S,ffalse)
 IMPLIES(x:S,y:S) -> IF(x:S,y:S,ttrue)
 NOT(x:S) -> IF(x:S,ffalse,ttrue)
 OR(x:S,y:S) -> IF(x:S,ttrue,y:S)
-> Rules:
 =(x:S,x:S) -> ttrue
 =(x:S,y:S) -> if(x:S,y:S,if(y:S,ffalse,ttrue))
 =(x:S,y:S) -> if(x:S,y:S,not(y:S))
 and(x:S,y:S) -> if(x:S,y:S,ffalse)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 if(x:S,x:S,if(x:S,ffalse,ttrue)) -> ttrue
 implies(x:S,y:S) -> if(x:S,y:S,ttrue)
 not(x:S) -> if(x:S,ffalse,ttrue)
 or(x:S,y:S) -> if(x:S,ttrue,y:S)

Problem 1: 

SCC Processor:
-> Pairs:
 =#(x:S,y:S) -> IF(x:S,y:S,if(y:S,ffalse,ttrue))
 =#(x:S,y:S) -> IF(x:S,y:S,not(y:S))
 =#(x:S,y:S) -> IF(y:S,ffalse,ttrue)
 =#(x:S,y:S) -> NOT(y:S)
 AND(x:S,y:S) -> IF(x:S,y:S,ffalse)
 IMPLIES(x:S,y:S) -> IF(x:S,y:S,ttrue)
 NOT(x:S) -> IF(x:S,ffalse,ttrue)
 OR(x:S,y:S) -> IF(x:S,ttrue,y:S)
-> Rules:
 =(x:S,x:S) -> ttrue
 =(x:S,y:S) -> if(x:S,y:S,if(y:S,ffalse,ttrue))
 =(x:S,y:S) -> if(x:S,y:S,not(y:S))
 and(x:S,y:S) -> if(x:S,y:S,ffalse)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 if(x:S,x:S,if(x:S,ffalse,ttrue)) -> ttrue
 implies(x:S,y:S) -> if(x:S,y:S,ttrue)
 not(x:S) -> if(x:S,ffalse,ttrue)
 or(x:S,y:S) -> if(x:S,ttrue,y:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.