MAYBE
*** Computating Strongly Quasi-Reducible Parts ***
TRS:
   [ eq(?x,?x) -> true,
     eq(0,s(?y)) -> false,
     eq(s(?x),0) -> false,
     not(true) -> false,
     not(false) -> true,
     neq(?x,?y) -> not(eq(?x,?y)),
     neq(?x,s(?x)) -> true ]
Constructors: {0,s,eq,not,true,false}
Defined function symbols: {neq}
Constructor subsystem:
   [ eq(0,s(?y)) -> false,
     eq(s(?x),0) -> false,
     not(true) -> false,
     not(false) -> true ]
Rule part & Conj Part:
   [ eq(0,s(?y)) -> false,
     eq(s(?x),0) -> false,
     not(true) -> false,
     not(false) -> true,
     neq(?x,?y) -> not(eq(?x,?y)) ]
   [ eq(?x,?x) -> true,
     neq(?x,s(?x)) -> true ]
Trying with:
R:
   [ eq(0,s(?y)) -> false,
     eq(s(?x),0) -> false,
     not(true) -> false,
     not(false) -> true,
     neq(?x,?y) -> not(eq(?x,?y)) ]
E:
   [ eq(?x,?x) = true,
     neq(?x,s(?x)) = true ]
*** Ground Confluence Check by Rewriting Induction ***
Sort: {Nat,Bool}
Signature:
   [ eq : Nat,Nat -> Bool,
     neq : Nat,Nat -> Bool,
     not : Bool -> Bool,
     true : Bool,
     false : Bool,
     s : Nat -> Nat,
     0 : Nat ]
Rule Part:
   [ eq(0,s(?y)) -> false,
     eq(s(?x),0) -> false,
     not(true) -> false,
     not(false) -> true,
     neq(?x,?y) -> not(eq(?x,?y)) ]
Conjecture Part:
   [ eq(?x,?x) = true,
     neq(?x,s(?x)) = true ]
Precedence (by weight): {(0,5),(s,2),(eq,3),(neq,7),(not,6),(true,1),(false,4)}
Rule part is confluent.
R0 is ground confluent.
Check conj part consists of inductive theorems of R0.
Rules:
   [ eq(0,s(?y)) -> false,
     eq(s(?x),0) -> false,
     not(true) -> false,
     not(false) -> true,
     neq(?x,?y) -> not(eq(?x,?y)) ]
Conjectures:
   [ eq(?x,?x) = true,
     neq(?x,s(?x)) = true ]
STEP 0
ES:
   [ eq(?x,?x) = true,
     neq(?x,s(?x)) = true ]
HS:
   [ ]
ES0:
   [ eq(?x,?x) = true,
     not(eq(?x,s(?x))) = true ]
HS0:
   [ ]
ES1:
   [ eq(?x,?x) = true,
     not(eq(?x,s(?x))) = true ]
HS1:
   [ ]
No equation to expand
Failed to prove conj part consists of inductive theorems of R0.
new/eq2.trs: Failure(unknown)
(15 msec.)