YES (ignored inputs)COMMENT translated from Cops 175 *** Computating Strongly Quasi-Reducible Parts *** TRS: [ +(0,?x) -> ?x, +(-(?x),?x) -> 0, +(+(?x,?y),?z) -> +(?x,+(?y,?z)), +(-(?x),+(?x,?y)) -> ?y, +(?x,0) -> ?x, -(0) -> 0, -(-(?x)) -> ?x, +(?x,-(?x)) -> 0, +(?x,+(-(?x),?y)) -> ?y, -(+(?x,?y)) -> +(-(?y),-(?x)), +(?x,?y) -> +(?y,?x) ] Constructors: {0} Defined function symbols: {+,-} Constructor subsystem: [ ] Rule part & Conj Part: [ +(?x,0) -> ?x, -(0) -> 0 ] [ +(0,?x) -> ?x, +(+(?x,?y),?z) -> +(?x,+(?y,?z)), -(-(?x)) -> ?x, -(+(?x,?y)) -> +(-(?y),-(?x)), +(-(?x),?x) -> 0, +(-(?x),+(?x,?y)) -> ?y, +(?x,-(?x)) -> 0, +(?x,+(-(?x),?y)) -> ?y, +(?x,?y) -> +(?y,?x) ] Rule part & Conj Part: [ +(0,?x) -> ?x, -(0) -> 0 ] [ +(?x,0) -> ?x, +(+(?x,?y),?z) -> +(?x,+(?y,?z)), -(-(?x)) -> ?x, -(+(?x,?y)) -> +(-(?y),-(?x)), +(-(?x),?x) -> 0, +(-(?x),+(?x,?y)) -> ?y, +(?x,-(?x)) -> 0, +(?x,+(-(?x),?y)) -> ?y, +(?x,?y) -> +(?y,?x) ] *** Ground Confluence Check by Rewriting Induction *** Sort: {Elem} Signature: [ + : Elem,Elem -> Elem, - : Elem -> Elem, 0 : Elem ] Rule Part: [ +(?x,0) -> ?x, -(0) -> 0 ] Conjecture Part: [ +(0,?x) = ?x, +(+(?x,?y),?z) = +(?x,+(?y,?z)), -(-(?x)) = ?x, -(+(?x,?y)) = +(-(?y),-(?x)), +(-(?x),?x) = 0, +(-(?x),+(?x,?y)) = ?y, +(?x,-(?x)) = 0, +(?x,+(-(?x),?y)) = ?y, +(?x,?y) = +(?y,?x) ] Precedence (by weight): {(+,0),(-,2),(0,1)} Rule part is confluent. R0 is ground confluent. Check conj part consists of inductive theorems of R0. Rules: [ +(?x,0) -> ?x, -(0) -> 0 ] Conjectures: [ +(0,?x) = ?x, +(+(?x,?y),?z) = +(?x,+(?y,?z)), -(-(?x)) = ?x, -(+(?x,?y)) = +(-(?y),-(?x)), +(-(?x),?x) = 0, +(-(?x),+(?x,?y)) = ?y, +(?x,-(?x)) = 0, +(?x,+(-(?x),?y)) = ?y, +(?x,?y) = +(?y,?x) ] STEP 0 ES: [ +(0,?x) = ?x, +(+(?x,?y),?z) = +(?x,+(?y,?z)), -(-(?x)) = ?x, -(+(?x,?y)) = +(-(?y),-(?x)), +(-(?x),?x) = 0, +(-(?x),+(?x,?y)) = ?y, +(?x,-(?x)) = 0, +(?x,+(-(?x),?y)) = ?y, +(?x,?y) = +(?y,?x) ] HS: [ ] ES0: [ +(0,?x) = ?x, +(+(?x,?y),?z) = +(?x,+(?y,?z)), -(-(?x)) = ?x, -(+(?x,?y)) = +(-(?y),-(?x)), +(-(?x),?x) = 0, +(-(?x),+(?x,?y)) = ?y, +(?x,-(?x)) = 0, +(?x,+(-(?x),?y)) = ?y, +(?x,?y) = +(?y,?x) ] HS0: [ ] ES1: [ +(0,?x) = ?x, +(+(?x,?y),?z) = +(?x,+(?y,?z)), -(-(?x)) = ?x, -(+(?x,?y)) = +(-(?y),-(?x)), +(-(?x),?x) = 0, +(-(?x),+(?x,?y)) = ?y, +(?x,-(?x)) = 0, +(?x,+(-(?x),?y)) = ?y, +(?x,?y) = +(?y,?x) ] HS1: [ ] Expand +(0,?x) = ?x [ 0 = 0 ] ES2: [ 0 = 0, +(?x,?y) = +(?y,?x), +(?x,+(-(?x),?y)) = ?y, +(?x,-(?x)) = 0, +(-(?x),+(?x,?y)) = ?y, +(-(?x),?x) = 0, -(+(?x,?y)) = +(-(?y),-(?x)), -(-(?x)) = ?x, +(+(?x,?y),?z) = +(?x,+(?y,?z)) ] HS2: [ +(0,?x) -> ?x ] STEP 1 ES: [ 0 = 0, +(?x,?y) = +(?y,?x), +(?x,+(-(?x),?y)) = ?y, +(?x,-(?x)) = 0, +(-(?x),+(?x,?y)) = ?y, +(-(?x),?x) = 0, -(+(?x,?y)) = +(-(?y),-(?x)), -(-(?x)) = ?x, +(+(?x,?y),?z) = +(?x,+(?y,?z)) ] HS: [ +(0,?x) -> ?x ] ES0: [ 0 = 0, +(?x,?y) = +(?y,?x), +(?x,+(-(?x),?y)) = ?y, +(?x,-(?x)) = 0, +(-(?x),+(?x,?y)) = ?y, +(-(?x),?x) = 0, -(+(?x,?y)) = +(-(?y),-(?x)), -(-(?x)) = ?x, +(+(?x,?y),?z) = +(?x,+(?y,?z)) ] HS0: [ +(0,?x) -> ?x ] ES1: [ +(?x,?y) = +(?y,?x), +(?x,+(-(?x),?y)) = ?y, +(?x,-(?x)) = 0, +(-(?x),+(?x,?y)) = ?y, +(-(?x),?x) = 0, -(+(?x,?y)) = +(-(?y),-(?x)), -(-(?x)) = ?x, +(+(?x,?y),?z) = +(?x,+(?y,?z)) ] HS1: [ +(0,?x) -> ?x ] Expand -(-(?x)) = ?x [ -(0) = 0 ] ES2: [ 0 = 0, +(+(?x,?y),?z) = +(?x,+(?y,?z)), +(?x,-(?x)) = 0, -(+(?x,?y)) = +(-(?y),-(?x)), +(-(?x),?x) = 0, +(-(?x),+(?x,?y)) = ?y, +(?x,?y) = +(?y,?x), +(?x,+(-(?x),?y)) = ?y ] HS2: [ -(-(?x)) -> ?x, +(0,?x) -> ?x ] STEP 2 ES: [ 0 = 0, +(+(?x,?y),?z) = +(?x,+(?y,?z)), +(?x,-(?x)) = 0, -(+(?x,?y)) = +(-(?y),-(?x)), +(-(?x),?x) = 0, +(-(?x),+(?x,?y)) = ?y, +(?x,?y) = +(?y,?x), +(?x,+(-(?x),?y)) = ?y ] HS: [ -(-(?x)) -> ?x, +(0,?x) -> ?x ] ES0: [ 0 = 0, +(+(?x,?y),?z) = +(?x,+(?y,?z)), +(?x,-(?x)) = 0, -(+(?x,?y)) = +(-(?y),-(?x)), +(-(?x),?x) = 0, +(-(?x),+(?x,?y)) = ?y, +(?x,?y) = +(?y,?x), +(?x,+(-(?x),?y)) = ?y ] HS0: [ -(-(?x)) -> ?x, +(0,?x) -> ?x ] ES1: [ +(+(?x,?y),?z) = +(?x,+(?y,?z)), +(?x,-(?x)) = 0, -(+(?x,?y)) = +(-(?y),-(?x)), +(-(?x),?x) = 0, +(-(?x),+(?x,?y)) = ?y, +(?x,?y) = +(?y,?x), +(?x,+(-(?x),?y)) = ?y ] HS1: [ -(-(?x)) -> ?x, +(0,?x) -> ?x ] Expand +(?x,-(?x)) = 0 [ +(0,0) = 0 ] ES2: [ 0 = 0, +(?x,+(-(?x),?y)) = ?y, +(?x,?y) = +(?y,?x), +(-(?x),+(?x,?y)) = ?y, +(-(?x),?x) = 0, -(+(?x,?y)) = +(-(?y),-(?x)), +(+(?x,?y),?z) = +(?x,+(?y,?z)) ] HS2: [ +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] STEP 3 ES: [ 0 = 0, +(?x,+(-(?x),?y)) = ?y, +(?x,?y) = +(?y,?x), +(-(?x),+(?x,?y)) = ?y, +(-(?x),?x) = 0, -(+(?x,?y)) = +(-(?y),-(?x)), +(+(?x,?y),?z) = +(?x,+(?y,?z)) ] HS: [ +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] ES0: [ 0 = 0, +(?x,+(-(?x),?y)) = ?y, +(?x,?y) = +(?y,?x), +(-(?x),+(?x,?y)) = ?y, +(-(?x),?x) = 0, -(+(?x,?y)) = +(-(?y),-(?x)), +(+(?x,?y),?z) = +(?x,+(?y,?z)) ] HS0: [ +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] ES1: [ +(?x,+(-(?x),?y)) = ?y, +(?x,?y) = +(?y,?x), +(-(?x),+(?x,?y)) = ?y, +(-(?x),?x) = 0, -(+(?x,?y)) = +(-(?y),-(?x)), +(+(?x,?y),?z) = +(?x,+(?y,?z)) ] HS1: [ +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] Expand +(-(?x),?x) = 0 [ +(0,0) = 0 ] ES2: [ 0 = 0, +(+(?x,?y),?z) = +(?x,+(?y,?z)), -(+(?x,?y)) = +(-(?y),-(?x)), +(?x,?y) = +(?y,?x), +(-(?x),+(?x,?y)) = ?y, +(?x,+(-(?x),?y)) = ?y ] HS2: [ +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] STEP 4 ES: [ 0 = 0, +(+(?x,?y),?z) = +(?x,+(?y,?z)), -(+(?x,?y)) = +(-(?y),-(?x)), +(?x,?y) = +(?y,?x), +(-(?x),+(?x,?y)) = ?y, +(?x,+(-(?x),?y)) = ?y ] HS: [ +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] ES0: [ 0 = 0, +(+(?x,?y),?z) = +(?x,+(?y,?z)), -(+(?x,?y)) = +(-(?y),-(?x)), +(?x,?y) = +(?y,?x), +(-(?x),+(?x,?y)) = ?y, +(?x,+(-(?x),?y)) = ?y ] HS0: [ +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] ES1: [ +(+(?x,?y),?z) = +(?x,+(?y,?z)), -(+(?x,?y)) = +(-(?y),-(?x)), +(?x,?y) = +(?y,?x), +(-(?x),+(?x,?y)) = ?y, +(?x,+(-(?x),?y)) = ?y ] HS1: [ +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] Expand +(?x,?y) = +(?y,?x) [ ?x_1 = +(0,?x_1) ] ES2: [ ?x_1 = +(0,?x_1), +(?x,+(-(?x),?y)) = ?y, +(-(?x),+(?x,?y)) = ?y, -(+(?x,?y)) = +(-(?y),-(?x)), +(+(?x,?y),?z) = +(?x,+(?y,?z)) ] HS2: [ +(?x,?y) -> +(?y,?x), +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] STEP 5 ES: [ ?x_1 = +(0,?x_1), +(?x,+(-(?x),?y)) = ?y, +(-(?x),+(?x,?y)) = ?y, -(+(?x,?y)) = +(-(?y),-(?x)), +(+(?x,?y),?z) = +(?x,+(?y,?z)) ] HS: [ +(?x,?y) -> +(?y,?x), +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] ES0: [ ?x_1 = ?x_1, +(?x,+(-(?x),?y)) = ?y, +(-(?x),+(?x,?y)) = ?y, -(+(?x,?y)) = +(-(?y),-(?x)), +(+(?x,?y),?z) = +(?x,+(?y,?z)) ] HS0: [ +(?x,?y) -> +(?y,?x), +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] ES1: [ +(?x,+(-(?x),?y)) = ?y, +(-(?x),+(?x,?y)) = ?y, -(+(?x,?y)) = +(-(?y),-(?x)), +(+(?x,?y),?z) = +(?x,+(?y,?z)) ] HS1: [ +(?x,?y) -> +(?y,?x), +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] Expand +(?x,+(-(?x),?y)) = ?y [ +(0,+(0,?y)) = ?y ] ES2: [ ?y = ?y, +(+(?x,?y),?z) = +(?x,+(?y,?z)), -(+(?x,?y)) = +(-(?y),-(?x)), +(-(?x),+(?x,?y)) = ?y ] HS2: [ +(?x,+(-(?x),?y)) -> ?y, +(?x,?y) -> +(?y,?x), +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] STEP 6 ES: [ ?y = ?y, +(+(?x,?y),?z) = +(?x,+(?y,?z)), -(+(?x,?y)) = +(-(?y),-(?x)), +(-(?x),+(?x,?y)) = ?y ] HS: [ +(?x,+(-(?x),?y)) -> ?y, +(?x,?y) -> +(?y,?x), +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] ES0: [ ?y = ?y, +(+(?x,?y),?z) = +(?x,+(?y,?z)), -(+(?x,?y)) = +(-(?y),-(?x)), +(-(?x),+(?x,?y)) = ?y ] HS0: [ +(?x,+(-(?x),?y)) -> ?y, +(?x,?y) -> +(?y,?x), +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] ES1: [ +(+(?x,?y),?z) = +(?x,+(?y,?z)), -(+(?x,?y)) = +(-(?y),-(?x)), +(-(?x),+(?x,?y)) = ?y ] HS1: [ +(?x,+(-(?x),?y)) -> ?y, +(?x,?y) -> +(?y,?x), +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] Expand +(-(?x),+(?x,?y)) = ?y [ +(0,+(0,?y)) = ?y ] ES2: [ ?y = ?y, -(+(?x,?y)) = +(-(?y),-(?x)), +(+(?x,?y),?z) = +(?x,+(?y,?z)) ] HS2: [ +(-(?x),+(?x,?y)) -> ?y, +(?x,+(-(?x),?y)) -> ?y, +(?x,?y) -> +(?y,?x), +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] STEP 7 ES: [ ?y = ?y, -(+(?x,?y)) = +(-(?y),-(?x)), +(+(?x,?y),?z) = +(?x,+(?y,?z)) ] HS: [ +(-(?x),+(?x,?y)) -> ?y, +(?x,+(-(?x),?y)) -> ?y, +(?x,?y) -> +(?y,?x), +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] ES0: [ ?y = ?y, -(+(?x,?y)) = +(-(?y),-(?x)), +(+(?x,?y),?z) = +(?x,+(?y,?z)) ] HS0: [ +(-(?x),+(?x,?y)) -> ?y, +(?x,+(-(?x),?y)) -> ?y, +(?x,?y) -> +(?y,?x), +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] ES1: [ -(+(?x,?y)) = +(-(?y),-(?x)), +(+(?x,?y),?z) = +(?x,+(?y,?z)) ] HS1: [ +(-(?x),+(?x,?y)) -> ?y, +(?x,+(-(?x),?y)) -> ?y, +(?x,?y) -> +(?y,?x), +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] Expand -(+(?x,?y)) = +(-(?y),-(?x)) [ -(?x_1) = +(-(0),-(?x_1)) ] ES2: [ -(?x_1) = +(-(0),-(?x_1)), +(+(?x,?y),?z) = +(?x,+(?y,?z)) ] HS2: [ -(+(?x,?y)) -> +(-(?y),-(?x)), +(-(?x),+(?x,?y)) -> ?y, +(?x,+(-(?x),?y)) -> ?y, +(?x,?y) -> +(?y,?x), +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] STEP 8 ES: [ -(?x_1) = +(-(0),-(?x_1)), +(+(?x,?y),?z) = +(?x,+(?y,?z)) ] HS: [ -(+(?x,?y)) -> +(-(?y),-(?x)), +(-(?x),+(?x,?y)) -> ?y, +(?x,+(-(?x),?y)) -> ?y, +(?x,?y) -> +(?y,?x), +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] ES0: [ -(?x_1) = -(?x_1), +(+(?x,?y),?z) = +(?x,+(?y,?z)) ] HS0: [ -(+(?x,?y)) -> +(-(?y),-(?x)), +(-(?x),+(?x,?y)) -> ?y, +(?x,+(-(?x),?y)) -> ?y, +(?x,?y) -> +(?y,?x), +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] ES1: [ +(+(?x,?y),?z) = +(?x,+(?y,?z)) ] HS1: [ -(+(?x,?y)) -> +(-(?y),-(?x)), +(-(?x),+(?x,?y)) -> ?y, +(?x,+(-(?x),?y)) -> ?y, +(?x,?y) -> +(?y,?x), +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] Expand +(+(?x,?y),?z) = +(?x,+(?y,?z)) [ +(?x_1,?z) = +(?x_1,+(0,?z)) ] ES2: [ +(?x_1,?z) = +(?x_1,+(0,?z)) ] HS2: [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)), -(+(?x,?y)) -> +(-(?y),-(?x)), +(-(?x),+(?x,?y)) -> ?y, +(?x,+(-(?x),?y)) -> ?y, +(?x,?y) -> +(?y,?x), +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] STEP 9 ES: [ +(?x_1,?z) = +(?x_1,+(0,?z)) ] HS: [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)), -(+(?x,?y)) -> +(-(?y),-(?x)), +(-(?x),+(?x,?y)) -> ?y, +(?x,+(-(?x),?y)) -> ?y, +(?x,?y) -> +(?y,?x), +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] ES0: [ +(?x_1,?z) = +(?x_1,?z) ] HS0: [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)), -(+(?x,?y)) -> +(-(?y),-(?x)), +(-(?x),+(?x,?y)) -> ?y, +(?x,+(-(?x),?y)) -> ?y, +(?x,?y) -> +(?y,?x), +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] ES1: [ ] HS1: [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)), -(+(?x,?y)) -> +(-(?y),-(?x)), +(-(?x),+(?x,?y)) -> ?y, +(?x,+(-(?x),?y)) -> ?y, +(?x,?y) -> +(?y,?x), +(-(?x),?x) -> 0, +(?x,-(?x)) -> 0, -(-(?x)) -> ?x, +(0,?x) -> ?x ] Conj part consisits of inductive theorems of R0. examples/fromCops/open/175.trs: Success(GCR) (7 msec.)