0.68/0.57 MAYBE 0.68/0.58 Input TRS: 0.68/0.58 AC symbols: _or_ _xor_ _and_ 0.68/0.58 1: U101(tt(),A,B) -> U102(isBool(B),A,B) 0.68/0.58 2: U102(tt(),A,B) -> _xor_(_and_(A,B),_xor_(A,B)) 0.68/0.58 3: U11(tt(),A) -> A 0.68/0.58 4: U111(tt()) -> false() 0.68/0.58 5: U121(tt(),A) -> A 0.68/0.58 6: U131(tt(),B,U',U) -> U132(isS(U'),B,U',U) 0.68/0.58 7: U132(tt(),B,U',U) -> U133(isS(U),B,U') 0.68/0.58 8: U133(tt(),B,U') -> U134(equal(_isNotEqualTo_(B,true()),true()),U') 0.68/0.58 9: U134(tt(),U') -> U' 0.68/0.58 10: U141(tt(),U) -> U142(isS(U),U) 0.68/0.58 11: U142(tt(),U) -> U 0.68/0.58 12: U151(tt(),V2) -> U152(isBool(V2)) 0.68/0.58 13: U152(tt()) -> tt() 0.68/0.58 14: U161(tt(),V2) -> U162(isBool(V2)) 0.68/0.58 15: U162(tt()) -> tt() 0.68/0.58 16: U171(tt(),V2) -> U172(isUniversal(V2)) 0.68/0.58 17: U172(tt()) -> tt() 0.68/0.58 18: U181(tt(),V2) -> U182(isUniversal(V2)) 0.68/0.58 19: U182(tt()) -> tt() 0.68/0.58 20: U191(tt(),V2) -> U192(isBool(V2)) 0.68/0.58 21: U192(tt()) -> tt() 0.68/0.58 22: U201(tt(),V2) -> U202(isBool(V2)) 0.68/0.58 23: U202(tt()) -> tt() 0.68/0.58 24: U21(tt(),A,B,C) -> U22(isBool(B),A,B,C) 0.68/0.58 25: U211(tt()) -> tt() 0.68/0.58 26: U22(tt(),A,B,C) -> U23(isBool(C),A,B,C) 0.68/0.58 27: U221(tt(),A) -> _xor_(A,true()) 0.68/0.58 28: U23(tt(),A,B,C) -> _xor_(_and_(A,B),_and_(A,C)) 0.68/0.58 29: U31(tt()) -> false() 0.68/0.58 30: U41(tt(),A) -> A 0.68/0.58 31: U51(tt(),A,B) -> U52(isBool(B),A,B) 0.68/0.58 32: U52(tt(),A,B) -> not_(_xor_(A,_and_(A,B))) 0.68/0.58 33: U61(tt(),U',U) -> U62(isS(U),U',U) 0.68/0.58 34: U62(tt(),U',U) -> U63(equal(_isNotEqualTo_(U,U'),true())) 0.68/0.58 35: U63(tt()) -> false() 0.68/0.58 36: U71(tt()) -> true() 0.68/0.58 37: U81(tt(),U',U) -> U82(isS(U),U',U) 0.68/0.58 38: U82(tt(),U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false(),true()) 0.68/0.58 39: U91(tt()) -> false() 0.68/0.58 40: _and_(A,A) -> U11(isBool(A),A) 0.68/0.58 41: _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C) 0.68/0.58 42: _and_(false(),A) -> U31(isBool(A)) 0.68/0.58 43: _and_(true(),A) -> U41(isBool(A),A) 0.68/0.58 44: _implies_(A,B) -> U51(isBool(A),A,B) 0.68/0.58 45: _isEqualTo_(U,U') -> U61(isS(U'),U',U) 0.68/0.58 46: _isEqualTo_(U,U) -> U71(isS(U)) 0.68/0.58 47: _isNotEqualTo_(U,U') -> U81(isS(U'),U',U) 0.68/0.58 48: _isNotEqualTo_(U,U) -> U91(isS(U)) 0.68/0.58 49: _or_(A,B) -> U101(isBool(A),A,B) 0.68/0.58 50: _xor_(A,A) -> U111(isBool(A)) 0.68/0.58 51: _xor_(false(),A) -> U121(isBool(A),A) 0.68/0.58 52: equal(X,X) -> tt() 0.68/0.58 53: if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U) 0.68/0.58 54: if_then_else_fi(true(),U,U') -> U141(isS(U'),U) 0.68/0.58 55: isBool(false()) -> tt() 0.68/0.58 56: isBool(true()) -> tt() 0.68/0.58 57: isBool(_and_(V1,V2)) -> U151(isBool(V1),V2) 0.68/0.58 58: isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2) 0.68/0.58 59: isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2) 0.68/0.58 60: isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2) 0.68/0.58 61: isBool(_or_(V1,V2)) -> U191(isBool(V1),V2) 0.68/0.58 62: isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2) 0.68/0.58 63: isBool(not_(V1)) -> U211(isBool(V1)) 0.68/0.58 64: not_(A) -> U221(isBool(A),A) 0.68/0.58 65: not_(false()) -> true() 0.68/0.58 66: not_(true()) -> false() 0.68/0.58 Number of strict rules: 66 0.68/0.58 Direct POLO(bPol) ... failed. 0.68/0.58 Uncurrying U181 U71 U91 U133 U142 U62 U172 U82 U81 U171 U182 U132 U141 U61 0.68/0.58 AC symbols: _or_ _xor_ _and_ 0.68/0.58 1: U101(tt(),A,B) -> U102(isBool(B),A,B) 0.68/0.58 2: U102(tt(),A,B) -> _xor_(_and_(A,B),_xor_(A,B)) 0.68/0.58 3: U11(tt(),A) -> A 0.68/0.58 4: U111(tt()) -> false() 0.68/0.58 5: U121(tt(),A) -> A 0.68/0.58 6: U131(tt(),B,U',U) -> U132^1_isS(U',B,U',U) 0.68/0.58 7: U132^1_tt(B,U',U) -> U133^1_isS(U,B,U') 0.68/0.58 8: U133^1_tt(B,U') -> U134(equal(_isNotEqualTo_(B,true()),true()),U') 0.68/0.58 9: U134(tt(),U') -> U' 0.68/0.58 10: U141^1_tt(U) -> U142^1_isS(U,U) 0.68/0.58 11: U142^1_tt(U) -> U 0.68/0.58 12: U151(tt(),V2) -> U152(isBool(V2)) 0.68/0.58 13: U152(tt()) -> tt() 0.68/0.58 14: U161(tt(),V2) -> U162(isBool(V2)) 0.68/0.58 15: U162(tt()) -> tt() 0.68/0.58 16: U171^1_tt(V2) -> U172^1_isUniversal(V2) 0.68/0.58 17: U172^1_tt() -> tt() 0.68/0.58 18: U181^1_tt(V2) -> U182^1_isUniversal(V2) 0.68/0.58 19: U182^1_tt() -> tt() 0.68/0.58 20: U191(tt(),V2) -> U192(isBool(V2)) 0.68/0.58 21: U192(tt()) -> tt() 0.68/0.58 22: U201(tt(),V2) -> U202(isBool(V2)) 0.68/0.58 23: U202(tt()) -> tt() 0.68/0.58 24: U21(tt(),A,B,C) -> U22(isBool(B),A,B,C) 0.68/0.58 25: U211(tt()) -> tt() 0.68/0.58 26: U22(tt(),A,B,C) -> U23(isBool(C),A,B,C) 0.68/0.58 27: U221(tt(),A) -> _xor_(A,true()) 0.68/0.58 28: U23(tt(),A,B,C) -> _xor_(_and_(A,B),_and_(A,C)) 0.68/0.58 29: U31(tt()) -> false() 0.68/0.58 30: U41(tt(),A) -> A 0.68/0.58 31: U51(tt(),A,B) -> U52(isBool(B),A,B) 0.68/0.58 32: U52(tt(),A,B) -> not_(_xor_(A,_and_(A,B))) 0.68/0.58 33: U61^1_tt(U',U) -> U62^1_isS(U,U',U) 0.68/0.58 34: U62^1_tt(U',U) -> U63(equal(_isNotEqualTo_(U,U'),true())) 0.68/0.58 35: U63(tt()) -> false() 0.68/0.58 36: U71^1_tt() -> true() 0.68/0.58 37: U81^1_tt(U',U) -> U82^1_isS(U,U',U) 0.68/0.58 38: U82^1_tt(U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false(),true()) 0.68/0.58 39: U91^1_tt() -> false() 0.68/0.58 40: _and_(A,A) -> U11(isBool(A),A) 0.68/0.58 41: _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C) 0.68/0.58 42: _and_(false(),A) -> U31(isBool(A)) 0.68/0.58 43: _and_(true(),A) -> U41(isBool(A),A) 0.68/0.58 44: _implies_(A,B) -> U51(isBool(A),A,B) 0.68/0.58 45: _isEqualTo_(U,U') -> U61^1_isS(U',U',U) 0.68/0.58 46: _isEqualTo_(U,U) -> U71^1_isS(U) 0.68/0.58 47: _isNotEqualTo_(U,U') -> U81^1_isS(U',U',U) 0.68/0.58 48: _isNotEqualTo_(U,U) -> U91^1_isS(U) 0.68/0.58 49: _or_(A,B) -> U101(isBool(A),A,B) 0.68/0.58 50: _xor_(A,A) -> U111(isBool(A)) 0.68/0.58 51: _xor_(false(),A) -> U121(isBool(A),A) 0.68/0.58 52: equal(X,X) -> tt() 0.68/0.58 53: if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U) 0.68/0.58 54: if_then_else_fi(true(),U,U') -> U141^1_isS(U',U) 0.68/0.58 55: isBool(false()) -> tt() 0.68/0.58 56: isBool(true()) -> tt() 0.68/0.58 57: isBool(_and_(V1,V2)) -> U151(isBool(V1),V2) 0.68/0.58 58: isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2) 0.68/0.58 59: isBool(_isEqualTo_(V1,V2)) -> U171^1_isUniversal(V1,V2) 0.68/0.58 60: isBool(_isNotEqualTo_(V1,V2)) -> U181^1_isUniversal(V1,V2) 0.68/0.58 61: isBool(_or_(V1,V2)) -> U191(isBool(V1),V2) 0.68/0.58 62: isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2) 0.68/0.58 63: isBool(not_(V1)) -> U211(isBool(V1)) 0.68/0.58 64: not_(A) -> U221(isBool(A),A) 0.68/0.58 65: not_(false()) -> true() 0.68/0.58 66: not_(true()) -> false() 0.68/0.58 67: U61(tt(),_2,_3) ->= U61^1_tt(_2,_3) 0.68/0.58 68: U61(isS(_1),_3,_4) ->= U61^1_isS(_1,_3,_4) 0.68/0.58 69: U141(tt(),_1) ->= U141^1_tt(_1) 0.68/0.58 70: U141(isS(_1),_2) ->= U141^1_isS(_1,_2) 0.68/0.58 71: U132(tt(),_3,_4,_5) ->= U132^1_tt(_3,_4,_5) 0.68/0.58 72: U132(isS(_1),_4,_5,_6) ->= U132^1_isS(_1,_4,_5,_6) 0.68/0.58 73: U182(tt()) ->= U182^1_tt() 0.68/0.58 74: U182(isUniversal(_1)) ->= U182^1_isUniversal(_1) 0.68/0.58 75: U171(tt(),_1) ->= U171^1_tt(_1) 0.68/0.58 76: U171(isUniversal(_1),_2) ->= U171^1_isUniversal(_1,_2) 0.68/0.58 77: U81(tt(),_2,_3) ->= U81^1_tt(_2,_3) 0.68/0.58 78: U81(isS(_1),_3,_4) ->= U81^1_isS(_1,_3,_4) 0.68/0.58 79: U82(tt(),_2,_3) ->= U82^1_tt(_2,_3) 0.68/0.58 80: U82(isS(_1),_3,_4) ->= U82^1_isS(_1,_3,_4) 0.68/0.58 81: U172(tt()) ->= U172^1_tt() 0.68/0.58 82: U172(isUniversal(_1)) ->= U172^1_isUniversal(_1) 0.68/0.58 83: U62(tt(),_2,_3) ->= U62^1_tt(_2,_3) 0.68/0.58 84: U62(isS(_1),_3,_4) ->= U62^1_isS(_1,_3,_4) 0.68/0.58 85: U142(tt(),_1) ->= U142^1_tt(_1) 0.68/0.58 86: U142(isS(_1),_2) ->= U142^1_isS(_1,_2) 0.68/0.58 87: U133(tt(),_2,_3) ->= U133^1_tt(_2,_3) 0.68/0.58 88: U133(isS(_1),_3,_4) ->= U133^1_isS(_1,_3,_4) 0.68/0.58 89: U91(tt()) ->= U91^1_tt() 0.68/0.58 90: U91(isS(_1)) ->= U91^1_isS(_1) 0.68/0.58 91: U71(tt()) ->= U71^1_tt() 0.68/0.58 92: U71(isS(_1)) ->= U71^1_isS(_1) 0.68/0.58 93: U181(tt(),_1) ->= U181^1_tt(_1) 0.68/0.58 94: U181(isUniversal(_1),_2) ->= U181^1_isUniversal(_1,_2) 0.68/0.58 Number of strict rules: 66 0.68/0.58 Direct POLO(bPol) ... failed. 0.68/0.58 Dependency Pairs: 0.68/0.58 #1: #U102(tt(),A,B) -> #_xor_(_and_(A,B),_xor_(A,B)) 0.68/0.58 #2: #U102(tt(),A,B) -> #_and_(A,B) 0.68/0.58 #3: #U102(tt(),A,B) -> #_xor_(A,B) 0.68/0.58 #4: #_and_(true(),A) -> #U41(isBool(A),A) 0.68/0.58 #5: #_and_(true(),A) -> #isBool(A) 0.68/0.58 #6: #U62(tt(),_2,_3) ->? #U62^1_tt(_2,_3) 0.68/0.58 #7: #_and_(false(),A) -> #U31(isBool(A)) 0.68/0.58 #8: #_and_(false(),A) -> #isBool(A) 0.68/0.58 #9: #_or_(x,_or_(y,z)) ->= #_or_(_or_(x,y),z) 0.68/0.58 #10: #_or_(x,_or_(y,z)) ->= #_or_(x,y) 0.68/0.58 #11: #_and_(A,_xor_(B,C)) -> #U21(isBool(A),A,B,C) 0.68/0.58 #12: #_and_(A,_xor_(B,C)) -> #isBool(A) 0.68/0.58 #13: #_and_(x,_and_(y,z)) ->= #_and_(_and_(x,y),z) 0.68/0.58 #14: #_and_(x,_and_(y,z)) ->= #_and_(x,y) 0.68/0.58 #15: #U71(tt()) ->? #U71^1_tt() 0.68/0.58 #16: #U182(tt()) ->? #U182^1_tt() 0.68/0.58 #17: #if_then_else_fi(B,U,U') -> #U131(isBool(B),B,U',U) 0.68/0.58 #18: #if_then_else_fi(B,U,U') -> #isBool(B) 0.68/0.58 #19: #U132(tt(),_3,_4,_5) ->? #U132^1_tt(_3,_4,_5) 0.68/0.58 #20: #U171(tt(),_1) ->? #U171^1_tt(_1) 0.68/0.58 #21: #isBool(_implies_(V1,V2)) -> #U161(isBool(V1),V2) 0.68/0.58 #22: #isBool(_implies_(V1,V2)) -> #isBool(V1) 0.68/0.58 #23: #isBool(_or_(V1,V2)) -> #U191(isBool(V1),V2) 0.68/0.58 #24: #isBool(_or_(V1,V2)) -> #isBool(V1) 0.68/0.58 #25: #U82^1_tt(U',U) -> #if_then_else_fi(_isEqualTo_(U,U'),false(),true()) 0.68/0.58 #26: #U82^1_tt(U',U) -> #_isEqualTo_(U,U') 0.68/0.58 #27: #U61(tt(),_2,_3) ->? #U61^1_tt(_2,_3) 0.68/0.58 #28: #_and_(A,A) -> #U11(isBool(A),A) 0.68/0.58 #29: #_and_(A,A) -> #isBool(A) 0.68/0.58 #30: #_xor_(false(),A) -> #U121(isBool(A),A) 0.68/0.58 #31: #_xor_(false(),A) -> #isBool(A) 0.68/0.58 #32: #isBool(_and_(V1,V2)) -> #U151(isBool(V1),V2) 0.68/0.58 #33: #isBool(_and_(V1,V2)) -> #isBool(V1) 0.68/0.58 #34: #U21(tt(),A,B,C) -> #U22(isBool(B),A,B,C) 0.68/0.58 #35: #U21(tt(),A,B,C) -> #isBool(B) 0.68/0.58 #36: #U172(tt()) ->? #U172^1_tt() 0.68/0.58 #37: #_xor_(x,_xor_(y,z)) ->= #_xor_(_xor_(x,y),z) 0.68/0.58 #38: #_xor_(x,_xor_(y,z)) ->= #_xor_(x,y) 0.68/0.58 #39: #U141(tt(),_1) ->? #U141^1_tt(_1) 0.68/0.58 #40: #U151(tt(),V2) -> #U152(isBool(V2)) 0.68/0.58 #41: #U151(tt(),V2) -> #isBool(V2) 0.68/0.58 #42: #U51(tt(),A,B) -> #U52(isBool(B),A,B) 0.68/0.58 #43: #U51(tt(),A,B) -> #isBool(B) 0.68/0.58 #44: #U82(tt(),_2,_3) ->? #U82^1_tt(_2,_3) 0.68/0.58 #45: #U91(tt()) ->? #U91^1_tt() 0.68/0.58 #46: #U161(tt(),V2) -> #U162(isBool(V2)) 0.68/0.58 #47: #U161(tt(),V2) -> #isBool(V2) 0.68/0.58 #48: #isBool(_xor_(V1,V2)) -> #U201(isBool(V1),V2) 0.68/0.58 #49: #isBool(_xor_(V1,V2)) -> #isBool(V1) 0.68/0.58 #50: #_or_(A,B) -> #U101(isBool(A),A,B) 0.68/0.58 #51: #_or_(A,B) -> #isBool(A) 0.68/0.58 #52: #U191(tt(),V2) -> #U192(isBool(V2)) 0.68/0.58 #53: #U191(tt(),V2) -> #isBool(V2) 0.68/0.58 #54: #U181(tt(),_1) ->? #U181^1_tt(_1) 0.68/0.58 #55: #not_(A) -> #U221(isBool(A),A) 0.68/0.58 #56: #not_(A) -> #isBool(A) 0.68/0.58 #57: #_implies_(A,B) -> #U51(isBool(A),A,B) 0.68/0.58 #58: #_implies_(A,B) -> #isBool(A) 0.68/0.58 #59: #U23(tt(),A,B,C) -> #_xor_(_and_(A,B),_and_(A,C)) 0.68/0.58 #60: #U23(tt(),A,B,C) -> #_and_(A,B) 0.68/0.58 #61: #U23(tt(),A,B,C) -> #_and_(A,C) 0.68/0.58 #62: #U201(tt(),V2) -> #U202(isBool(V2)) 0.68/0.58 #63: #U201(tt(),V2) -> #isBool(V2) 0.68/0.58 #64: #U62^1_tt(U',U) -> #U63(equal(_isNotEqualTo_(U,U'),true())) 0.68/0.58 #65: #U62^1_tt(U',U) -> #equal(_isNotEqualTo_(U,U'),true()) 0.68/0.58 #66: #U62^1_tt(U',U) -> #_isNotEqualTo_(U,U') 0.68/0.58 #67: #U133(tt(),_2,_3) ->? #U133^1_tt(_2,_3) 0.68/0.58 #68: #U221(tt(),A) -> #_xor_(A,true()) 0.68/0.58 #69: #U52(tt(),A,B) -> #not_(_xor_(A,_and_(A,B))) 0.68/0.58 #70: #U52(tt(),A,B) -> #_xor_(A,_and_(A,B)) 0.68/0.58 #71: #U52(tt(),A,B) -> #_and_(A,B) 0.68/0.58 #72: #isBool(not_(V1)) -> #U211(isBool(V1)) 0.68/0.58 #73: #isBool(not_(V1)) -> #isBool(V1) 0.68/0.58 #74: #U22(tt(),A,B,C) -> #U23(isBool(C),A,B,C) 0.68/0.58 #75: #U22(tt(),A,B,C) -> #isBool(C) 0.68/0.58 #76: #U142(tt(),_1) ->? #U142^1_tt(_1) 0.68/0.58 #77: #U81(tt(),_2,_3) ->? #U81^1_tt(_2,_3) 0.68/0.58 #78: #U101(tt(),A,B) -> #U102(isBool(B),A,B) 0.68/0.58 #79: #U101(tt(),A,B) -> #isBool(B) 0.68/0.58 #80: #U133^1_tt(B,U') -> #U134(equal(_isNotEqualTo_(B,true()),true()),U') 0.68/0.58 #81: #U133^1_tt(B,U') -> #equal(_isNotEqualTo_(B,true()),true()) 0.68/0.58 #82: #U133^1_tt(B,U') -> #_isNotEqualTo_(B,true()) 0.68/0.58 #83: #_xor_(A,A) -> #U111(isBool(A)) 0.68/0.58 #84: #_xor_(A,A) -> #isBool(A) 0.68/0.58 Number of SCCs: 4, DPs: 24 0.68/0.58 SCC { #9 #10 } 0.68/0.58 only weak rules. 0.68/0.58 Number of SCCs: 3, DPs: 22 0.68/0.58 SCC { #37 #38 } 0.68/0.58 only weak rules. 0.68/0.58 Number of SCCs: 2, DPs: 20 0.68/0.58 SCC { #11 #13 #14 #34 #60 #61 #74 } 0.68/0.58 POLO(Sum)... POLO(max)... QLPOS... POLO(mSum)... QWPOpS(mSum)... Mat2b... failed. 0.68/0.58 Finding a loop... failed. 0.68/0.58 EOF