Input TRS: 1: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2)) 2: active(U12(tt(),V2)) -> mark(U13(isNat(V2))) 3: active(U13(tt())) -> mark(tt()) 4: active(U21(tt(),V1)) -> mark(U22(isNat(V1))) 5: active(U22(tt())) -> mark(tt()) 6: active(U31(tt(),N)) -> mark(N) 7: active(U41(tt(),M,N)) -> mark(s(plus(N,M))) 8: active(and(tt(),X)) -> mark(X) 9: active(isNat(0())) -> mark(tt()) 10: active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) 11: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) 12: active(isNatKind(0())) -> mark(tt()) 13: active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2))) 14: active(isNatKind(s(V1))) -> mark(isNatKind(V1)) 15: active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N)) 16: active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) 17: mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) 18: mark(tt()) -> active(tt()) 19: mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) 20: mark(isNat(X)) -> active(isNat(X)) 21: mark(U13(X)) -> active(U13(mark(X))) 22: mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) 23: mark(U22(X)) -> active(U22(mark(X))) 24: mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) 25: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) 26: mark(s(X)) -> active(s(mark(X))) 27: mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) 28: mark(and(X1,X2)) -> active(and(mark(X1),X2)) 29: mark(0()) -> active(0()) 30: mark(isNatKind(X)) -> active(isNatKind(X)) 31: U11(mark(X1),X2,X3) -> U11(X1,X2,X3) 32: U11(X1,mark(X2),X3) -> U11(X1,X2,X3) 33: U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) 34: U11(active(X1),X2,X3) -> U11(X1,X2,X3) 35: U11(X1,active(X2),X3) -> U11(X1,X2,X3) 36: U11(X1,X2,active(X3)) -> U11(X1,X2,X3) 37: U12(mark(X1),X2) -> U12(X1,X2) 38: U12(X1,mark(X2)) -> U12(X1,X2) 39: U12(active(X1),X2) -> U12(X1,X2) 40: U12(X1,active(X2)) -> U12(X1,X2) 41: isNat(mark(X)) -> isNat(X) 42: isNat(active(X)) -> isNat(X) 43: U13(mark(X)) -> U13(X) 44: U13(active(X)) -> U13(X) 45: U21(mark(X1),X2) -> U21(X1,X2) 46: U21(X1,mark(X2)) -> U21(X1,X2) 47: U21(active(X1),X2) -> U21(X1,X2) 48: U21(X1,active(X2)) -> U21(X1,X2) 49: U22(mark(X)) -> U22(X) 50: U22(active(X)) -> U22(X) 51: U31(mark(X1),X2) -> U31(X1,X2) 52: U31(X1,mark(X2)) -> U31(X1,X2) 53: U31(active(X1),X2) -> U31(X1,X2) 54: U31(X1,active(X2)) -> U31(X1,X2) 55: U41(mark(X1),X2,X3) -> U41(X1,X2,X3) 56: U41(X1,mark(X2),X3) -> U41(X1,X2,X3) 57: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) 58: U41(active(X1),X2,X3) -> U41(X1,X2,X3) 59: U41(X1,active(X2),X3) -> U41(X1,X2,X3) 60: U41(X1,X2,active(X3)) -> U41(X1,X2,X3) 61: s(mark(X)) -> s(X) 62: s(active(X)) -> s(X) 63: plus(mark(X1),X2) -> plus(X1,X2) 64: plus(X1,mark(X2)) -> plus(X1,X2) 65: plus(active(X1),X2) -> plus(X1,X2) 66: plus(X1,active(X2)) -> plus(X1,X2) 67: and(mark(X1),X2) -> and(X1,X2) 68: and(X1,mark(X2)) -> and(X1,X2) 69: and(active(X1),X2) -> and(X1,X2) 70: and(X1,active(X2)) -> and(X1,X2) 71: isNatKind(mark(X)) -> isNatKind(X) 72: isNatKind(active(X)) -> isNatKind(X) Number of strict rules: 72 Direct Order(PosReal,>,Poly) ... failed. Freezing ... failed. Dependency Pairs: #1: #active(U12(tt(),V2)) -> #mark(U13(isNat(V2))) #2: #active(U12(tt(),V2)) -> #U13(isNat(V2)) #3: #active(U12(tt(),V2)) -> #isNat(V2) #4: #U13(mark(X)) -> #U13(X) #5: #mark(0()) -> #active(0()) #6: #U11(X1,active(X2),X3) -> #U11(X1,X2,X3) #7: #plus(X1,active(X2)) -> #plus(X1,X2) #8: #U21(X1,mark(X2)) -> #U21(X1,X2) #9: #isNat(active(X)) -> #isNat(X) #10: #isNat(mark(X)) -> #isNat(X) #11: #U12(mark(X1),X2) -> #U12(X1,X2) #12: #U21(active(X1),X2) -> #U21(X1,X2) #13: #U31(active(X1),X2) -> #U31(X1,X2) #14: #isNatKind(mark(X)) -> #isNatKind(X) #15: #U21(X1,active(X2)) -> #U21(X1,X2) #16: #U41(active(X1),X2,X3) -> #U41(X1,X2,X3) #17: #s(mark(X)) -> #s(X) #18: #U12(X1,mark(X2)) -> #U12(X1,X2) #19: #active(U31(tt(),N)) -> #mark(N) #20: #U41(X1,active(X2),X3) -> #U41(X1,X2,X3) #21: #U41(mark(X1),X2,X3) -> #U41(X1,X2,X3) #22: #and(mark(X1),X2) -> #and(X1,X2) #23: #U12(X1,active(X2)) -> #U12(X1,X2) #24: #U31(mark(X1),X2) -> #U31(X1,X2) #25: #active(isNatKind(plus(V1,V2))) -> #mark(and(isNatKind(V1),isNatKind(V2))) #26: #active(isNatKind(plus(V1,V2))) -> #and(isNatKind(V1),isNatKind(V2)) #27: #active(isNatKind(plus(V1,V2))) -> #isNatKind(V1) #28: #active(isNatKind(plus(V1,V2))) -> #isNatKind(V2) #29: #active(isNat(0())) -> #mark(tt()) #30: #active(isNat(s(V1))) -> #mark(U21(isNatKind(V1),V1)) #31: #active(isNat(s(V1))) -> #U21(isNatKind(V1),V1) #32: #active(isNat(s(V1))) -> #isNatKind(V1) #33: #U41(X1,X2,mark(X3)) -> #U41(X1,X2,X3) #34: #mark(U31(X1,X2)) -> #active(U31(mark(X1),X2)) #35: #mark(U31(X1,X2)) -> #U31(mark(X1),X2) #36: #mark(U31(X1,X2)) -> #mark(X1) #37: #and(X1,active(X2)) -> #and(X1,X2) #38: #mark(U22(X)) -> #active(U22(mark(X))) #39: #mark(U22(X)) -> #U22(mark(X)) #40: #mark(U22(X)) -> #mark(X) #41: #U21(mark(X1),X2) -> #U21(X1,X2) #42: #and(active(X1),X2) -> #and(X1,X2) #43: #active(isNatKind(0())) -> #mark(tt()) #44: #U11(mark(X1),X2,X3) -> #U11(X1,X2,X3) #45: #U41(X1,mark(X2),X3) -> #U41(X1,X2,X3) #46: #active(isNatKind(s(V1))) -> #mark(isNatKind(V1)) #47: #active(isNatKind(s(V1))) -> #isNatKind(V1) #48: #s(active(X)) -> #s(X) #49: #mark(isNatKind(X)) -> #active(isNatKind(X)) #50: #U31(X1,mark(X2)) -> #U31(X1,X2) #51: #U22(mark(X)) -> #U22(X) #52: #mark(U41(X1,X2,X3)) -> #active(U41(mark(X1),X2,X3)) #53: #mark(U41(X1,X2,X3)) -> #U41(mark(X1),X2,X3) #54: #mark(U41(X1,X2,X3)) -> #mark(X1) #55: #mark(isNat(X)) -> #active(isNat(X)) #56: #active(U41(tt(),M,N)) -> #mark(s(plus(N,M))) #57: #active(U41(tt(),M,N)) -> #s(plus(N,M)) #58: #active(U41(tt(),M,N)) -> #plus(N,M) #59: #U12(active(X1),X2) -> #U12(X1,X2) #60: #active(isNat(plus(V1,V2))) -> #mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)) #61: #active(isNat(plus(V1,V2))) -> #U11(and(isNatKind(V1),isNatKind(V2)),V1,V2) #62: #active(isNat(plus(V1,V2))) -> #and(isNatKind(V1),isNatKind(V2)) #63: #active(isNat(plus(V1,V2))) -> #isNatKind(V1) #64: #active(isNat(plus(V1,V2))) -> #isNatKind(V2) #65: #plus(X1,mark(X2)) -> #plus(X1,X2) #66: #U11(X1,X2,mark(X3)) -> #U11(X1,X2,X3) #67: #isNatKind(active(X)) -> #isNatKind(X) #68: #active(U22(tt())) -> #mark(tt()) #69: #U13(active(X)) -> #U13(X) #70: #plus(active(X1),X2) -> #plus(X1,X2) #71: #mark(and(X1,X2)) -> #active(and(mark(X1),X2)) #72: #mark(and(X1,X2)) -> #and(mark(X1),X2) #73: #mark(and(X1,X2)) -> #mark(X1) #74: #mark(U21(X1,X2)) -> #active(U21(mark(X1),X2)) #75: #mark(U21(X1,X2)) -> #U21(mark(X1),X2) #76: #mark(U21(X1,X2)) -> #mark(X1) #77: #U11(active(X1),X2,X3) -> #U11(X1,X2,X3) #78: #mark(plus(X1,X2)) -> #active(plus(mark(X1),mark(X2))) #79: #mark(plus(X1,X2)) -> #plus(mark(X1),mark(X2)) #80: #mark(plus(X1,X2)) -> #mark(X1) #81: #mark(plus(X1,X2)) -> #mark(X2) #82: #U41(X1,X2,active(X3)) -> #U41(X1,X2,X3) #83: #mark(U11(X1,X2,X3)) -> #active(U11(mark(X1),X2,X3)) #84: #mark(U11(X1,X2,X3)) -> #U11(mark(X1),X2,X3) #85: #mark(U11(X1,X2,X3)) -> #mark(X1) #86: #U11(X1,mark(X2),X3) -> #U11(X1,X2,X3) #87: #mark(U12(X1,X2)) -> #active(U12(mark(X1),X2)) #88: #mark(U12(X1,X2)) -> #U12(mark(X1),X2) #89: #mark(U12(X1,X2)) -> #mark(X1) #90: #plus(mark(X1),X2) -> #plus(X1,X2) #91: #mark(s(X)) -> #active(s(mark(X))) #92: #mark(s(X)) -> #s(mark(X)) #93: #mark(s(X)) -> #mark(X) #94: #and(X1,mark(X2)) -> #and(X1,X2) #95: #U11(X1,X2,active(X3)) -> #U11(X1,X2,X3) #96: #mark(U13(X)) -> #active(U13(mark(X))) #97: #mark(U13(X)) -> #U13(mark(X)) #98: #mark(U13(X)) -> #mark(X) #99: #active(plus(N,s(M))) -> #mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)) #100: #active(plus(N,s(M))) -> #U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) #101: #active(plus(N,s(M))) -> #and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) #102: #active(plus(N,s(M))) -> #and(isNat(M),isNatKind(M)) #103: #active(plus(N,s(M))) -> #isNat(M) #104: #active(plus(N,s(M))) -> #isNatKind(M) #105: #active(plus(N,s(M))) -> #and(isNat(N),isNatKind(N)) #106: #active(plus(N,s(M))) -> #isNat(N) #107: #active(plus(N,s(M))) -> #isNatKind(N) #108: #active(U13(tt())) -> #mark(tt()) #109: #active(U11(tt(),V1,V2)) -> #mark(U12(isNat(V1),V2)) #110: #active(U11(tt(),V1,V2)) -> #U12(isNat(V1),V2) #111: #active(U11(tt(),V1,V2)) -> #isNat(V1) #112: #U31(X1,active(X2)) -> #U31(X1,X2) #113: #active(and(tt(),X)) -> #mark(X) #114: #active(plus(N,0())) -> #mark(U31(and(isNat(N),isNatKind(N)),N)) #115: #active(plus(N,0())) -> #U31(and(isNat(N),isNatKind(N)),N) #116: #active(plus(N,0())) -> #and(isNat(N),isNatKind(N)) #117: #active(plus(N,0())) -> #isNat(N) #118: #active(plus(N,0())) -> #isNatKind(N) #119: #active(U21(tt(),V1)) -> #mark(U22(isNat(V1))) #120: #active(U21(tt(),V1)) -> #U22(isNat(V1)) #121: #active(U21(tt(),V1)) -> #isNat(V1) #122: #U22(active(X)) -> #U22(X) #123: #mark(tt()) -> #active(tt()) Number of SCCs: 13, DPs: 74, edges: 460 SCC { #51 #122 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: 0 U11(x1,x2,x3) weight: 0 s(x1) weight: 0 #isNat(x1) weight: 0 and(x1,x2) weight: 0 #plus(x1,x2) weight: 0 #U13(x1) weight: 0 U12(x1,x2) weight: 0 #U12(x1,x2) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 isNat(x1) weight: 0 plus(x1,x2) weight: 0 #U11(x1,x2,x3) weight: 0 active(x1) weight: (/ 1 2) + x1 U31(x1,x2) weight: 0 #U41(x1,x2,x3) weight: 0 #active(x1) weight: 0 #U21(x1,x2) weight: 0 #U22(x1) weight: x1 tt() weight: 0 U13(x1) weight: 0 U22(x1) weight: 0 #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: 0 #U31(x1,x2) weight: 0 #and(x1,x2) weight: 0 Usable rules: { } Removed DPs: #51 #122 Number of SCCs: 12, DPs: 72, edges: 456 SCC { #9 #10 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: 0 U11(x1,x2,x3) weight: 0 s(x1) weight: 0 #isNat(x1) weight: x1 and(x1,x2) weight: 0 #plus(x1,x2) weight: 0 #U13(x1) weight: 0 U12(x1,x2) weight: 0 #U12(x1,x2) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 isNat(x1) weight: 0 plus(x1,x2) weight: 0 #U11(x1,x2,x3) weight: 0 active(x1) weight: (/ 1 2) + x1 U31(x1,x2) weight: 0 #U41(x1,x2,x3) weight: 0 #active(x1) weight: 0 #U21(x1,x2) weight: 0 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: 0 U22(x1) weight: 0 #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: 0 #U31(x1,x2) weight: 0 #and(x1,x2) weight: 0 Usable rules: { } Removed DPs: #9 #10 Number of SCCs: 11, DPs: 70, edges: 452 SCC { #4 #69 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: 0 U11(x1,x2,x3) weight: 0 s(x1) weight: 0 #isNat(x1) weight: 0 and(x1,x2) weight: 0 #plus(x1,x2) weight: 0 #U13(x1) weight: x1 U12(x1,x2) weight: 0 #U12(x1,x2) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 isNat(x1) weight: 0 plus(x1,x2) weight: 0 #U11(x1,x2,x3) weight: 0 active(x1) weight: (/ 1 2) + x1 U31(x1,x2) weight: 0 #U41(x1,x2,x3) weight: 0 #active(x1) weight: 0 #U21(x1,x2) weight: 0 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: 0 U22(x1) weight: 0 #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: 0 #U31(x1,x2) weight: 0 #and(x1,x2) weight: 0 Usable rules: { } Removed DPs: #4 #69 Number of SCCs: 10, DPs: 68, edges: 448 SCC { #17 #48 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: 0 U11(x1,x2,x3) weight: 0 s(x1) weight: 0 #isNat(x1) weight: 0 and(x1,x2) weight: 0 #plus(x1,x2) weight: 0 #U13(x1) weight: 0 U12(x1,x2) weight: 0 #U12(x1,x2) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: x1 mark(x1) weight: (/ 1 2) + x1 isNat(x1) weight: 0 plus(x1,x2) weight: 0 #U11(x1,x2,x3) weight: 0 active(x1) weight: (/ 1 2) + x1 U31(x1,x2) weight: 0 #U41(x1,x2,x3) weight: 0 #active(x1) weight: 0 #U21(x1,x2) weight: 0 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: 0 U22(x1) weight: 0 #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: 0 #U31(x1,x2) weight: 0 #and(x1,x2) weight: 0 Usable rules: { } Removed DPs: #17 #48 Number of SCCs: 9, DPs: 66, edges: 444 SCC { #14 #67 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: 0 U11(x1,x2,x3) weight: 0 s(x1) weight: 0 #isNat(x1) weight: 0 and(x1,x2) weight: 0 #plus(x1,x2) weight: 0 #U13(x1) weight: 0 U12(x1,x2) weight: 0 #U12(x1,x2) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 isNat(x1) weight: 0 plus(x1,x2) weight: 0 #U11(x1,x2,x3) weight: 0 active(x1) weight: (/ 1 2) + x1 U31(x1,x2) weight: 0 #U41(x1,x2,x3) weight: 0 #active(x1) weight: 0 #U21(x1,x2) weight: 0 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: 0 U22(x1) weight: 0 #isNatKind(x1) weight: x1 U41(x1,x2,x3) weight: 0 #U31(x1,x2) weight: 0 #and(x1,x2) weight: 0 Usable rules: { } Removed DPs: #14 #67 Number of SCCs: 8, DPs: 64, edges: 440 SCC { #13 #24 #50 #112 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: 0 U11(x1,x2,x3) weight: 0 s(x1) weight: 0 #isNat(x1) weight: 0 and(x1,x2) weight: 0 #plus(x1,x2) weight: 0 #U13(x1) weight: 0 U12(x1,x2) weight: 0 #U12(x1,x2) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 isNat(x1) weight: 0 plus(x1,x2) weight: 0 #U11(x1,x2,x3) weight: 0 active(x1) weight: (/ 1 2) + x1 U31(x1,x2) weight: 0 #U41(x1,x2,x3) weight: 0 #active(x1) weight: 0 #U21(x1,x2) weight: 0 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: 0 U22(x1) weight: 0 #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: 0 #U31(x1,x2) weight: x1 #and(x1,x2) weight: 0 Usable rules: { } Removed DPs: #13 #24 Number of SCCs: 8, DPs: 62, edges: 428 SCC { #50 #112 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: 0 U11(x1,x2,x3) weight: 0 s(x1) weight: 0 #isNat(x1) weight: 0 and(x1,x2) weight: 0 #plus(x1,x2) weight: 0 #U13(x1) weight: 0 U12(x1,x2) weight: 0 #U12(x1,x2) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 isNat(x1) weight: 0 plus(x1,x2) weight: 0 #U11(x1,x2,x3) weight: 0 active(x1) weight: (/ 1 2) + x1 U31(x1,x2) weight: 0 #U41(x1,x2,x3) weight: 0 #active(x1) weight: 0 #U21(x1,x2) weight: 0 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: 0 U22(x1) weight: 0 #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: 0 #U31(x1,x2) weight: x2 #and(x1,x2) weight: 0 Usable rules: { } Removed DPs: #50 #112 Number of SCCs: 7, DPs: 60, edges: 424 SCC { #7 #65 #70 #90 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: 0 U11(x1,x2,x3) weight: 0 s(x1) weight: 0 #isNat(x1) weight: 0 and(x1,x2) weight: 0 #plus(x1,x2) weight: x1 #U13(x1) weight: 0 U12(x1,x2) weight: 0 #U12(x1,x2) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 isNat(x1) weight: 0 plus(x1,x2) weight: 0 #U11(x1,x2,x3) weight: 0 active(x1) weight: (/ 1 2) + x1 U31(x1,x2) weight: 0 #U41(x1,x2,x3) weight: 0 #active(x1) weight: 0 #U21(x1,x2) weight: 0 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: 0 U22(x1) weight: 0 #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: 0 #U31(x1,x2) weight: 0 #and(x1,x2) weight: 0 Usable rules: { } Removed DPs: #70 #90 Number of SCCs: 8, DPs: 58, edges: 412 SCC { #7 #65 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: 0 U11(x1,x2,x3) weight: 0 s(x1) weight: 0 #isNat(x1) weight: 0 and(x1,x2) weight: 0 #plus(x1,x2) weight: x2 #U13(x1) weight: 0 U12(x1,x2) weight: 0 #U12(x1,x2) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 isNat(x1) weight: 0 plus(x1,x2) weight: 0 #U11(x1,x2,x3) weight: 0 active(x1) weight: (/ 1 2) + x1 U31(x1,x2) weight: 0 #U41(x1,x2,x3) weight: 0 #active(x1) weight: 0 #U21(x1,x2) weight: 0 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: 0 U22(x1) weight: 0 #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: 0 #U31(x1,x2) weight: 0 #and(x1,x2) weight: 0 Usable rules: { } Removed DPs: #7 #65 Number of SCCs: 7, DPs: 56, edges: 408 SCC { #11 #18 #23 #59 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: 0 U11(x1,x2,x3) weight: 0 s(x1) weight: 0 #isNat(x1) weight: 0 and(x1,x2) weight: 0 #plus(x1,x2) weight: 0 #U13(x1) weight: 0 U12(x1,x2) weight: 0 #U12(x1,x2) weight: x2 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 isNat(x1) weight: 0 plus(x1,x2) weight: 0 #U11(x1,x2,x3) weight: 0 active(x1) weight: (/ 1 2) + x1 U31(x1,x2) weight: 0 #U41(x1,x2,x3) weight: 0 #active(x1) weight: 0 #U21(x1,x2) weight: 0 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: 0 U22(x1) weight: 0 #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: 0 #U31(x1,x2) weight: 0 #and(x1,x2) weight: 0 Usable rules: { } Removed DPs: #18 #23 Number of SCCs: 8, DPs: 54, edges: 396 SCC { #11 #59 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: 0 U11(x1,x2,x3) weight: 0 s(x1) weight: 0 #isNat(x1) weight: 0 and(x1,x2) weight: 0 #plus(x1,x2) weight: 0 #U13(x1) weight: 0 U12(x1,x2) weight: 0 #U12(x1,x2) weight: x1 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 isNat(x1) weight: 0 plus(x1,x2) weight: 0 #U11(x1,x2,x3) weight: 0 active(x1) weight: (/ 1 2) + x1 U31(x1,x2) weight: 0 #U41(x1,x2,x3) weight: 0 #active(x1) weight: 0 #U21(x1,x2) weight: 0 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: 0 U22(x1) weight: 0 #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: 0 #U31(x1,x2) weight: 0 #and(x1,x2) weight: 0 Usable rules: { } Removed DPs: #11 #59 Number of SCCs: 7, DPs: 52, edges: 392 SCC { #8 #12 #15 #41 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: 0 U11(x1,x2,x3) weight: 0 s(x1) weight: 0 #isNat(x1) weight: 0 and(x1,x2) weight: 0 #plus(x1,x2) weight: 0 #U13(x1) weight: 0 U12(x1,x2) weight: 0 #U12(x1,x2) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 isNat(x1) weight: 0 plus(x1,x2) weight: 0 #U11(x1,x2,x3) weight: 0 active(x1) weight: (/ 1 2) + x1 U31(x1,x2) weight: 0 #U41(x1,x2,x3) weight: 0 #active(x1) weight: 0 #U21(x1,x2) weight: x2 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: 0 U22(x1) weight: 0 #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: 0 #U31(x1,x2) weight: 0 #and(x1,x2) weight: 0 Usable rules: { } Removed DPs: #8 #15 Number of SCCs: 8, DPs: 50, edges: 380 SCC { #12 #41 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: 0 U11(x1,x2,x3) weight: 0 s(x1) weight: 0 #isNat(x1) weight: 0 and(x1,x2) weight: 0 #plus(x1,x2) weight: 0 #U13(x1) weight: 0 U12(x1,x2) weight: 0 #U12(x1,x2) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 isNat(x1) weight: 0 plus(x1,x2) weight: 0 #U11(x1,x2,x3) weight: 0 active(x1) weight: (/ 1 2) + x1 U31(x1,x2) weight: 0 #U41(x1,x2,x3) weight: 0 #active(x1) weight: 0 #U21(x1,x2) weight: x1 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: 0 U22(x1) weight: 0 #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: 0 #U31(x1,x2) weight: 0 #and(x1,x2) weight: 0 Usable rules: { } Removed DPs: #12 #41 Number of SCCs: 7, DPs: 48, edges: 376 SCC { #22 #37 #42 #94 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: 0 U11(x1,x2,x3) weight: 0 s(x1) weight: 0 #isNat(x1) weight: 0 and(x1,x2) weight: 0 #plus(x1,x2) weight: 0 #U13(x1) weight: 0 U12(x1,x2) weight: 0 #U12(x1,x2) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 isNat(x1) weight: 0 plus(x1,x2) weight: 0 #U11(x1,x2,x3) weight: 0 active(x1) weight: (/ 1 2) + x1 U31(x1,x2) weight: 0 #U41(x1,x2,x3) weight: 0 #active(x1) weight: 0 #U21(x1,x2) weight: 0 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: 0 U22(x1) weight: 0 #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: 0 #U31(x1,x2) weight: 0 #and(x1,x2) weight: x2 Usable rules: { } Removed DPs: #37 #94 Number of SCCs: 8, DPs: 46, edges: 364 SCC { #22 #42 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: 0 U11(x1,x2,x3) weight: 0 s(x1) weight: 0 #isNat(x1) weight: 0 and(x1,x2) weight: 0 #plus(x1,x2) weight: 0 #U13(x1) weight: 0 U12(x1,x2) weight: 0 #U12(x1,x2) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 isNat(x1) weight: 0 plus(x1,x2) weight: 0 #U11(x1,x2,x3) weight: 0 active(x1) weight: (/ 1 2) + x1 U31(x1,x2) weight: 0 #U41(x1,x2,x3) weight: 0 #active(x1) weight: 0 #U21(x1,x2) weight: 0 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: 0 U22(x1) weight: 0 #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: 0 #U31(x1,x2) weight: 0 #and(x1,x2) weight: x1 Usable rules: { } Removed DPs: #22 #42 Number of SCCs: 7, DPs: 44, edges: 360 SCC { #16 #20 #21 #33 #45 #82 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: 0 U11(x1,x2,x3) weight: 0 s(x1) weight: 0 #isNat(x1) weight: 0 and(x1,x2) weight: 0 #plus(x1,x2) weight: 0 #U13(x1) weight: 0 U12(x1,x2) weight: 0 #U12(x1,x2) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 isNat(x1) weight: 0 plus(x1,x2) weight: 0 #U11(x1,x2,x3) weight: 0 active(x1) weight: (/ 1 2) + x1 U31(x1,x2) weight: 0 #U41(x1,x2,x3) weight: x1 + x3 #active(x1) weight: 0 #U21(x1,x2) weight: 0 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: 0 U22(x1) weight: 0 #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: 0 #U31(x1,x2) weight: 0 #and(x1,x2) weight: 0 Usable rules: { } Removed DPs: #16 #21 #33 #82 Number of SCCs: 8, DPs: 40, edges: 328 SCC { #20 #45 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: 0 U11(x1,x2,x3) weight: 0 s(x1) weight: 0 #isNat(x1) weight: 0 and(x1,x2) weight: 0 #plus(x1,x2) weight: 0 #U13(x1) weight: 0 U12(x1,x2) weight: 0 #U12(x1,x2) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 isNat(x1) weight: 0 plus(x1,x2) weight: 0 #U11(x1,x2,x3) weight: 0 active(x1) weight: (/ 1 2) + x1 U31(x1,x2) weight: 0 #U41(x1,x2,x3) weight: x2 #active(x1) weight: 0 #U21(x1,x2) weight: 0 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: 0 U22(x1) weight: 0 #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: 0 #U31(x1,x2) weight: 0 #and(x1,x2) weight: 0 Usable rules: { } Removed DPs: #20 #45 Number of SCCs: 7, DPs: 38, edges: 324 SCC { #6 #44 #66 #77 #86 #95 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: 0 U11(x1,x2,x3) weight: 0 s(x1) weight: 0 #isNat(x1) weight: 0 and(x1,x2) weight: 0 #plus(x1,x2) weight: 0 #U13(x1) weight: 0 U12(x1,x2) weight: 0 #U12(x1,x2) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 isNat(x1) weight: 0 plus(x1,x2) weight: 0 #U11(x1,x2,x3) weight: x1 + x3 active(x1) weight: (/ 1 2) + x1 U31(x1,x2) weight: 0 #U41(x1,x2,x3) weight: 0 #active(x1) weight: 0 #U21(x1,x2) weight: 0 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: 0 U22(x1) weight: 0 #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: 0 #U31(x1,x2) weight: 0 #and(x1,x2) weight: 0 Usable rules: { } Removed DPs: #44 #66 #77 #95 Number of SCCs: 8, DPs: 34, edges: 292 SCC { #6 #86 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: 0 U11(x1,x2,x3) weight: 0 s(x1) weight: 0 #isNat(x1) weight: 0 and(x1,x2) weight: 0 #plus(x1,x2) weight: 0 #U13(x1) weight: 0 U12(x1,x2) weight: 0 #U12(x1,x2) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 isNat(x1) weight: 0 plus(x1,x2) weight: 0 #U11(x1,x2,x3) weight: x2 active(x1) weight: (/ 1 2) + x1 U31(x1,x2) weight: 0 #U41(x1,x2,x3) weight: 0 #active(x1) weight: 0 #U21(x1,x2) weight: 0 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: 0 U22(x1) weight: 0 #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: 0 #U31(x1,x2) weight: 0 #and(x1,x2) weight: 0 Usable rules: { } Removed DPs: #6 #86 Number of SCCs: 7, DPs: 32, edges: 288 SCC { #1 #19 #25 #30 #34 #36 #40 #46 #49 #52 #54..56 #60 #71 #73 #74 #76 #78 #80 #81 #83 #85 #87 #89 #93 #98 #99 #109 #113 #114 #119 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: x1 U11(x1,x2,x3) weight: x1 s(x1) weight: (/ 1 4) + x1 #isNat(x1) weight: 0 and(x1,x2) weight: x1 + x2 #plus(x1,x2) weight: 0 #U13(x1) weight: 0 U12(x1,x2) weight: x1 #U12(x1,x2) weight: 0 #mark(x1) weight: x1 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: x1 isNat(x1) weight: 0 plus(x1,x2) weight: (/ 1 2) + x1 + x2 #U11(x1,x2,x3) weight: 0 active(x1) weight: x1 U31(x1,x2) weight: (/ 1 4) + x1 + x2 #U41(x1,x2,x3) weight: 0 #active(x1) weight: x1 #U21(x1,x2) weight: 0 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: x1 U22(x1) weight: x1 #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: (/ 3 4) + x1 + x2 + x3 #U31(x1,x2) weight: 0 #and(x1,x2) weight: 0 Usable rules: { 1..72 } Removed DPs: #19 #36 #54 #80 #81 #93 #114 Number of SCCs: 8, DPs: 20, edges: 103 SCC { #1 #25 #30 #40 #46 #49 #55 #60 #71 #73 #74 #76 #83 #85 #87 #89 #98 #109 #113 #119 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: 0 U21(x1,x2) weight: (/ 3 16) + x1 + x2 U11(x1,x2,x3) weight: (/ 5 16) + x1 + x2 + x3 s(x1) weight: (/ 3 16) + x1 #isNat(x1) weight: 0 and(x1,x2) weight: x1 + x2 #plus(x1,x2) weight: 0 #U13(x1) weight: 0 U12(x1,x2) weight: (/ 3 16) + x1 + x2 #U12(x1,x2) weight: 0 #mark(x1) weight: x1 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: x1 isNat(x1) weight: (/ 1 16) + x1 plus(x1,x2) weight: (/ 5 16) + x1 + x2 #U11(x1,x2,x3) weight: 0 active(x1) weight: x1 U31(x1,x2) weight: (/ 1 16) + x2 #U41(x1,x2,x3) weight: 0 #active(x1) weight: x1 #U21(x1,x2) weight: 0 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: (/ 1 16) + x1 U22(x1) weight: (/ 1 16) + x1 #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: (/ 1 2) + x2 + x3 #U31(x1,x2) weight: 0 #and(x1,x2) weight: 0 Usable rules: { 1..72 } Removed DPs: #1 #30 #40 #60 #76 #85 #89 #98 #109 #119 Number of SCCs: 8, DPs: 6, edges: 12 SCC { #25 #46 #49 #71 #73 #113 } Removing DPs: Order(PosReal,>,Sum)... succeeded. isNatKind(x1) weight: (/ 1 16) + x1 U21(x1,x2) weight: (/ 1 16) + x1 U11(x1,x2,x3) weight: (/ 1 16) + x1 s(x1) weight: (/ 3 16) + x1 #isNat(x1) weight: 0 and(x1,x2) weight: (/ 1 8) + x1 + x2 #plus(x1,x2) weight: 0 #U13(x1) weight: 0 U12(x1,x2) weight: (/ 1 16) #U12(x1,x2) weight: 0 #mark(x1) weight: (/ 1 16) + x1 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: x1 isNat(x1) weight: x1 plus(x1,x2) weight: (/ 5 16) + x1 + x2 #U11(x1,x2,x3) weight: 0 active(x1) weight: x1 U31(x1,x2) weight: (/ 1 16) + x2 #U41(x1,x2,x3) weight: 0 #active(x1) weight: x1 #U21(x1,x2) weight: 0 #U22(x1) weight: 0 tt() weight: 0 U13(x1) weight: (/ 1 16) U22(x1) weight: (/ 1 16) #isNatKind(x1) weight: 0 U41(x1,x2,x3) weight: (/ 1 2) + x2 + x3 #U31(x1,x2) weight: 0 #and(x1,x2) weight: 0 Usable rules: { 1..72 } Removed DPs: #25 #46 #49 #71 #73 #113 Number of SCCs: 7, DPs: 0, edges: 0 YES