Input TRS: 1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) 2: active(sqr(0())) -> mark(0()) 3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) 4: active(dbl(0())) -> mark(0()) 5: active(dbl(s(X))) -> mark(s(s(dbl(X)))) 6: active(add(0(),X)) -> mark(X) 7: active(add(s(X),Y)) -> mark(s(add(X,Y))) 8: active(first(0(),X)) -> mark(nil()) 9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) 10: mark(terms(X)) -> active(terms(mark(X))) 11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) 12: mark(recip(X)) -> active(recip(mark(X))) 13: mark(sqr(X)) -> active(sqr(mark(X))) 14: mark(s(X)) -> active(s(mark(X))) 15: mark(0()) -> active(0()) 16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) 17: mark(dbl(X)) -> active(dbl(mark(X))) 18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) 19: mark(nil()) -> active(nil()) 20: terms(mark(X)) -> terms(X) 21: terms(active(X)) -> terms(X) 22: cons(mark(X1),X2) -> cons(X1,X2) 23: cons(X1,mark(X2)) -> cons(X1,X2) 24: cons(active(X1),X2) -> cons(X1,X2) 25: cons(X1,active(X2)) -> cons(X1,X2) 26: recip(mark(X)) -> recip(X) 27: recip(active(X)) -> recip(X) 28: sqr(mark(X)) -> sqr(X) 29: sqr(active(X)) -> sqr(X) 30: s(mark(X)) -> s(X) 31: s(active(X)) -> s(X) 32: add(mark(X1),X2) -> add(X1,X2) 33: add(X1,mark(X2)) -> add(X1,X2) 34: add(active(X1),X2) -> add(X1,X2) 35: add(X1,active(X2)) -> add(X1,X2) 36: dbl(mark(X)) -> dbl(X) 37: dbl(active(X)) -> dbl(X) 38: first(mark(X1),X2) -> first(X1,X2) 39: first(X1,mark(X2)) -> first(X1,X2) 40: first(active(X1),X2) -> first(X1,X2) 41: first(X1,active(X2)) -> first(X1,X2) Number of strict rules: 41 Direct Order(PosReal,>,Poly) ... failed. Freezing ... failed. Dependency Pairs: #1: #active(sqr(0())) -> #mark(0()) #2: #sqr(active(X)) -> #sqr(X) #3: #add(X1,active(X2)) -> #add(X1,X2) #4: #first(X1,active(X2)) -> #first(X1,X2) #5: #dbl(active(X)) -> #dbl(X) #6: #first(mark(X1),X2) -> #first(X1,X2) #7: #active(add(0(),X)) -> #mark(X) #8: #first(active(X1),X2) -> #first(X1,X2) #9: #mark(sqr(X)) -> #active(sqr(mark(X))) #10: #mark(sqr(X)) -> #sqr(mark(X)) #11: #mark(sqr(X)) -> #mark(X) #12: #active(first(s(X),cons(Y,Z))) -> #mark(cons(Y,first(X,Z))) #13: #active(first(s(X),cons(Y,Z))) -> #cons(Y,first(X,Z)) #14: #active(first(s(X),cons(Y,Z))) -> #first(X,Z) #15: #mark(cons(X1,X2)) -> #active(cons(mark(X1),X2)) #16: #mark(cons(X1,X2)) -> #cons(mark(X1),X2) #17: #mark(cons(X1,X2)) -> #mark(X1) #18: #cons(active(X1),X2) -> #cons(X1,X2) #19: #cons(X1,mark(X2)) -> #cons(X1,X2) #20: #mark(recip(X)) -> #active(recip(mark(X))) #21: #mark(recip(X)) -> #recip(mark(X)) #22: #mark(recip(X)) -> #mark(X) #23: #s(active(X)) -> #s(X) #24: #mark(s(X)) -> #active(s(mark(X))) #25: #mark(s(X)) -> #s(mark(X)) #26: #mark(s(X)) -> #mark(X) #27: #s(mark(X)) -> #s(X) #28: #cons(X1,active(X2)) -> #cons(X1,X2) #29: #terms(mark(X)) -> #terms(X) #30: #active(add(s(X),Y)) -> #mark(s(add(X,Y))) #31: #active(add(s(X),Y)) -> #s(add(X,Y)) #32: #active(add(s(X),Y)) -> #add(X,Y) #33: #first(X1,mark(X2)) -> #first(X1,X2) #34: #mark(terms(X)) -> #active(terms(mark(X))) #35: #mark(terms(X)) -> #terms(mark(X)) #36: #mark(terms(X)) -> #mark(X) #37: #add(X1,mark(X2)) -> #add(X1,X2) #38: #active(dbl(s(X))) -> #mark(s(s(dbl(X)))) #39: #active(dbl(s(X))) -> #s(s(dbl(X))) #40: #active(dbl(s(X))) -> #s(dbl(X)) #41: #active(dbl(s(X))) -> #dbl(X) #42: #sqr(mark(X)) -> #sqr(X) #43: #cons(mark(X1),X2) -> #cons(X1,X2) #44: #add(active(X1),X2) -> #add(X1,X2) #45: #recip(active(X)) -> #recip(X) #46: #mark(dbl(X)) -> #active(dbl(mark(X))) #47: #mark(dbl(X)) -> #dbl(mark(X)) #48: #mark(dbl(X)) -> #mark(X) #49: #add(mark(X1),X2) -> #add(X1,X2) #50: #mark(nil()) -> #active(nil()) #51: #recip(mark(X)) -> #recip(X) #52: #dbl(mark(X)) -> #dbl(X) #53: #terms(active(X)) -> #terms(X) #54: #mark(add(X1,X2)) -> #active(add(mark(X1),mark(X2))) #55: #mark(add(X1,X2)) -> #add(mark(X1),mark(X2)) #56: #mark(add(X1,X2)) -> #mark(X1) #57: #mark(add(X1,X2)) -> #mark(X2) #58: #active(sqr(s(X))) -> #mark(s(add(sqr(X),dbl(X)))) #59: #active(sqr(s(X))) -> #s(add(sqr(X),dbl(X))) #60: #active(sqr(s(X))) -> #add(sqr(X),dbl(X)) #61: #active(sqr(s(X))) -> #sqr(X) #62: #active(sqr(s(X))) -> #dbl(X) #63: #active(terms(N)) -> #mark(cons(recip(sqr(N)),terms(s(N)))) #64: #active(terms(N)) -> #cons(recip(sqr(N)),terms(s(N))) #65: #active(terms(N)) -> #recip(sqr(N)) #66: #active(terms(N)) -> #sqr(N) #67: #active(terms(N)) -> #terms(s(N)) #68: #active(terms(N)) -> #s(N) #69: #active(first(0(),X)) -> #mark(nil()) #70: #mark(0()) -> #active(0()) #71: #active(dbl(0())) -> #mark(0()) #72: #mark(first(X1,X2)) -> #active(first(mark(X1),mark(X2))) #73: #mark(first(X1,X2)) -> #first(mark(X1),mark(X2)) #74: #mark(first(X1,X2)) -> #mark(X1) #75: #mark(first(X1,X2)) -> #mark(X2) Number of SCCs: 9, DPs: 43, edges: 244 SCC { #2 #42 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: 0 recip(x1) weight: 0 #recip(x1) weight: 0 dbl(x1) weight: 0 #dbl(x1) weight: 0 #terms(x1) weight: 0 #sqr(x1) weight: x1 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 #first(x1,x2) weight: 0 nil() weight: 0 mark(x1) weight: (/ 1 2) + x1 first(x1,x2) weight: 0 active(x1) weight: (/ 1 2) + x1 cons(x1,x2) weight: 0 #active(x1) weight: 0 #add(x1,x2) weight: 0 add(x1,x2) weight: 0 sqr(x1) weight: 0 terms(x1) weight: 0 Usable rules: { } Removed DPs: #2 #42 Number of SCCs: 8, DPs: 41, edges: 240 SCC { #29 #53 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: 0 recip(x1) weight: 0 #recip(x1) weight: 0 dbl(x1) weight: 0 #dbl(x1) weight: 0 #terms(x1) weight: x1 #sqr(x1) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 #first(x1,x2) weight: 0 nil() weight: 0 mark(x1) weight: (/ 1 2) + x1 first(x1,x2) weight: 0 active(x1) weight: (/ 1 2) + x1 cons(x1,x2) weight: 0 #active(x1) weight: 0 #add(x1,x2) weight: 0 add(x1,x2) weight: 0 sqr(x1) weight: 0 terms(x1) weight: 0 Usable rules: { } Removed DPs: #29 #53 Number of SCCs: 7, DPs: 39, edges: 236 SCC { #5 #52 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: 0 recip(x1) weight: 0 #recip(x1) weight: 0 dbl(x1) weight: 0 #dbl(x1) weight: x1 #terms(x1) weight: 0 #sqr(x1) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 #first(x1,x2) weight: 0 nil() weight: 0 mark(x1) weight: (/ 1 2) + x1 first(x1,x2) weight: 0 active(x1) weight: (/ 1 2) + x1 cons(x1,x2) weight: 0 #active(x1) weight: 0 #add(x1,x2) weight: 0 add(x1,x2) weight: 0 sqr(x1) weight: 0 terms(x1) weight: 0 Usable rules: { } Removed DPs: #5 #52 Number of SCCs: 6, DPs: 37, edges: 232 SCC { #23 #27 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: 0 recip(x1) weight: 0 #recip(x1) weight: 0 dbl(x1) weight: 0 #dbl(x1) weight: 0 #terms(x1) weight: 0 #sqr(x1) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: x1 #first(x1,x2) weight: 0 nil() weight: 0 mark(x1) weight: (/ 1 2) + x1 first(x1,x2) weight: 0 active(x1) weight: (/ 1 2) + x1 cons(x1,x2) weight: 0 #active(x1) weight: 0 #add(x1,x2) weight: 0 add(x1,x2) weight: 0 sqr(x1) weight: 0 terms(x1) weight: 0 Usable rules: { } Removed DPs: #23 #27 Number of SCCs: 5, DPs: 35, edges: 228 SCC { #45 #51 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: 0 recip(x1) weight: 0 #recip(x1) weight: x1 dbl(x1) weight: 0 #dbl(x1) weight: 0 #terms(x1) weight: 0 #sqr(x1) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 #first(x1,x2) weight: 0 nil() weight: 0 mark(x1) weight: (/ 1 2) + x1 first(x1,x2) weight: 0 active(x1) weight: (/ 1 2) + x1 cons(x1,x2) weight: 0 #active(x1) weight: 0 #add(x1,x2) weight: 0 add(x1,x2) weight: 0 sqr(x1) weight: 0 terms(x1) weight: 0 Usable rules: { } Removed DPs: #45 #51 Number of SCCs: 4, DPs: 33, edges: 224 SCC { #18 #19 #28 #43 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: x2 s(x1) weight: 0 recip(x1) weight: 0 #recip(x1) weight: 0 dbl(x1) weight: 0 #dbl(x1) weight: 0 #terms(x1) weight: 0 #sqr(x1) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 #first(x1,x2) weight: 0 nil() weight: 0 mark(x1) weight: (/ 1 2) + x1 first(x1,x2) weight: 0 active(x1) weight: (/ 1 2) + x1 cons(x1,x2) weight: 0 #active(x1) weight: 0 #add(x1,x2) weight: 0 add(x1,x2) weight: 0 sqr(x1) weight: 0 terms(x1) weight: 0 Usable rules: { } Removed DPs: #19 #28 Number of SCCs: 4, DPs: 31, edges: 212 SCC { #18 #43 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: x1 s(x1) weight: 0 recip(x1) weight: 0 #recip(x1) weight: 0 dbl(x1) weight: 0 #dbl(x1) weight: 0 #terms(x1) weight: 0 #sqr(x1) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 #first(x1,x2) weight: 0 nil() weight: 0 mark(x1) weight: (/ 1 2) + x1 first(x1,x2) weight: 0 active(x1) weight: (/ 1 2) + x1 cons(x1,x2) weight: 0 #active(x1) weight: 0 #add(x1,x2) weight: 0 add(x1,x2) weight: 0 sqr(x1) weight: 0 terms(x1) weight: 0 Usable rules: { } Removed DPs: #18 #43 Number of SCCs: 3, DPs: 29, edges: 208 SCC { #4 #6 #8 #33 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: 0 recip(x1) weight: 0 #recip(x1) weight: 0 dbl(x1) weight: 0 #dbl(x1) weight: 0 #terms(x1) weight: 0 #sqr(x1) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 #first(x1,x2) weight: x1 nil() weight: 0 mark(x1) weight: (/ 1 2) + x1 first(x1,x2) weight: 0 active(x1) weight: (/ 1 2) + x1 cons(x1,x2) weight: 0 #active(x1) weight: 0 #add(x1,x2) weight: 0 add(x1,x2) weight: 0 sqr(x1) weight: 0 terms(x1) weight: 0 Usable rules: { } Removed DPs: #6 #8 Number of SCCs: 4, DPs: 27, edges: 196 SCC { #4 #33 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: 0 recip(x1) weight: 0 #recip(x1) weight: 0 dbl(x1) weight: 0 #dbl(x1) weight: 0 #terms(x1) weight: 0 #sqr(x1) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 #first(x1,x2) weight: x2 nil() weight: 0 mark(x1) weight: (/ 1 2) + x1 first(x1,x2) weight: 0 active(x1) weight: (/ 1 2) + x1 cons(x1,x2) weight: 0 #active(x1) weight: 0 #add(x1,x2) weight: 0 add(x1,x2) weight: 0 sqr(x1) weight: 0 terms(x1) weight: 0 Usable rules: { } Removed DPs: #4 #33 Number of SCCs: 3, DPs: 25, edges: 192 SCC { #3 #37 #44 #49 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: 0 recip(x1) weight: 0 #recip(x1) weight: 0 dbl(x1) weight: 0 #dbl(x1) weight: 0 #terms(x1) weight: 0 #sqr(x1) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 #first(x1,x2) weight: 0 nil() weight: 0 mark(x1) weight: (/ 1 2) + x1 first(x1,x2) weight: 0 active(x1) weight: (/ 1 2) + x1 cons(x1,x2) weight: 0 #active(x1) weight: 0 #add(x1,x2) weight: x1 add(x1,x2) weight: 0 sqr(x1) weight: 0 terms(x1) weight: 0 Usable rules: { } Removed DPs: #44 #49 Number of SCCs: 4, DPs: 23, edges: 180 SCC { #3 #37 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: 0 recip(x1) weight: 0 #recip(x1) weight: 0 dbl(x1) weight: 0 #dbl(x1) weight: 0 #terms(x1) weight: 0 #sqr(x1) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 #first(x1,x2) weight: 0 nil() weight: 0 mark(x1) weight: (/ 1 2) + x1 first(x1,x2) weight: 0 active(x1) weight: (/ 1 2) + x1 cons(x1,x2) weight: 0 #active(x1) weight: 0 #add(x1,x2) weight: x2 add(x1,x2) weight: 0 sqr(x1) weight: 0 terms(x1) weight: 0 Usable rules: { } Removed DPs: #3 #37 Number of SCCs: 3, DPs: 21, edges: 176 SCC { #7 #9 #11 #12 #17 #22 #26 #30 #34 #36 #38 #46 #48 #54 #56..58 #63 #72 #74 #75 } Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: x1 recip(x1) weight: (/ 1 8) + x1 #recip(x1) weight: 0 dbl(x1) weight: (/ 1 8) + x1 #dbl(x1) weight: 0 #terms(x1) weight: 0 #sqr(x1) weight: 0 #mark(x1) weight: x1 0() weight: (/ 1 8) #s(x1) weight: 0 #first(x1,x2) weight: 0 nil() weight: (/ 1 2) mark(x1) weight: x1 first(x1,x2) weight: max{(/ 1 2) + x2, (/ 1 4) + x1} active(x1) weight: x1 cons(x1,x2) weight: max{x2, (/ 1 8) + x1} #active(x1) weight: x1 #add(x1,x2) weight: 0 add(x1,x2) weight: max{(/ 23977 8) + x2, x1} sqr(x1) weight: (/ 23979 8) + x1 terms(x1) weight: (/ 11991 4) + x1 Usable rules: { 1..41 } Removed DPs: #7 #11 #17 #22 #36 #48 #57 #74 #75 Number of SCCs: 4, DPs: 8, edges: 16 SCC { #9 #26 #30 #38 #46 #54 #56 #58 } Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded. #cons(x1,x2) status: [x1,x2] precedence above: s(x1) status: [x1] precedence above: recip(x1) status: [] precedence above: #recip(x1) status: [] precedence above: dbl(x1) status: [x1] precedence above: s #dbl(x1) status: [] precedence above: #terms(x1) status: [] precedence above: #sqr(x1) status: [] precedence above: #mark(x1) status: [x1] precedence above: s dbl #active add 0() status: [] precedence above: nil #s(x1) status: [] precedence above: #first(x1,x2) status: [x2] precedence above: nil() status: [] precedence above: mark(x1) status: x1 first(x1,x2) status: x1 active(x1) status: x1 cons(x1,x2) status: x2 #active(x1) status: [x1] precedence above: s dbl #mark add #add(x1,x2) status: [x2] precedence above: add(x1,x2) status: [x1,x2] precedence above: s sqr(x1) status: [x1] precedence above: s dbl #mark #active add terms(x1) status: [] precedence above: recip cons Usable rules: { 1..41 } Removed DPs: #26 #30 #38 #56 #58 Number of SCCs: 3, DPs: 0, edges: 0 YES