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: active(half(0())) -> mark(0()) 11: active(half(s(0()))) -> mark(0()) 12: active(half(s(s(X)))) -> mark(s(half(X))) 13: active(half(dbl(X))) -> mark(X) 14: mark(terms(X)) -> active(terms(mark(X))) 15: mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) 16: mark(recip(X)) -> active(recip(mark(X))) 17: mark(sqr(X)) -> active(sqr(mark(X))) 18: mark(s(X)) -> active(s(mark(X))) 19: mark(0()) -> active(0()) 20: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) 21: mark(dbl(X)) -> active(dbl(mark(X))) 22: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) 23: mark(nil()) -> active(nil()) 24: mark(half(X)) -> active(half(mark(X))) 25: terms(mark(X)) -> terms(X) 26: terms(active(X)) -> terms(X) 27: cons(mark(X1),X2) -> cons(X1,X2) 28: cons(X1,mark(X2)) -> cons(X1,X2) 29: cons(active(X1),X2) -> cons(X1,X2) 30: cons(X1,active(X2)) -> cons(X1,X2) 31: recip(mark(X)) -> recip(X) 32: recip(active(X)) -> recip(X) 33: sqr(mark(X)) -> sqr(X) 34: sqr(active(X)) -> sqr(X) 35: s(mark(X)) -> s(X) 36: s(active(X)) -> s(X) 37: add(mark(X1),X2) -> add(X1,X2) 38: add(X1,mark(X2)) -> add(X1,X2) 39: add(active(X1),X2) -> add(X1,X2) 40: add(X1,active(X2)) -> add(X1,X2) 41: dbl(mark(X)) -> dbl(X) 42: dbl(active(X)) -> dbl(X) 43: first(mark(X1),X2) -> first(X1,X2) 44: first(X1,mark(X2)) -> first(X1,X2) 45: first(active(X1),X2) -> first(X1,X2) 46: first(X1,active(X2)) -> first(X1,X2) 47: half(mark(X)) -> half(X) 48: half(active(X)) -> half(X) Number of strict rules: 48 Direct Order(PosReal,>,Poly) ... failed. Freezing ... failed. Dependency Pairs: #1: #active(sqr(0())) -> #mark(0()) #2: #first(mark(X1),X2) -> #first(X1,X2) #3: #cons(active(X1),X2) -> #cons(X1,X2) #4: #s(mark(X)) -> #s(X) #5: #first(X1,active(X2)) -> #first(X1,X2) #6: #dbl(active(X)) -> #dbl(X) #7: #dbl(mark(X)) -> #dbl(X) #8: #add(mark(X1),X2) -> #add(X1,X2) #9: #half(mark(X)) -> #half(X) #10: #half(active(X)) -> #half(X) #11: #add(X1,mark(X2)) -> #add(X1,X2) #12: #active(add(0(),X)) -> #mark(X) #13: #add(X1,active(X2)) -> #add(X1,X2) #14: #active(half(dbl(X))) -> #mark(X) #15: #active(first(s(X),cons(Y,Z))) -> #mark(cons(Y,first(X,Z))) #16: #active(first(s(X),cons(Y,Z))) -> #cons(Y,first(X,Z)) #17: #active(first(s(X),cons(Y,Z))) -> #first(X,Z) #18: #active(half(s(0()))) -> #mark(0()) #19: #mark(half(X)) -> #active(half(mark(X))) #20: #mark(half(X)) -> #half(mark(X)) #21: #mark(half(X)) -> #mark(X) #22: #mark(nil()) -> #active(nil()) #23: #first(active(X1),X2) -> #first(X1,X2) #24: #active(half(s(s(X)))) -> #mark(s(half(X))) #25: #active(half(s(s(X)))) -> #s(half(X)) #26: #active(half(s(s(X)))) -> #half(X) #27: #recip(mark(X)) -> #recip(X) #28: #mark(terms(X)) -> #active(terms(mark(X))) #29: #mark(terms(X)) -> #terms(mark(X)) #30: #mark(terms(X)) -> #mark(X) #31: #cons(X1,active(X2)) -> #cons(X1,X2) #32: #terms(mark(X)) -> #terms(X) #33: #mark(add(X1,X2)) -> #active(add(mark(X1),mark(X2))) #34: #mark(add(X1,X2)) -> #add(mark(X1),mark(X2)) #35: #mark(add(X1,X2)) -> #mark(X1) #36: #mark(add(X1,X2)) -> #mark(X2) #37: #active(add(s(X),Y)) -> #mark(s(add(X,Y))) #38: #active(add(s(X),Y)) -> #s(add(X,Y)) #39: #active(add(s(X),Y)) -> #add(X,Y) #40: #add(active(X1),X2) -> #add(X1,X2) #41: #active(half(0())) -> #mark(0()) #42: #sqr(mark(X)) -> #sqr(X) #43: #active(dbl(s(X))) -> #mark(s(s(dbl(X)))) #44: #active(dbl(s(X))) -> #s(s(dbl(X))) #45: #active(dbl(s(X))) -> #s(dbl(X)) #46: #active(dbl(s(X))) -> #dbl(X) #47: #first(X1,mark(X2)) -> #first(X1,X2) #48: #cons(X1,mark(X2)) -> #cons(X1,X2) #49: #mark(first(X1,X2)) -> #active(first(mark(X1),mark(X2))) #50: #mark(first(X1,X2)) -> #first(mark(X1),mark(X2)) #51: #mark(first(X1,X2)) -> #mark(X1) #52: #mark(first(X1,X2)) -> #mark(X2) #53: #sqr(active(X)) -> #sqr(X) #54: #cons(mark(X1),X2) -> #cons(X1,X2) #55: #mark(sqr(X)) -> #active(sqr(mark(X))) #56: #mark(sqr(X)) -> #sqr(mark(X)) #57: #mark(sqr(X)) -> #mark(X) #58: #recip(active(X)) -> #recip(X) #59: #mark(0()) -> #active(0()) #60: #terms(active(X)) -> #terms(X) #61: #s(active(X)) -> #s(X) #62: #mark(dbl(X)) -> #active(dbl(mark(X))) #63: #mark(dbl(X)) -> #dbl(mark(X)) #64: #mark(dbl(X)) -> #mark(X) #65: #mark(recip(X)) -> #active(recip(mark(X))) #66: #mark(recip(X)) -> #recip(mark(X)) #67: #mark(recip(X)) -> #mark(X) #68: #active(sqr(s(X))) -> #mark(s(add(sqr(X),dbl(X)))) #69: #active(sqr(s(X))) -> #s(add(sqr(X),dbl(X))) #70: #active(sqr(s(X))) -> #add(sqr(X),dbl(X)) #71: #active(sqr(s(X))) -> #sqr(X) #72: #active(sqr(s(X))) -> #dbl(X) #73: #active(terms(N)) -> #mark(cons(recip(sqr(N)),terms(s(N)))) #74: #active(terms(N)) -> #cons(recip(sqr(N)),terms(s(N))) #75: #active(terms(N)) -> #recip(sqr(N)) #76: #active(terms(N)) -> #sqr(N) #77: #active(terms(N)) -> #terms(s(N)) #78: #active(terms(N)) -> #s(N) #79: #active(first(0(),X)) -> #mark(nil()) #80: #mark(cons(X1,X2)) -> #active(cons(mark(X1),X2)) #81: #mark(cons(X1,X2)) -> #cons(mark(X1),X2) #82: #mark(cons(X1,X2)) -> #mark(X1) #83: #active(dbl(0())) -> #mark(0()) #84: #mark(s(X)) -> #active(s(mark(X))) #85: #mark(s(X)) -> #s(mark(X)) #86: #mark(s(X)) -> #mark(X) Number of SCCs: 10, DPs: 49, edges: 307 SCC { #9 #10 } 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 #half(x1) weight: x1 half(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: #9 #10 Number of SCCs: 9, DPs: 47, edges: 303 SCC { #4 #61 } 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 #half(x1) weight: 0 half(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: #4 #61 Number of SCCs: 8, DPs: 45, edges: 299 SCC { #27 #58 } 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 #half(x1) weight: 0 half(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: #27 #58 Number of SCCs: 7, DPs: 43, edges: 295 SCC { #6 #7 } 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 #half(x1) weight: 0 half(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: #6 #7 Number of SCCs: 6, DPs: 41, edges: 291 SCC { #42 #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: 0 #half(x1) weight: 0 half(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: #42 #53 Number of SCCs: 5, DPs: 39, edges: 287 SCC { #32 #60 } 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 #half(x1) weight: 0 half(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: #32 #60 Number of SCCs: 4, DPs: 37, edges: 283 SCC { #2 #5 #23 #47 } 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 #half(x1) weight: 0 half(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: #5 #47 Number of SCCs: 4, DPs: 35, edges: 271 SCC { #2 #23 } 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 #half(x1) weight: 0 half(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: #2 #23 Number of SCCs: 3, DPs: 33, edges: 267 SCC { #3 #31 #48 #54 } 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 #half(x1) weight: 0 half(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: #3 #54 Number of SCCs: 4, DPs: 31, edges: 255 SCC { #31 #48 } 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 #half(x1) weight: 0 half(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: #31 #48 Number of SCCs: 3, DPs: 29, edges: 251 SCC { #8 #11 #13 #40 } 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 #half(x1) weight: 0 half(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: #11 #13 Number of SCCs: 4, DPs: 27, edges: 239 SCC { #8 #40 } 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 #half(x1) weight: 0 half(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: #8 #40 Number of SCCs: 3, DPs: 25, edges: 235 SCC { #12 #14 #15 #19 #21 #24 #28 #30 #33 #35..37 #43 #49 #51 #52 #55 #57 #62 #64 #67 #68 #73 #82 #86 } 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: (/ 49073 8) + x1 #dbl(x1) weight: 0 #terms(x1) weight: 0 #half(x1) weight: 0 half(x1) weight: (/ 233937 8) + x1 #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{(/ 244897 8) + x2, x1} sqr(x1) weight: (/ 293971 8) + x1 terms(x1) weight: (/ 146987 4) + x1 Usable rules: { 1..48 } Removed DPs: #12 #14 #21 #30 #36 #51 #52 #57 #64 #67 #82 Number of SCCs: 4, DPs: 10, edges: 20 SCC { #19 #24 #33 #35 #37 #43 #55 #62 #68 #86 } Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded. #cons(x1,x2) status: [] precedence above: s(x1) status: [x1] precedence above: half 0 nil first cons recip(x1) status: x1 #recip(x1) status: [] precedence above: dbl(x1) status: [x1] precedence above: s half 0 nil first cons #dbl(x1) status: [] precedence above: #terms(x1) status: [] precedence above: #half(x1) status: [] precedence above: half(x1) status: [x1] precedence above: s 0 nil first cons #sqr(x1) status: [] precedence above: #mark(x1) status: [x1] precedence above: s half 0 nil first cons #active 0() status: [] precedence above: nil #s(x1) status: [] precedence above: #first(x1,x2) status: [x1,x2] precedence above: nil() status: [] precedence above: mark(x1) status: x1 first(x1,x2) status: [x1] precedence above: nil cons active(x1) status: x1 cons(x1,x2) status: [] precedence above: #active(x1) status: [x1] precedence above: s half #mark 0 nil first cons #add(x1,x2) status: [x2] precedence above: add(x1,x2) status: [x2,x1] precedence above: s half #mark 0 nil first cons #active sqr(x1) status: [x1] precedence above: s dbl half #mark 0 nil mark first active cons #active add terms(x1) status: [x1] precedence above: s dbl half #mark 0 nil mark first active cons #active add sqr Usable rules: { 1..48 } Removed DPs: #24 #35 #37 #43 #68 #86 Number of SCCs: 3, DPs: 0, edges: 0 YES