YES We show the termination of the TRS R: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) active(oddNs()) -> mark(incr(pairNs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(take(|0|(),XS)) -> mark(nil()) active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) active(zip(nil(),XS)) -> mark(nil()) active(zip(X,nil())) -> mark(nil()) active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) active(tail(cons(X,XS))) -> mark(XS) active(repItems(nil())) -> mark(nil()) active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(zip(X1,X2)) -> zip(active(X1),X2) active(zip(X1,X2)) -> zip(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(repItems(X)) -> repItems(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) zip(mark(X1),X2) -> mark(zip(X1,X2)) zip(X1,mark(X2)) -> mark(zip(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) tail(mark(X)) -> mark(tail(X)) repItems(mark(X)) -> mark(repItems(X)) proper(pairNs()) -> ok(pairNs()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(|0|()) -> ok(|0|()) proper(incr(X)) -> incr(proper(X)) proper(oddNs()) -> ok(oddNs()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(tail(X)) -> tail(proper(X)) proper(repItems(X)) -> repItems(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) tail(ok(X)) -> ok(tail(X)) repItems(ok(X)) -> ok(repItems(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: active#(pairNs()) -> cons#(|0|(),incr(oddNs())) p2: active#(pairNs()) -> incr#(oddNs()) p3: active#(oddNs()) -> incr#(pairNs()) p4: active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) p5: active#(incr(cons(X,XS))) -> s#(X) p6: active#(incr(cons(X,XS))) -> incr#(XS) p7: active#(take(s(N),cons(X,XS))) -> cons#(X,take(N,XS)) p8: active#(take(s(N),cons(X,XS))) -> take#(N,XS) p9: active#(zip(cons(X,XS),cons(Y,YS))) -> cons#(pair(X,Y),zip(XS,YS)) p10: active#(zip(cons(X,XS),cons(Y,YS))) -> pair#(X,Y) p11: active#(zip(cons(X,XS),cons(Y,YS))) -> zip#(XS,YS) p12: active#(repItems(cons(X,XS))) -> cons#(X,cons(X,repItems(XS))) p13: active#(repItems(cons(X,XS))) -> cons#(X,repItems(XS)) p14: active#(repItems(cons(X,XS))) -> repItems#(XS) p15: active#(cons(X1,X2)) -> cons#(active(X1),X2) p16: active#(cons(X1,X2)) -> active#(X1) p17: active#(incr(X)) -> incr#(active(X)) p18: active#(incr(X)) -> active#(X) p19: active#(s(X)) -> s#(active(X)) p20: active#(s(X)) -> active#(X) p21: active#(take(X1,X2)) -> take#(active(X1),X2) p22: active#(take(X1,X2)) -> active#(X1) p23: active#(take(X1,X2)) -> take#(X1,active(X2)) p24: active#(take(X1,X2)) -> active#(X2) p25: active#(zip(X1,X2)) -> zip#(active(X1),X2) p26: active#(zip(X1,X2)) -> active#(X1) p27: active#(zip(X1,X2)) -> zip#(X1,active(X2)) p28: active#(zip(X1,X2)) -> active#(X2) p29: active#(pair(X1,X2)) -> pair#(active(X1),X2) p30: active#(pair(X1,X2)) -> active#(X1) p31: active#(pair(X1,X2)) -> pair#(X1,active(X2)) p32: active#(pair(X1,X2)) -> active#(X2) p33: active#(tail(X)) -> tail#(active(X)) p34: active#(tail(X)) -> active#(X) p35: active#(repItems(X)) -> repItems#(active(X)) p36: active#(repItems(X)) -> active#(X) p37: cons#(mark(X1),X2) -> cons#(X1,X2) p38: incr#(mark(X)) -> incr#(X) p39: s#(mark(X)) -> s#(X) p40: take#(mark(X1),X2) -> take#(X1,X2) p41: take#(X1,mark(X2)) -> take#(X1,X2) p42: zip#(mark(X1),X2) -> zip#(X1,X2) p43: zip#(X1,mark(X2)) -> zip#(X1,X2) p44: pair#(mark(X1),X2) -> pair#(X1,X2) p45: pair#(X1,mark(X2)) -> pair#(X1,X2) p46: tail#(mark(X)) -> tail#(X) p47: repItems#(mark(X)) -> repItems#(X) p48: proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) p49: proper#(cons(X1,X2)) -> proper#(X1) p50: proper#(cons(X1,X2)) -> proper#(X2) p51: proper#(incr(X)) -> incr#(proper(X)) p52: proper#(incr(X)) -> proper#(X) p53: proper#(s(X)) -> s#(proper(X)) p54: proper#(s(X)) -> proper#(X) p55: proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) p56: proper#(take(X1,X2)) -> proper#(X1) p57: proper#(take(X1,X2)) -> proper#(X2) p58: proper#(zip(X1,X2)) -> zip#(proper(X1),proper(X2)) p59: proper#(zip(X1,X2)) -> proper#(X1) p60: proper#(zip(X1,X2)) -> proper#(X2) p61: proper#(pair(X1,X2)) -> pair#(proper(X1),proper(X2)) p62: proper#(pair(X1,X2)) -> proper#(X1) p63: proper#(pair(X1,X2)) -> proper#(X2) p64: proper#(tail(X)) -> tail#(proper(X)) p65: proper#(tail(X)) -> proper#(X) p66: proper#(repItems(X)) -> repItems#(proper(X)) p67: proper#(repItems(X)) -> proper#(X) p68: cons#(ok(X1),ok(X2)) -> cons#(X1,X2) p69: incr#(ok(X)) -> incr#(X) p70: s#(ok(X)) -> s#(X) p71: take#(ok(X1),ok(X2)) -> take#(X1,X2) p72: zip#(ok(X1),ok(X2)) -> zip#(X1,X2) p73: pair#(ok(X1),ok(X2)) -> pair#(X1,X2) p74: tail#(ok(X)) -> tail#(X) p75: repItems#(ok(X)) -> repItems#(X) p76: top#(mark(X)) -> top#(proper(X)) p77: top#(mark(X)) -> proper#(X) p78: top#(ok(X)) -> top#(active(X)) p79: top#(ok(X)) -> active#(X) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The estimated dependency graph contains the following SCCs: {p76, p78} {p16, p18, p20, p22, p24, p26, p28, p30, p32, p34, p36} {p49, p50, p52, p54, p56, p57, p59, p60, p62, p63, p65, p67} {p37, p68} {p39, p70} {p38, p69} {p40, p41, p71} {p44, p45, p73} {p42, p43, p72} {p47, p75} {p46, p74} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: top#(ok(X)) -> top#(active(X)) p2: top#(mark(X)) -> top#(proper(X)) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53 Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: standard order interpretations: top#_A(x1) = x1 ok_A(x1) = x1 active_A(x1) = x1 mark_A(x1) = x1 proper_A(x1) = x1 cons_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,1),(1,0)) x2 incr_A(x1) = x1 s_A(x1) = ((1,0),(1,1)) x1 + (0,2) take_A(x1,x2) = ((1,1),(0,1)) x1 + x2 + (0,1) zip_A(x1,x2) = x1 + x2 + (1,1) pair_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (0,1) tail_A(x1) = ((1,1),(1,1)) x1 + (1,1) repItems_A(x1) = ((1,1),(1,1)) x1 + (2,1) pairNs_A() = (1,1) |0|_A() = (0,2) oddNs_A() = (1,1) nil_A() = (1,3) precedence: top# = ok = active = mark = proper = cons = incr = s = take = zip = pair = tail = repItems = pairNs = |0| = oddNs = nil partial status: pi(top#) = [] pi(ok) = [] pi(active) = [] pi(mark) = [] pi(proper) = [] pi(cons) = [] pi(incr) = [] pi(s) = [] pi(take) = [] pi(zip) = [] pi(pair) = [] pi(tail) = [] pi(repItems) = [] pi(pairNs) = [] pi(|0|) = [] pi(oddNs) = [] pi(nil) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: standard order interpretations: top#_A(x1) = ((0,1),(1,1)) x1 + (1,1) ok_A(x1) = ((1,0),(1,0)) x1 + (0,2) active_A(x1) = ((1,0),(1,0)) x1 + (0,2) mark_A(x1) = ((1,0),(1,0)) x1 + (7,8) proper_A(x1) = ((1,0),(1,0)) x1 + (0,5) cons_A(x1,x2) = ((1,0),(1,0)) x1 + (2,4) incr_A(x1) = x1 + (9,9) s_A(x1) = x1 + (1,1) take_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (0,2) zip_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (8,10) pair_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (1,3) tail_A(x1) = ((1,0),(1,0)) x1 + (3,5) repItems_A(x1) = ((1,0),(1,0)) x1 + (8,10) pairNs_A() = (10,9) |0|_A() = (0,1) oddNs_A() = (27,27) nil_A() = (1,2) precedence: proper > active = tail > take = oddNs > pairNs = |0| > zip = pair = nil > cons > incr > mark = s = repItems > ok > top# partial status: pi(top#) = [] pi(ok) = [] pi(active) = [] pi(mark) = [] pi(proper) = [] pi(cons) = [] pi(incr) = [1] pi(s) = [1] pi(take) = [] pi(zip) = [] pi(pair) = [] pi(tail) = [] pi(repItems) = [] pi(pairNs) = [] pi(|0|) = [] pi(oddNs) = [] pi(nil) = [] The next rules are strictly ordered: p2 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: top#(ok(X)) -> top#(active(X)) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: top#(ok(X)) -> top#(active(X)) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r46, r47, r48, r49, r50, r51, r52, r53 Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: top#_A(x1) = ((1,0),(0,0)) x1 ok_A(x1) = x1 + (2,14) active_A(x1) = ((1,0),(1,1)) x1 + (0,12) cons_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (19,10) mark_A(x1) = ((0,0),(1,0)) x1 + (4,11) incr_A(x1) = ((1,0),(1,1)) x1 + (20,25) s_A(x1) = ((1,0),(1,1)) x1 + (3,15) take_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (34,16) zip_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (20,17) pair_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (18,13) tail_A(x1) = ((1,0),(1,1)) x1 + (5,15) repItems_A(x1) = x1 + (1,9) pairNs_A() = (7,85) |0|_A() = (6,18) oddNs_A() = (28,118) nil_A() = (5,17) precedence: top# = ok = active = cons = mark = incr = s = take = zip = pair = tail = repItems = pairNs = |0| = oddNs = nil partial status: pi(top#) = [] pi(ok) = [] pi(active) = [] pi(cons) = [] pi(mark) = [] pi(incr) = [] pi(s) = [] pi(take) = [] pi(zip) = [] pi(pair) = [] pi(tail) = [] pi(repItems) = [] pi(pairNs) = [] pi(|0|) = [] pi(oddNs) = [] pi(nil) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: top#_A(x1) = (0,0) ok_A(x1) = ((1,0),(1,1)) x1 + (8,14) active_A(x1) = ((1,0),(1,1)) x1 + (1,1) cons_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (9,15) mark_A(x1) = (6,11) incr_A(x1) = ((1,0),(1,1)) x1 + (2,2) s_A(x1) = ((1,0),(1,1)) x1 + (7,13) take_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (10,16) zip_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (10,16) pair_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (11,42) tail_A(x1) = ((1,0),(1,1)) x1 + (7,12) repItems_A(x1) = ((1,0),(1,1)) x1 + (7,12) pairNs_A() = (1,1) |0|_A() = (8,13) oddNs_A() = (4,5) nil_A() = (7,12) precedence: top# = ok = cons > active = mark = incr = s = take = zip = pair = tail = repItems = pairNs = |0| = oddNs = nil partial status: pi(top#) = [] pi(ok) = [1] pi(active) = [1] pi(cons) = [1, 2] pi(mark) = [] pi(incr) = [1] pi(s) = [1] pi(take) = [1, 2] pi(zip) = [1, 2] pi(pair) = [1, 2] pi(tail) = [1] pi(repItems) = [1] pi(pairNs) = [] pi(|0|) = [] pi(oddNs) = [] pi(nil) = [] The next rules are strictly ordered: p1 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: active#(repItems(X)) -> active#(X) p2: active#(tail(X)) -> active#(X) p3: active#(pair(X1,X2)) -> active#(X2) p4: active#(pair(X1,X2)) -> active#(X1) p5: active#(zip(X1,X2)) -> active#(X2) p6: active#(zip(X1,X2)) -> active#(X1) p7: active#(take(X1,X2)) -> active#(X2) p8: active#(take(X1,X2)) -> active#(X1) p9: active#(s(X)) -> active#(X) p10: active#(incr(X)) -> active#(X) p11: active#(cons(X1,X2)) -> active#(X1) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: active#_A(x1) = ((1,0),(1,1)) x1 + (2,2) repItems_A(x1) = ((1,0),(1,1)) x1 + (3,3) tail_A(x1) = x1 + (1,3) pair_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (3,3) zip_A(x1,x2) = x1 + x2 + (3,3) take_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (1,3) s_A(x1) = ((1,0),(0,0)) x1 + (3,1) incr_A(x1) = ((1,0),(0,0)) x1 + (3,1) cons_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (3,3) precedence: s > tail = take > incr > active# = zip > pair > repItems = cons partial status: pi(active#) = [1] pi(repItems) = [1] pi(tail) = [] pi(pair) = [] pi(zip) = [1] pi(take) = [] pi(s) = [] pi(incr) = [] pi(cons) = [1] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: active#_A(x1) = (0,0) repItems_A(x1) = (1,1) tail_A(x1) = (1,1) pair_A(x1,x2) = (1,1) zip_A(x1,x2) = (1,1) take_A(x1,x2) = (1,1) s_A(x1) = (1,1) incr_A(x1) = (1,1) cons_A(x1,x2) = (1,1) precedence: active# = zip = take = cons > repItems > pair = s = incr > tail partial status: pi(active#) = [] pi(repItems) = [] pi(tail) = [] pi(pair) = [] pi(zip) = [] pi(take) = [] pi(s) = [] pi(incr) = [] pi(cons) = [] The next rules are strictly ordered: p1, p2, p4, p5, p6, p7, p8 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: active#(pair(X1,X2)) -> active#(X2) p2: active#(s(X)) -> active#(X) p3: active#(incr(X)) -> active#(X) p4: active#(cons(X1,X2)) -> active#(X1) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: active#(pair(X1,X2)) -> active#(X2) p2: active#(cons(X1,X2)) -> active#(X1) p3: active#(incr(X)) -> active#(X) p4: active#(s(X)) -> active#(X) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: active#_A(x1) = x1 pair_A(x1,x2) = x2 + (1,1) cons_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (1,1) incr_A(x1) = x1 s_A(x1) = ((1,0),(0,0)) x1 + (1,0) precedence: pair = cons = incr > s > active# partial status: pi(active#) = [1] pi(pair) = [] pi(cons) = [1, 2] pi(incr) = [1] pi(s) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: active#_A(x1) = (0,0) pair_A(x1,x2) = (1,1) cons_A(x1,x2) = ((1,0),(0,0)) x1 + (1,1) incr_A(x1) = ((1,0),(0,0)) x1 + (1,1) s_A(x1) = (1,1) precedence: active# = pair = cons = incr = s partial status: pi(active#) = [] pi(pair) = [] pi(cons) = [] pi(incr) = [] pi(s) = [] The next rules are strictly ordered: p1, p2, p3 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: active#(s(X)) -> active#(X) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: active#(s(X)) -> active#(X) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: active#_A(x1) = ((1,0),(1,1)) x1 + (2,2) s_A(x1) = ((1,0),(0,0)) x1 + (1,1) precedence: active# = s partial status: pi(active#) = [] pi(s) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: active#_A(x1) = ((1,0),(0,0)) x1 s_A(x1) = (1,1) precedence: s > active# partial status: pi(active#) = [] pi(s) = [] The next rules are strictly ordered: p1 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: proper#(repItems(X)) -> proper#(X) p2: proper#(tail(X)) -> proper#(X) p3: proper#(pair(X1,X2)) -> proper#(X2) p4: proper#(pair(X1,X2)) -> proper#(X1) p5: proper#(zip(X1,X2)) -> proper#(X2) p6: proper#(zip(X1,X2)) -> proper#(X1) p7: proper#(take(X1,X2)) -> proper#(X2) p8: proper#(take(X1,X2)) -> proper#(X1) p9: proper#(s(X)) -> proper#(X) p10: proper#(incr(X)) -> proper#(X) p11: proper#(cons(X1,X2)) -> proper#(X2) p12: proper#(cons(X1,X2)) -> proper#(X1) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: proper#_A(x1) = ((1,0),(0,0)) x1 repItems_A(x1) = ((1,0),(1,1)) x1 + (1,1) tail_A(x1) = ((1,0),(0,0)) x1 + (1,1) pair_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,1) zip_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (1,1) take_A(x1,x2) = x1 + x2 + (1,1) s_A(x1) = x1 incr_A(x1) = ((1,0),(0,0)) x1 + (1,1) cons_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (1,1) precedence: cons > incr > s > repItems > tail = pair > proper# = zip = take partial status: pi(proper#) = [] pi(repItems) = [] pi(tail) = [] pi(pair) = [] pi(zip) = [1] pi(take) = [1, 2] pi(s) = [1] pi(incr) = [] pi(cons) = [1] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: proper#_A(x1) = (0,0) repItems_A(x1) = (1,1) tail_A(x1) = (1,1) pair_A(x1,x2) = (1,1) zip_A(x1,x2) = x1 + (1,1) take_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (1,1) s_A(x1) = (1,1) incr_A(x1) = (1,1) cons_A(x1,x2) = x1 + (1,1) precedence: proper# = repItems = zip > tail > cons > pair = s > incr > take partial status: pi(proper#) = [] pi(repItems) = [] pi(tail) = [] pi(pair) = [] pi(zip) = [1] pi(take) = [1] pi(s) = [] pi(incr) = [] pi(cons) = [] The next rules are strictly ordered: p7 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: proper#(repItems(X)) -> proper#(X) p2: proper#(tail(X)) -> proper#(X) p3: proper#(pair(X1,X2)) -> proper#(X2) p4: proper#(pair(X1,X2)) -> proper#(X1) p5: proper#(zip(X1,X2)) -> proper#(X2) p6: proper#(zip(X1,X2)) -> proper#(X1) p7: proper#(take(X1,X2)) -> proper#(X1) p8: proper#(s(X)) -> proper#(X) p9: proper#(incr(X)) -> proper#(X) p10: proper#(cons(X1,X2)) -> proper#(X2) p11: proper#(cons(X1,X2)) -> proper#(X1) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: proper#(repItems(X)) -> proper#(X) p2: proper#(cons(X1,X2)) -> proper#(X1) p3: proper#(cons(X1,X2)) -> proper#(X2) p4: proper#(incr(X)) -> proper#(X) p5: proper#(s(X)) -> proper#(X) p6: proper#(take(X1,X2)) -> proper#(X1) p7: proper#(zip(X1,X2)) -> proper#(X1) p8: proper#(zip(X1,X2)) -> proper#(X2) p9: proper#(pair(X1,X2)) -> proper#(X1) p10: proper#(pair(X1,X2)) -> proper#(X2) p11: proper#(tail(X)) -> proper#(X) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: proper#_A(x1) = ((1,0),(1,1)) x1 + (1,2) repItems_A(x1) = ((1,0),(1,1)) x1 + (2,3) cons_A(x1,x2) = x1 + x2 + (2,3) incr_A(x1) = x1 + (2,3) s_A(x1) = x1 + (2,3) take_A(x1,x2) = x1 + x2 + (2,1) zip_A(x1,x2) = x1 + x2 + (2,3) pair_A(x1,x2) = x1 + x2 + (2,1) tail_A(x1) = ((1,0),(1,1)) x1 + (2,3) precedence: zip > incr > proper# = take > s > cons > repItems > pair = tail partial status: pi(proper#) = [1] pi(repItems) = [1] pi(cons) = [] pi(incr) = [1] pi(s) = [1] pi(take) = [1, 2] pi(zip) = [1, 2] pi(pair) = [2] pi(tail) = [1] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: proper#_A(x1) = (0,0) repItems_A(x1) = (1,1) cons_A(x1,x2) = (1,1) incr_A(x1) = ((1,0),(0,0)) x1 + (1,1) s_A(x1) = ((1,0),(1,0)) x1 + (1,0) take_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,0) zip_A(x1,x2) = x1 + x2 + (1,1) pair_A(x1,x2) = (1,1) tail_A(x1) = (1,1) precedence: cons = incr = s = pair = tail > proper# = repItems = take = zip partial status: pi(proper#) = [] pi(repItems) = [] pi(cons) = [] pi(incr) = [] pi(s) = [] pi(take) = [2] pi(zip) = [] pi(pair) = [] pi(tail) = [] The next rules are strictly ordered: p3 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: proper#(repItems(X)) -> proper#(X) p2: proper#(cons(X1,X2)) -> proper#(X1) p3: proper#(incr(X)) -> proper#(X) p4: proper#(s(X)) -> proper#(X) p5: proper#(take(X1,X2)) -> proper#(X1) p6: proper#(zip(X1,X2)) -> proper#(X1) p7: proper#(zip(X1,X2)) -> proper#(X2) p8: proper#(pair(X1,X2)) -> proper#(X1) p9: proper#(pair(X1,X2)) -> proper#(X2) p10: proper#(tail(X)) -> proper#(X) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: proper#(repItems(X)) -> proper#(X) p2: proper#(tail(X)) -> proper#(X) p3: proper#(pair(X1,X2)) -> proper#(X2) p4: proper#(pair(X1,X2)) -> proper#(X1) p5: proper#(zip(X1,X2)) -> proper#(X2) p6: proper#(zip(X1,X2)) -> proper#(X1) p7: proper#(take(X1,X2)) -> proper#(X1) p8: proper#(s(X)) -> proper#(X) p9: proper#(incr(X)) -> proper#(X) p10: proper#(cons(X1,X2)) -> proper#(X1) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: proper#_A(x1) = x1 + (0,2) repItems_A(x1) = x1 + (1,1) tail_A(x1) = ((1,0),(1,1)) x1 + (1,3) pair_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (1,3) zip_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (1,1) take_A(x1,x2) = x1 + x2 + (1,3) s_A(x1) = x1 + (1,3) incr_A(x1) = x1 + (1,1) cons_A(x1,x2) = x1 + x2 + (1,3) precedence: tail > zip > pair > repItems > take = incr > proper# = s = cons partial status: pi(proper#) = [1] pi(repItems) = [1] pi(tail) = [1] pi(pair) = [1, 2] pi(zip) = [1, 2] pi(take) = [1, 2] pi(s) = [1] pi(incr) = [1] pi(cons) = [1] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: proper#_A(x1) = ((1,0),(0,0)) x1 repItems_A(x1) = ((1,0),(0,0)) x1 + (1,1) tail_A(x1) = ((1,0),(0,0)) x1 + (1,1) pair_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (1,1) zip_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,1) take_A(x1,x2) = x1 + x2 + (1,1) s_A(x1) = (1,1) incr_A(x1) = (1,1) cons_A(x1,x2) = ((1,0),(0,0)) x1 + (1,1) precedence: proper# = pair = zip > repItems > take = s = incr > cons > tail partial status: pi(proper#) = [] pi(repItems) = [] pi(tail) = [] pi(pair) = [] pi(zip) = [] pi(take) = [2] pi(s) = [] pi(incr) = [] pi(cons) = [] The next rules are strictly ordered: p1, p2, p3, p4, p5, p6, p7, p8, p9, p10 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: cons#(mark(X1),X2) -> cons#(X1,X2) p2: cons#(ok(X1),ok(X2)) -> cons#(X1,X2) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: cons#_A(x1,x2) = ((1,0),(1,0)) x2 + (2,2) mark_A(x1) = (1,1) ok_A(x1) = ((1,0),(0,0)) x1 + (3,0) precedence: cons# > mark > ok partial status: pi(cons#) = [] pi(mark) = [] pi(ok) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: cons#_A(x1,x2) = (0,0) mark_A(x1) = (1,1) ok_A(x1) = (1,1) precedence: mark > cons# = ok partial status: pi(cons#) = [] pi(mark) = [] pi(ok) = [] The next rules are strictly ordered: p2 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: cons#(mark(X1),X2) -> cons#(X1,X2) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: cons#(mark(X1),X2) -> cons#(X1,X2) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: cons#_A(x1,x2) = ((1,0),(1,1)) x1 + (1,2) mark_A(x1) = ((1,0),(0,0)) x1 + (2,1) precedence: mark > cons# partial status: pi(cons#) = [1] pi(mark) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: cons#_A(x1,x2) = (1,1) mark_A(x1) = (2,2) precedence: cons# > mark partial status: pi(cons#) = [] pi(mark) = [] The next rules are strictly ordered: p1 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: s#(mark(X)) -> s#(X) p2: s#(ok(X)) -> s#(X) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: s#_A(x1) = x1 + (1,2) mark_A(x1) = x1 + (2,1) ok_A(x1) = ((1,0),(1,1)) x1 + (2,1) precedence: s# = mark > ok partial status: pi(s#) = [1] pi(mark) = [1] pi(ok) = [1] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: s#_A(x1) = ((1,0),(1,0)) x1 + (0,2) mark_A(x1) = ((1,0),(0,0)) x1 + (1,1) ok_A(x1) = ((1,0),(0,0)) x1 + (1,1) precedence: s# = mark = ok partial status: pi(s#) = [] pi(mark) = [] pi(ok) = [] The next rules are strictly ordered: p1, p2 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: incr#(mark(X)) -> incr#(X) p2: incr#(ok(X)) -> incr#(X) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: incr#_A(x1) = x1 + (1,2) mark_A(x1) = x1 + (2,1) ok_A(x1) = ((1,0),(1,1)) x1 + (2,1) precedence: incr# = mark = ok partial status: pi(incr#) = [1] pi(mark) = [1] pi(ok) = [1] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: incr#_A(x1) = ((1,0),(1,0)) x1 + (0,2) mark_A(x1) = ((1,0),(0,0)) x1 + (1,1) ok_A(x1) = ((1,0),(0,0)) x1 + (1,1) precedence: incr# = mark = ok partial status: pi(incr#) = [] pi(mark) = [] pi(ok) = [] The next rules are strictly ordered: p1, p2 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: take#(mark(X1),X2) -> take#(X1,X2) p2: take#(ok(X1),ok(X2)) -> take#(X1,X2) p3: take#(X1,mark(X2)) -> take#(X1,X2) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: take#_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (1,2) mark_A(x1) = ((1,0),(1,0)) x1 + (2,1) ok_A(x1) = x1 + (2,3) precedence: take# = mark = ok partial status: pi(take#) = [] pi(mark) = [] pi(ok) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: take#_A(x1,x2) = ((0,0),(1,0)) x2 + (2,2) mark_A(x1) = (3,1) ok_A(x1) = (1,0) precedence: take# > mark > ok partial status: pi(take#) = [] pi(mark) = [] pi(ok) = [] The next rules are strictly ordered: p1, p2, p3 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: pair#(mark(X1),X2) -> pair#(X1,X2) p2: pair#(ok(X1),ok(X2)) -> pair#(X1,X2) p3: pair#(X1,mark(X2)) -> pair#(X1,X2) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: pair#_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (1,2) mark_A(x1) = ((1,0),(1,0)) x1 + (2,1) ok_A(x1) = x1 + (2,3) precedence: pair# = mark = ok partial status: pi(pair#) = [] pi(mark) = [] pi(ok) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: pair#_A(x1,x2) = ((0,0),(1,0)) x2 + (2,2) mark_A(x1) = (3,1) ok_A(x1) = (1,0) precedence: pair# > mark > ok partial status: pi(pair#) = [] pi(mark) = [] pi(ok) = [] The next rules are strictly ordered: p1, p2, p3 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: zip#(mark(X1),X2) -> zip#(X1,X2) p2: zip#(ok(X1),ok(X2)) -> zip#(X1,X2) p3: zip#(X1,mark(X2)) -> zip#(X1,X2) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: zip#_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (1,2) mark_A(x1) = ((1,0),(1,0)) x1 + (2,1) ok_A(x1) = x1 + (2,3) precedence: zip# = mark = ok partial status: pi(zip#) = [] pi(mark) = [] pi(ok) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: zip#_A(x1,x2) = ((0,0),(1,0)) x2 + (2,2) mark_A(x1) = (3,1) ok_A(x1) = (1,0) precedence: zip# > mark > ok partial status: pi(zip#) = [] pi(mark) = [] pi(ok) = [] The next rules are strictly ordered: p1, p2, p3 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: repItems#(mark(X)) -> repItems#(X) p2: repItems#(ok(X)) -> repItems#(X) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: repItems#_A(x1) = x1 + (1,2) mark_A(x1) = x1 + (2,1) ok_A(x1) = ((1,0),(1,1)) x1 + (2,1) precedence: repItems# = mark = ok partial status: pi(repItems#) = [1] pi(mark) = [1] pi(ok) = [1] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: repItems#_A(x1) = ((1,0),(1,0)) x1 + (0,2) mark_A(x1) = ((1,0),(0,0)) x1 + (1,1) ok_A(x1) = ((1,0),(0,0)) x1 + (1,1) precedence: repItems# = mark = ok partial status: pi(repItems#) = [] pi(mark) = [] pi(ok) = [] The next rules are strictly ordered: p1, p2 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: tail#(mark(X)) -> tail#(X) p2: tail#(ok(X)) -> tail#(X) and R consists of: r1: active(pairNs()) -> mark(cons(|0|(),incr(oddNs()))) r2: active(oddNs()) -> mark(incr(pairNs())) r3: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) r4: active(take(|0|(),XS)) -> mark(nil()) r5: active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) r6: active(zip(nil(),XS)) -> mark(nil()) r7: active(zip(X,nil())) -> mark(nil()) r8: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) r9: active(tail(cons(X,XS))) -> mark(XS) r10: active(repItems(nil())) -> mark(nil()) r11: active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) r12: active(cons(X1,X2)) -> cons(active(X1),X2) r13: active(incr(X)) -> incr(active(X)) r14: active(s(X)) -> s(active(X)) r15: active(take(X1,X2)) -> take(active(X1),X2) r16: active(take(X1,X2)) -> take(X1,active(X2)) r17: active(zip(X1,X2)) -> zip(active(X1),X2) r18: active(zip(X1,X2)) -> zip(X1,active(X2)) r19: active(pair(X1,X2)) -> pair(active(X1),X2) r20: active(pair(X1,X2)) -> pair(X1,active(X2)) r21: active(tail(X)) -> tail(active(X)) r22: active(repItems(X)) -> repItems(active(X)) r23: cons(mark(X1),X2) -> mark(cons(X1,X2)) r24: incr(mark(X)) -> mark(incr(X)) r25: s(mark(X)) -> mark(s(X)) r26: take(mark(X1),X2) -> mark(take(X1,X2)) r27: take(X1,mark(X2)) -> mark(take(X1,X2)) r28: zip(mark(X1),X2) -> mark(zip(X1,X2)) r29: zip(X1,mark(X2)) -> mark(zip(X1,X2)) r30: pair(mark(X1),X2) -> mark(pair(X1,X2)) r31: pair(X1,mark(X2)) -> mark(pair(X1,X2)) r32: tail(mark(X)) -> mark(tail(X)) r33: repItems(mark(X)) -> mark(repItems(X)) r34: proper(pairNs()) -> ok(pairNs()) r35: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) r36: proper(|0|()) -> ok(|0|()) r37: proper(incr(X)) -> incr(proper(X)) r38: proper(oddNs()) -> ok(oddNs()) r39: proper(s(X)) -> s(proper(X)) r40: proper(take(X1,X2)) -> take(proper(X1),proper(X2)) r41: proper(nil()) -> ok(nil()) r42: proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) r43: proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) r44: proper(tail(X)) -> tail(proper(X)) r45: proper(repItems(X)) -> repItems(proper(X)) r46: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) r47: incr(ok(X)) -> ok(incr(X)) r48: s(ok(X)) -> ok(s(X)) r49: take(ok(X1),ok(X2)) -> ok(take(X1,X2)) r50: zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) r51: pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) r52: tail(ok(X)) -> ok(tail(X)) r53: repItems(ok(X)) -> ok(repItems(X)) r54: top(mark(X)) -> top(proper(X)) r55: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: tail#_A(x1) = x1 + (1,2) mark_A(x1) = x1 + (2,1) ok_A(x1) = ((1,0),(1,1)) x1 + (2,1) precedence: tail# = mark = ok partial status: pi(tail#) = [1] pi(mark) = [1] pi(ok) = [1] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: lexicographic order interpretations: tail#_A(x1) = ((1,0),(1,0)) x1 + (0,2) mark_A(x1) = ((1,0),(0,0)) x1 + (1,1) ok_A(x1) = ((1,0),(0,0)) x1 + (1,1) precedence: tail# = mark = ok partial status: pi(tail#) = [] pi(mark) = [] pi(ok) = [] The next rules are strictly ordered: p1, p2 We remove them from the problem. Then no dependency pair remains.