YES We show the termination of the TRS R: app(app(append(),nil()),l) -> l app(app(append(),app(app(cons(),h),t)),l) -> app(app(cons(),h),app(app(append(),t),l)) app(app(map(),f),nil()) -> nil() app(app(map(),f),app(app(cons(),h),t)) -> app(app(cons(),app(f,h)),app(app(map(),f),t)) app(app(append(),app(app(append(),l1),l2)),l3) -> app(app(append(),l1),app(app(append(),l2),l3)) app(app(map(),f),app(app(append(),l1),l2)) -> app(app(append(),app(app(map(),f),l1)),app(app(map(),f),l2)) Instead we prove termination of the uncurried system: app(append_0(),y) -> append_1(y) app(append_1(x1),y) -> append_2(x1,y) app(cons_0(),y) -> cons_1(y) app(cons_1(x1),y) -> cons_2(x1,y) app(map_0(),y) -> map_1(y) app(map_1(x1),y) -> map_2(x1,y) append_2(nil_0(),l) -> l append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l)) map_2(f,nil_0()) -> nil_0() map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3)) map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2)) -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: app#(append_1(x1),y) -> append_2#(x1,y) p2: app#(map_1(x1),y) -> map_2#(x1,y) p3: append_2#(cons_2(h,t),l) -> append_2#(t,l) p4: map_2#(f,cons_2(h,t)) -> app#(f,h) p5: map_2#(f,cons_2(h,t)) -> map_2#(f,t) p6: append_2#(append_2(l1,l2),l3) -> append_2#(l1,append_2(l2,l3)) p7: append_2#(append_2(l1,l2),l3) -> append_2#(l2,l3) p8: map_2#(f,append_2(l1,l2)) -> append_2#(map_2(f,l1),map_2(f,l2)) p9: map_2#(f,append_2(l1,l2)) -> map_2#(f,l1) p10: map_2#(f,append_2(l1,l2)) -> map_2#(f,l2) and R consists of: r1: app(append_0(),y) -> append_1(y) r2: app(append_1(x1),y) -> append_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: append_2(nil_0(),l) -> l r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l)) r9: map_2(f,nil_0()) -> nil_0() r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3)) r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2)) The estimated dependency graph contains the following SCCs: {p2, p4, p5, p9, p10} {p3, p6, p7} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: map_2#(f,cons_2(h,t)) -> app#(f,h) p2: app#(map_1(x1),y) -> map_2#(x1,y) p3: map_2#(f,append_2(l1,l2)) -> map_2#(f,l2) p4: map_2#(f,append_2(l1,l2)) -> map_2#(f,l1) p5: map_2#(f,cons_2(h,t)) -> map_2#(f,t) and R consists of: r1: app(append_0(),y) -> append_1(y) r2: app(append_1(x1),y) -> append_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: append_2(nil_0(),l) -> l r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l)) r9: map_2(f,nil_0()) -> nil_0() r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3)) r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2)) The set of usable rules consists of r1, r3, r5, r7, r8, r9, r11 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: map_2#_A(x1,x2) = x2 + 2 cons_2_A(x1,x2) = x1 + x2 + 4 app#_A(x1,x2) = x2 + 3 map_1_A(x1) = 1 append_2_A(x1,x2) = x1 + x2 + 5 app_A(x1,x2) = x2 + 2 append_0_A() = 2 append_1_A(x1) = 1 cons_0_A() = 2 cons_1_A(x1) = 1 map_0_A() = 2 nil_0_A() = 1 map_2_A(x1,x2) = x1 + x2 + 6 base order 2: precedence: app > append_2 = append_0 > cons_2 = append_1 = cons_0 = map_0 > cons_1 = map_2 > map_2# = map_1 = nil_0 > app# partial status: pi(map_2#) = [] pi(cons_2) = [1] pi(app#) = [] pi(map_1) = [] pi(append_2) = [] pi(app) = [] pi(append_0) = [] pi(append_1) = [] pi(cons_0) = [] pi(cons_1) = [] pi(map_0) = [] pi(nil_0) = [] pi(map_2) = [2] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: app#(map_1(x1),y) -> map_2#(x1,y) p2: map_2#(f,append_2(l1,l2)) -> map_2#(f,l2) p3: map_2#(f,append_2(l1,l2)) -> map_2#(f,l1) p4: map_2#(f,cons_2(h,t)) -> map_2#(f,t) and R consists of: r1: app(append_0(),y) -> append_1(y) r2: app(append_1(x1),y) -> append_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: append_2(nil_0(),l) -> l r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l)) r9: map_2(f,nil_0()) -> nil_0() r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3)) r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2)) The estimated dependency graph contains the following SCCs: {p2, p3, p4} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: map_2#(f,append_2(l1,l2)) -> map_2#(f,l2) p2: map_2#(f,cons_2(h,t)) -> map_2#(f,t) p3: map_2#(f,append_2(l1,l2)) -> map_2#(f,l1) and R consists of: r1: app(append_0(),y) -> append_1(y) r2: app(append_1(x1),y) -> append_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: append_2(nil_0(),l) -> l r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l)) r9: map_2(f,nil_0()) -> nil_0() r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3)) r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2)) The set of usable rules consists of r1, r3, r5, r7, r9 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: map_2#_A(x1,x2) = x2 append_2_A(x1,x2) = x1 + x2 + 4 cons_2_A(x1,x2) = x1 + x2 + 3 app_A(x1,x2) = x2 + 2 append_0_A() = 2 append_1_A(x1) = 1 cons_0_A() = 2 cons_1_A(x1) = 1 map_0_A() = 2 map_1_A(x1) = 1 nil_0_A() = 1 map_2_A(x1,x2) = x1 + x2 + 5 base order 2: precedence: map_2# = append_2 = cons_2 = cons_0 > cons_1 = map_2 > app = append_0 = map_0 = nil_0 > append_1 = map_1 partial status: pi(map_2#) = [] pi(append_2) = [1] pi(cons_2) = [2] pi(app) = [] pi(append_0) = [] pi(append_1) = [] pi(cons_0) = [] pi(cons_1) = [] pi(map_0) = [] pi(map_1) = [] pi(nil_0) = [] pi(map_2) = [2] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: map_2#(f,cons_2(h,t)) -> map_2#(f,t) p2: map_2#(f,append_2(l1,l2)) -> map_2#(f,l1) and R consists of: r1: app(append_0(),y) -> append_1(y) r2: app(append_1(x1),y) -> append_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: append_2(nil_0(),l) -> l r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l)) r9: map_2(f,nil_0()) -> nil_0() r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3)) r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2)) The estimated dependency graph contains the following SCCs: {p1, p2} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: map_2#(f,cons_2(h,t)) -> map_2#(f,t) p2: map_2#(f,append_2(l1,l2)) -> map_2#(f,l1) and R consists of: r1: app(append_0(),y) -> append_1(y) r2: app(append_1(x1),y) -> append_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: append_2(nil_0(),l) -> l r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l)) r9: map_2(f,nil_0()) -> nil_0() r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3)) r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2)) The set of usable rules consists of r7, r9 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: map_2#_A(x1,x2) = x2 cons_2_A(x1,x2) = x1 + x2 + 2 append_2_A(x1,x2) = x1 + x2 + 3 nil_0_A() = 1 map_2_A(x1,x2) = x1 + x2 + 4 base order 2: precedence: map_2 > append_2 = nil_0 > cons_2 > map_2# partial status: pi(map_2#) = [] pi(cons_2) = [] pi(append_2) = [2] pi(nil_0) = [] pi(map_2) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: map_2#(f,append_2(l1,l2)) -> map_2#(f,l1) and R consists of: r1: app(append_0(),y) -> append_1(y) r2: app(append_1(x1),y) -> append_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: append_2(nil_0(),l) -> l r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l)) r9: map_2(f,nil_0()) -> nil_0() r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3)) r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2)) The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: map_2#(f,append_2(l1,l2)) -> map_2#(f,l1) and R consists of: r1: app(append_0(),y) -> append_1(y) r2: app(append_1(x1),y) -> append_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: append_2(nil_0(),l) -> l r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l)) r9: map_2(f,nil_0()) -> nil_0() r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3)) r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2)) The set of usable rules consists of r1, r3, r5, r7, r8, r9, r11 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: map_2#_A(x1,x2) = x2 append_2_A(x1,x2) = x1 + x2 + 4 app_A(x1,x2) = x2 + 3 append_0_A() = 2 append_1_A(x1) = 1 cons_0_A() = 3 cons_1_A(x1) = 2 map_0_A() = 2 map_1_A(x1) = 1 nil_0_A() = 1 cons_2_A(x1,x2) = x1 + x2 + 1 map_2_A(x1,x2) = x1 + x2 + 5 base order 2: precedence: append_0 > append_2 = app = append_1 = cons_1 = map_0 = cons_2 = map_2 > map_2# = cons_0 = map_1 = nil_0 partial status: pi(map_2#) = [] pi(append_2) = [] pi(app) = [2] pi(append_0) = [] pi(append_1) = [] pi(cons_0) = [] pi(cons_1) = [] pi(map_0) = [] pi(map_1) = [] pi(nil_0) = [] pi(cons_2) = [] pi(map_2) = [2] The following rules are removed: p1 Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: append_2#(cons_2(h,t),l) -> append_2#(t,l) p2: append_2#(append_2(l1,l2),l3) -> append_2#(l2,l3) p3: append_2#(append_2(l1,l2),l3) -> append_2#(l1,append_2(l2,l3)) and R consists of: r1: app(append_0(),y) -> append_1(y) r2: app(append_1(x1),y) -> append_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: append_2(nil_0(),l) -> l r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l)) r9: map_2(f,nil_0()) -> nil_0() r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3)) r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2)) The set of usable rules consists of r1, r3, r5, r7, r8, r9, r11 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: append_2#_A(x1,x2) = x1 + x2 + 5 cons_2_A(x1,x2) = x1 + x2 + 2 append_2_A(x1,x2) = x1 + x2 + 4 app_A(x1,x2) = x2 + 3 append_0_A() = 2 append_1_A(x1) = 1 cons_0_A() = 2 cons_1_A(x1) = 1 map_0_A() = 2 map_1_A(x1) = 1 nil_0_A() = 1 map_2_A(x1,x2) = x1 + x2 + 5 base order 2: precedence: append_2# = map_2 > append_2 > cons_2 = app = append_0 = cons_0 = map_0 = nil_0 > append_1 = cons_1 = map_1 partial status: pi(append_2#) = [] pi(cons_2) = [2] pi(append_2) = [1, 2] pi(app) = [2] pi(append_0) = [] pi(append_1) = [] pi(cons_0) = [] pi(cons_1) = [] pi(map_0) = [] pi(map_1) = [] pi(nil_0) = [] pi(map_2) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: append_2#(append_2(l1,l2),l3) -> append_2#(l2,l3) p2: append_2#(append_2(l1,l2),l3) -> append_2#(l1,append_2(l2,l3)) and R consists of: r1: app(append_0(),y) -> append_1(y) r2: app(append_1(x1),y) -> append_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: append_2(nil_0(),l) -> l r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l)) r9: map_2(f,nil_0()) -> nil_0() r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3)) r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2)) The estimated dependency graph contains the following SCCs: {p1, p2} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: append_2#(append_2(l1,l2),l3) -> append_2#(l2,l3) p2: append_2#(append_2(l1,l2),l3) -> append_2#(l1,append_2(l2,l3)) and R consists of: r1: app(append_0(),y) -> append_1(y) r2: app(append_1(x1),y) -> append_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: append_2(nil_0(),l) -> l r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l)) r9: map_2(f,nil_0()) -> nil_0() r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3)) r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2)) The set of usable rules consists of r1, r3, r5, r7, r8, r9, r11 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: append_2#_A(x1,x2) = x1 + x2 + 6 append_2_A(x1,x2) = x1 + x2 + 5 app_A(x1,x2) = x2 + 3 append_0_A() = 2 append_1_A(x1) = 1 cons_0_A() = 1 cons_1_A(x1) = 2 map_0_A() = 2 map_1_A(x1) = 1 nil_0_A() = 1 cons_2_A(x1,x2) = x1 + x2 + 4 map_2_A(x1,x2) = x1 + x2 + 6 base order 2: precedence: append_2# > append_2 = cons_2 = map_2 > app = append_0 = cons_0 = map_0 = nil_0 > append_1 = cons_1 = map_1 partial status: pi(append_2#) = [] pi(append_2) = [] pi(app) = [2] pi(append_0) = [] pi(append_1) = [] pi(cons_0) = [] pi(cons_1) = [] pi(map_0) = [] pi(map_1) = [] pi(nil_0) = [] pi(cons_2) = [] pi(map_2) = [2] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: append_2#(append_2(l1,l2),l3) -> append_2#(l1,append_2(l2,l3)) and R consists of: r1: app(append_0(),y) -> append_1(y) r2: app(append_1(x1),y) -> append_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: append_2(nil_0(),l) -> l r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l)) r9: map_2(f,nil_0()) -> nil_0() r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3)) r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2)) The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: append_2#(append_2(l1,l2),l3) -> append_2#(l1,append_2(l2,l3)) and R consists of: r1: app(append_0(),y) -> append_1(y) r2: app(append_1(x1),y) -> append_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: append_2(nil_0(),l) -> l r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l)) r9: map_2(f,nil_0()) -> nil_0() r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3)) r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2)) The set of usable rules consists of r1, r3, r5, r7, r8, r9, r11 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: append_2#_A(x1,x2) = x1 + x2 + 1 append_2_A(x1,x2) = x1 + x2 + 4 app_A(x1,x2) = x2 + 2 append_0_A() = 2 append_1_A(x1) = 1 cons_0_A() = 2 cons_1_A(x1) = 1 map_0_A() = 2 map_1_A(x1) = 1 nil_0_A() = 1 cons_2_A(x1,x2) = x1 + x2 + 3 map_2_A(x1,x2) = x1 + x2 + 5 base order 2: precedence: append_2# > append_2 = cons_0 = map_2 > nil_0 = cons_2 > cons_1 > app = append_0 = map_0 > append_1 = map_1 partial status: pi(append_2#) = [1] pi(append_2) = [1] pi(app) = [] pi(append_0) = [] pi(append_1) = [] pi(cons_0) = [] pi(cons_1) = [] pi(map_0) = [] pi(map_1) = [] pi(nil_0) = [] pi(cons_2) = [1] pi(map_2) = [] The following rules are removed: p1 Then no dependency pair remains.