YES We show the termination of the TRS R: app(app(twice(),f),x) -> app(f,app(f,x)) 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(fmap(),nil()),x) -> nil() app(app(fmap(),app(app(cons(),f),t_f())),x) -> app(app(cons(),app(f,x)),app(app(fmap(),t_f()),x)) Instead we prove termination of the uncurried system: app(cons_0(),y) -> cons_1(y) app(cons_1(x1),y) -> cons_2(x1,y) app(fmap_0(),y) -> fmap_1(y) app(fmap_1(x1),y) -> fmap_2(x1,y) app(map_0(),y) -> map_1(y) app(map_1(x1),y) -> map_2(x1,y) app(twice_0(),y) -> twice_1(y) app(twice_1(x1),y) -> twice_2(x1,y) twice_2(f,x) -> app(f,app(f,x)) map_2(f,nil_0()) -> nil_0() map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) fmap_2(nil_0(),x) -> nil_0() fmap_2(cons_2(f,t_f_0()),x) -> cons_2(app(f,x),fmap_2(t_f_0(),x)) -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: app#(fmap_1(x1),y) -> fmap_2#(x1,y) p2: app#(map_1(x1),y) -> map_2#(x1,y) p3: app#(twice_1(x1),y) -> twice_2#(x1,y) p4: twice_2#(f,x) -> app#(f,app(f,x)) p5: twice_2#(f,x) -> app#(f,x) p6: map_2#(f,cons_2(h,t)) -> app#(f,h) p7: map_2#(f,cons_2(h,t)) -> map_2#(f,t) p8: fmap_2#(cons_2(f,t_f_0()),x) -> app#(f,x) p9: fmap_2#(cons_2(f,t_f_0()),x) -> fmap_2#(t_f_0(),x) and R consists of: r1: app(cons_0(),y) -> cons_1(y) r2: app(cons_1(x1),y) -> cons_2(x1,y) r3: app(fmap_0(),y) -> fmap_1(y) r4: app(fmap_1(x1),y) -> fmap_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: app(twice_0(),y) -> twice_1(y) r8: app(twice_1(x1),y) -> twice_2(x1,y) r9: twice_2(f,x) -> app(f,app(f,x)) r10: map_2(f,nil_0()) -> nil_0() r11: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r12: fmap_2(nil_0(),x) -> nil_0() r13: fmap_2(cons_2(f,t_f_0()),x) -> cons_2(app(f,x),fmap_2(t_f_0(),x)) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: app#(fmap_1(x1),y) -> fmap_2#(x1,y) p2: fmap_2#(cons_2(f,t_f_0()),x) -> app#(f,x) p3: app#(twice_1(x1),y) -> twice_2#(x1,y) p4: twice_2#(f,x) -> app#(f,x) p5: app#(map_1(x1),y) -> map_2#(x1,y) p6: map_2#(f,cons_2(h,t)) -> map_2#(f,t) p7: map_2#(f,cons_2(h,t)) -> app#(f,h) p8: twice_2#(f,x) -> app#(f,app(f,x)) and R consists of: r1: app(cons_0(),y) -> cons_1(y) r2: app(cons_1(x1),y) -> cons_2(x1,y) r3: app(fmap_0(),y) -> fmap_1(y) r4: app(fmap_1(x1),y) -> fmap_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: app(twice_0(),y) -> twice_1(y) r8: app(twice_1(x1),y) -> twice_2(x1,y) r9: twice_2(f,x) -> app(f,app(f,x)) r10: map_2(f,nil_0()) -> nil_0() r11: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r12: fmap_2(nil_0(),x) -> nil_0() r13: fmap_2(cons_2(f,t_f_0()),x) -> cons_2(app(f,x),fmap_2(t_f_0(),x)) The set of usable rules consists of r1, r12 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: app#_A(x1,x2) = x1 + 1 fmap_1_A(x1) = x1 + 1 fmap_2#_A(x1,x2) = x1 + 1 cons_2_A(x1,x2) = x1 + x2 + 3 t_f_0_A() = 2 twice_1_A(x1) = x1 + 2 twice_2#_A(x1,x2) = x1 + 2 map_1_A(x1) = x1 + 2 map_2#_A(x1,x2) = x1 + 2 app_A(x1,x2) = 0 cons_0_A() = 0 cons_1_A(x1) = 0 fmap_2_A(x1,x2) = x1 nil_0_A() = 1 base order 2: precedence: t_f_0 = twice_1 > app# = app = cons_0 = fmap_2 > fmap_1 = twice_2# > fmap_2# = map_1 = nil_0 > cons_2 = map_2# > cons_1 partial status: pi(app#) = [1] pi(fmap_1) = [1] pi(fmap_2#) = [] pi(cons_2) = [2] pi(t_f_0) = [] pi(twice_1) = [] pi(twice_2#) = [] pi(map_1) = [] pi(map_2#) = [] pi(app) = [] pi(cons_0) = [] pi(cons_1) = [] pi(fmap_2) = [] pi(nil_0) = [] The following rules are removed: p7 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: app#(fmap_1(x1),y) -> fmap_2#(x1,y) p2: fmap_2#(cons_2(f,t_f_0()),x) -> app#(f,x) p3: app#(twice_1(x1),y) -> twice_2#(x1,y) p4: twice_2#(f,x) -> app#(f,x) p5: app#(map_1(x1),y) -> map_2#(x1,y) p6: map_2#(f,cons_2(h,t)) -> map_2#(f,t) p7: twice_2#(f,x) -> app#(f,app(f,x)) and R consists of: r1: app(cons_0(),y) -> cons_1(y) r2: app(cons_1(x1),y) -> cons_2(x1,y) r3: app(fmap_0(),y) -> fmap_1(y) r4: app(fmap_1(x1),y) -> fmap_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: app(twice_0(),y) -> twice_1(y) r8: app(twice_1(x1),y) -> twice_2(x1,y) r9: twice_2(f,x) -> app(f,app(f,x)) r10: map_2(f,nil_0()) -> nil_0() r11: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r12: fmap_2(nil_0(),x) -> nil_0() r13: fmap_2(cons_2(f,t_f_0()),x) -> cons_2(app(f,x),fmap_2(t_f_0(),x)) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p7} {p6} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: app#(fmap_1(x1),y) -> fmap_2#(x1,y) p2: fmap_2#(cons_2(f,t_f_0()),x) -> app#(f,x) p3: app#(twice_1(x1),y) -> twice_2#(x1,y) p4: twice_2#(f,x) -> app#(f,app(f,x)) p5: twice_2#(f,x) -> app#(f,x) and R consists of: r1: app(cons_0(),y) -> cons_1(y) r2: app(cons_1(x1),y) -> cons_2(x1,y) r3: app(fmap_0(),y) -> fmap_1(y) r4: app(fmap_1(x1),y) -> fmap_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: app(twice_0(),y) -> twice_1(y) r8: app(twice_1(x1),y) -> twice_2(x1,y) r9: twice_2(f,x) -> app(f,app(f,x)) r10: map_2(f,nil_0()) -> nil_0() r11: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r12: fmap_2(nil_0(),x) -> nil_0() r13: fmap_2(cons_2(f,t_f_0()),x) -> cons_2(app(f,x),fmap_2(t_f_0(),x)) The set of usable rules consists of r5, r12 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: app#_A(x1,x2) = x1 + 8 fmap_1_A(x1) = x1 + 3 fmap_2#_A(x1,x2) = x1 + 4 cons_2_A(x1,x2) = x1 + x2 + 2 t_f_0_A() = 7 twice_1_A(x1) = x1 + 10 twice_2#_A(x1,x2) = x1 + 9 app_A(x1,x2) = 3 map_0_A() = 2 map_1_A(x1) = 1 fmap_2_A(x1,x2) = 3 nil_0_A() = 1 base order 2: precedence: nil_0 > cons_2 = app = map_0 > fmap_2# = twice_2# = map_1 > app# = twice_1 > t_f_0 > fmap_2 > fmap_1 partial status: pi(app#) = [] pi(fmap_1) = [] pi(fmap_2#) = [1] pi(cons_2) = [2] pi(t_f_0) = [] pi(twice_1) = [] pi(twice_2#) = [1] pi(app) = [] pi(map_0) = [] pi(map_1) = [] pi(fmap_2) = [] pi(nil_0) = [] The following rules are removed: p5 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: app#(fmap_1(x1),y) -> fmap_2#(x1,y) p2: fmap_2#(cons_2(f,t_f_0()),x) -> app#(f,x) p3: app#(twice_1(x1),y) -> twice_2#(x1,y) p4: twice_2#(f,x) -> app#(f,app(f,x)) and R consists of: r1: app(cons_0(),y) -> cons_1(y) r2: app(cons_1(x1),y) -> cons_2(x1,y) r3: app(fmap_0(),y) -> fmap_1(y) r4: app(fmap_1(x1),y) -> fmap_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: app(twice_0(),y) -> twice_1(y) r8: app(twice_1(x1),y) -> twice_2(x1,y) r9: twice_2(f,x) -> app(f,app(f,x)) r10: map_2(f,nil_0()) -> nil_0() r11: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r12: fmap_2(nil_0(),x) -> nil_0() r13: fmap_2(cons_2(f,t_f_0()),x) -> cons_2(app(f,x),fmap_2(t_f_0(),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: app#(fmap_1(x1),y) -> fmap_2#(x1,y) p2: fmap_2#(cons_2(f,t_f_0()),x) -> app#(f,x) p3: app#(twice_1(x1),y) -> twice_2#(x1,y) p4: twice_2#(f,x) -> app#(f,app(f,x)) and R consists of: r1: app(cons_0(),y) -> cons_1(y) r2: app(cons_1(x1),y) -> cons_2(x1,y) r3: app(fmap_0(),y) -> fmap_1(y) r4: app(fmap_1(x1),y) -> fmap_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: app(twice_0(),y) -> twice_1(y) r8: app(twice_1(x1),y) -> twice_2(x1,y) r9: twice_2(f,x) -> app(f,app(f,x)) r10: map_2(f,nil_0()) -> nil_0() r11: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r12: fmap_2(nil_0(),x) -> nil_0() r13: fmap_2(cons_2(f,t_f_0()),x) -> cons_2(app(f,x),fmap_2(t_f_0(),x)) The set of usable rules consists of r1, r2, r3, r5, r7, r12 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: app#_A(x1,x2) = x1 + 3 fmap_1_A(x1) = x1 + 2 fmap_2#_A(x1,x2) = x1 cons_2_A(x1,x2) = x1 + x2 + 4 t_f_0_A() = 2 twice_1_A(x1) = x1 + 5 twice_2#_A(x1,x2) = x1 + 4 app_A(x1,x2) = x1 + x2 + 5 cons_0_A() = 1 cons_1_A(x1) = x1 + 5 fmap_0_A() = 1 map_0_A() = 1 map_1_A(x1) = x1 + 2 twice_0_A() = 1 fmap_2_A(x1,x2) = x1 + x2 + 5 nil_0_A() = 1 base order 2: precedence: app > fmap_2# = cons_2 = twice_2# = fmap_2 > fmap_1 = t_f_0 = twice_1 = cons_1 > app# = cons_0 = fmap_0 = map_1 = twice_0 = nil_0 > map_0 partial status: pi(app#) = [] pi(fmap_1) = [1] pi(fmap_2#) = [1] pi(cons_2) = [] pi(t_f_0) = [] pi(twice_1) = [1] pi(twice_2#) = [1] pi(app) = [] pi(cons_0) = [] pi(cons_1) = [1] pi(fmap_0) = [] pi(map_0) = [] pi(map_1) = [1] pi(twice_0) = [] pi(fmap_2) = [1] pi(nil_0) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: fmap_2#(cons_2(f,t_f_0()),x) -> app#(f,x) p2: app#(twice_1(x1),y) -> twice_2#(x1,y) p3: twice_2#(f,x) -> app#(f,app(f,x)) and R consists of: r1: app(cons_0(),y) -> cons_1(y) r2: app(cons_1(x1),y) -> cons_2(x1,y) r3: app(fmap_0(),y) -> fmap_1(y) r4: app(fmap_1(x1),y) -> fmap_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: app(twice_0(),y) -> twice_1(y) r8: app(twice_1(x1),y) -> twice_2(x1,y) r9: twice_2(f,x) -> app(f,app(f,x)) r10: map_2(f,nil_0()) -> nil_0() r11: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r12: fmap_2(nil_0(),x) -> nil_0() r13: fmap_2(cons_2(f,t_f_0()),x) -> cons_2(app(f,x),fmap_2(t_f_0(),x)) The estimated dependency graph contains the following SCCs: {p2, p3} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: app#(twice_1(x1),y) -> twice_2#(x1,y) p2: twice_2#(f,x) -> app#(f,app(f,x)) and R consists of: r1: app(cons_0(),y) -> cons_1(y) r2: app(cons_1(x1),y) -> cons_2(x1,y) r3: app(fmap_0(),y) -> fmap_1(y) r4: app(fmap_1(x1),y) -> fmap_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: app(twice_0(),y) -> twice_1(y) r8: app(twice_1(x1),y) -> twice_2(x1,y) r9: twice_2(f,x) -> app(f,app(f,x)) r10: map_2(f,nil_0()) -> nil_0() r11: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r12: fmap_2(nil_0(),x) -> nil_0() r13: fmap_2(cons_2(f,t_f_0()),x) -> cons_2(app(f,x),fmap_2(t_f_0(),x)) The set of usable rules consists of r12 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: app#_A(x1,x2) = x1 twice_1_A(x1) = x1 + 2 twice_2#_A(x1,x2) = x1 + 1 app_A(x1,x2) = 0 fmap_2_A(x1,x2) = 0 nil_0_A() = 0 base order 2: precedence: app# = twice_1 = app > twice_2# = fmap_2 > nil_0 partial status: pi(app#) = [] pi(twice_1) = [1] pi(twice_2#) = [] pi(app) = [] pi(fmap_2) = [] pi(nil_0) = [] The following rules are removed: p2 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: app#(twice_1(x1),y) -> twice_2#(x1,y) and R consists of: r1: app(cons_0(),y) -> cons_1(y) r2: app(cons_1(x1),y) -> cons_2(x1,y) r3: app(fmap_0(),y) -> fmap_1(y) r4: app(fmap_1(x1),y) -> fmap_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: app(twice_0(),y) -> twice_1(y) r8: app(twice_1(x1),y) -> twice_2(x1,y) r9: twice_2(f,x) -> app(f,app(f,x)) r10: map_2(f,nil_0()) -> nil_0() r11: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r12: fmap_2(nil_0(),x) -> nil_0() r13: fmap_2(cons_2(f,t_f_0()),x) -> cons_2(app(f,x),fmap_2(t_f_0(),x)) The estimated dependency graph contains the following SCCs: (no SCCs) -- 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) and R consists of: r1: app(cons_0(),y) -> cons_1(y) r2: app(cons_1(x1),y) -> cons_2(x1,y) r3: app(fmap_0(),y) -> fmap_1(y) r4: app(fmap_1(x1),y) -> fmap_2(x1,y) r5: app(map_0(),y) -> map_1(y) r6: app(map_1(x1),y) -> map_2(x1,y) r7: app(twice_0(),y) -> twice_1(y) r8: app(twice_1(x1),y) -> twice_2(x1,y) r9: twice_2(f,x) -> app(f,app(f,x)) r10: map_2(f,nil_0()) -> nil_0() r11: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t)) r12: fmap_2(nil_0(),x) -> nil_0() r13: fmap_2(cons_2(f,t_f_0()),x) -> cons_2(app(f,x),fmap_2(t_f_0(),x)) The set of usable rules consists of r1, r2, r3, r5, r7, r12 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 + 3 app_A(x1,x2) = x1 + x2 + 5 cons_0_A() = 1 cons_1_A(x1) = x1 + 2 fmap_0_A() = 1 fmap_1_A(x1) = x1 + 2 map_0_A() = 1 map_1_A(x1) = x1 + 2 twice_0_A() = 1 twice_1_A(x1) = x1 + 2 fmap_2_A(x1,x2) = x1 + x2 + 4 nil_0_A() = 1 base order 2: precedence: fmap_2 > nil_0 > app > cons_2 = fmap_1 = map_1 = twice_1 > map_2# = cons_1 = fmap_0 = map_0 = twice_0 > cons_0 partial status: pi(map_2#) = [] pi(cons_2) = [1, 2] pi(app) = [] pi(cons_0) = [] pi(cons_1) = [1] pi(fmap_0) = [] pi(fmap_1) = [1] pi(map_0) = [] pi(map_1) = [1] pi(twice_0) = [] pi(twice_1) = [1] pi(fmap_2) = [] pi(nil_0) = [] The following rules are removed: p1 Then no dependency pair remains.