*** 1 Progress [(O(1),O(n^1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: +Full(0(),y) -> y +Full(S(x),y) -> +Full(x,S(y)) f(x) -> *(x,x) goal(xs) -> map(xs) map(Cons(x,xs)) -> Cons(f(x),map(xs)) map(Nil()) -> Nil() Weak DP Rules: Weak TRS Rules: *(x,0()) -> 0() *(x,S(0())) -> x *(x,S(S(y))) -> +(x,*(x,S(y))) *(0(),y) -> 0() Signature: {*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1} Obligation: Innermost basic terms: {*,+Full,f,goal,map}/{+,0,Cons,Nil,S} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(+) = {2}, uargs(Cons) = {1,2} Following symbols are considered usable: {} TcT has computed the following interpretation: p(*) = [12] x1 + [3] p(+) = [1] x2 + [0] p(+Full) = [3] x2 + [0] p(0) = [0] p(Cons) = [1] x1 + [1] x2 + [0] p(Nil) = [0] p(S) = [8] p(f) = [12] x1 + [1] p(goal) = [13] x1 + [1] p(map) = [13] x1 + [12] Following rules are strictly oriented: map(Nil()) = [12] > [0] = Nil() Following rules are (at-least) weakly oriented: *(x,0()) = [12] x + [3] >= [0] = 0() *(x,S(0())) = [12] x + [3] >= [1] x + [0] = x *(x,S(S(y))) = [12] x + [3] >= [12] x + [3] = +(x,*(x,S(y))) *(0(),y) = [3] >= [0] = 0() +Full(0(),y) = [3] y + [0] >= [1] y + [0] = y +Full(S(x),y) = [3] y + [0] >= [24] = +Full(x,S(y)) f(x) = [12] x + [1] >= [12] x + [3] = *(x,x) goal(xs) = [13] xs + [1] >= [13] xs + [12] = map(xs) map(Cons(x,xs)) = [13] x + [13] xs + [12] >= [12] x + [13] xs + [13] = Cons(f(x),map(xs)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1 Progress [(O(1),O(n^1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: +Full(0(),y) -> y +Full(S(x),y) -> +Full(x,S(y)) f(x) -> *(x,x) goal(xs) -> map(xs) map(Cons(x,xs)) -> Cons(f(x),map(xs)) Weak DP Rules: Weak TRS Rules: *(x,0()) -> 0() *(x,S(0())) -> x *(x,S(S(y))) -> +(x,*(x,S(y))) *(0(),y) -> 0() map(Nil()) -> Nil() Signature: {*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1} Obligation: Innermost basic terms: {*,+Full,f,goal,map}/{+,0,Cons,Nil,S} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(+) = {2}, uargs(Cons) = {1,2} Following symbols are considered usable: {} TcT has computed the following interpretation: p(*) = [2] x1 + [12] p(+) = [1] x2 + [0] p(+Full) = [8] x2 + [1] p(0) = [4] p(Cons) = [1] x1 + [1] x2 + [3] p(Nil) = [4] p(S) = [0] p(f) = [2] x1 + [4] p(goal) = [7] x1 + [0] p(map) = [7] x1 + [2] Following rules are strictly oriented: +Full(0(),y) = [8] y + [1] > [1] y + [0] = y map(Cons(x,xs)) = [7] x + [7] xs + [23] > [2] x + [7] xs + [9] = Cons(f(x),map(xs)) Following rules are (at-least) weakly oriented: *(x,0()) = [2] x + [12] >= [4] = 0() *(x,S(0())) = [2] x + [12] >= [1] x + [0] = x *(x,S(S(y))) = [2] x + [12] >= [2] x + [12] = +(x,*(x,S(y))) *(0(),y) = [20] >= [4] = 0() +Full(S(x),y) = [8] y + [1] >= [1] = +Full(x,S(y)) f(x) = [2] x + [4] >= [2] x + [12] = *(x,x) goal(xs) = [7] xs + [0] >= [7] xs + [2] = map(xs) map(Nil()) = [30] >= [4] = Nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1 Progress [(O(1),O(n^1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: +Full(S(x),y) -> +Full(x,S(y)) f(x) -> *(x,x) goal(xs) -> map(xs) Weak DP Rules: Weak TRS Rules: *(x,0()) -> 0() *(x,S(0())) -> x *(x,S(S(y))) -> +(x,*(x,S(y))) *(0(),y) -> 0() +Full(0(),y) -> y map(Cons(x,xs)) -> Cons(f(x),map(xs)) map(Nil()) -> Nil() Signature: {*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1} Obligation: Innermost basic terms: {*,+Full,f,goal,map}/{+,0,Cons,Nil,S} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(+) = {2}, uargs(Cons) = {1,2} Following symbols are considered usable: {} TcT has computed the following interpretation: p(*) = [1] x1 + [0] p(+) = [1] x2 + [0] p(+Full) = [8] x1 + [1] x2 + [5] p(0) = [0] p(Cons) = [1] x1 + [1] x2 + [0] p(Nil) = [1] p(S) = [1] x1 + [0] p(f) = [2] x1 + [0] p(goal) = [4] x1 + [10] p(map) = [4] x1 + [1] Following rules are strictly oriented: goal(xs) = [4] xs + [10] > [4] xs + [1] = map(xs) Following rules are (at-least) weakly oriented: *(x,0()) = [1] x + [0] >= [0] = 0() *(x,S(0())) = [1] x + [0] >= [1] x + [0] = x *(x,S(S(y))) = [1] x + [0] >= [1] x + [0] = +(x,*(x,S(y))) *(0(),y) = [0] >= [0] = 0() +Full(0(),y) = [1] y + [5] >= [1] y + [0] = y +Full(S(x),y) = [8] x + [1] y + [5] >= [8] x + [1] y + [5] = +Full(x,S(y)) f(x) = [2] x + [0] >= [1] x + [0] = *(x,x) map(Cons(x,xs)) = [4] x + [4] xs + [1] >= [2] x + [4] xs + [1] = Cons(f(x),map(xs)) map(Nil()) = [5] >= [1] = Nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1 Progress [(O(1),O(n^1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: +Full(S(x),y) -> +Full(x,S(y)) f(x) -> *(x,x) Weak DP Rules: Weak TRS Rules: *(x,0()) -> 0() *(x,S(0())) -> x *(x,S(S(y))) -> +(x,*(x,S(y))) *(0(),y) -> 0() +Full(0(),y) -> y goal(xs) -> map(xs) map(Cons(x,xs)) -> Cons(f(x),map(xs)) map(Nil()) -> Nil() Signature: {*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1} Obligation: Innermost basic terms: {*,+Full,f,goal,map}/{+,0,Cons,Nil,S} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(+) = {2}, uargs(Cons) = {1,2} Following symbols are considered usable: {} TcT has computed the following interpretation: p(*) = [1] x1 + [2] x2 + [0] p(+) = [1] x2 + [8] p(+Full) = [4] x1 + [5] x2 + [1] p(0) = [4] p(Cons) = [1] x1 + [1] x2 + [8] p(Nil) = [3] p(S) = [1] x1 + [4] p(f) = [3] x1 + [8] p(goal) = [3] x1 + [4] p(map) = [3] x1 + [4] Following rules are strictly oriented: f(x) = [3] x + [8] > [3] x + [0] = *(x,x) Following rules are (at-least) weakly oriented: *(x,0()) = [1] x + [8] >= [4] = 0() *(x,S(0())) = [1] x + [16] >= [1] x + [0] = x *(x,S(S(y))) = [1] x + [2] y + [16] >= [1] x + [2] y + [16] = +(x,*(x,S(y))) *(0(),y) = [2] y + [4] >= [4] = 0() +Full(0(),y) = [5] y + [17] >= [1] y + [0] = y +Full(S(x),y) = [4] x + [5] y + [17] >= [4] x + [5] y + [21] = +Full(x,S(y)) goal(xs) = [3] xs + [4] >= [3] xs + [4] = map(xs) map(Cons(x,xs)) = [3] x + [3] xs + [28] >= [3] x + [3] xs + [20] = Cons(f(x),map(xs)) map(Nil()) = [13] >= [3] = Nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.1 Progress [(O(1),O(n^1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: +Full(S(x),y) -> +Full(x,S(y)) Weak DP Rules: Weak TRS Rules: *(x,0()) -> 0() *(x,S(0())) -> x *(x,S(S(y))) -> +(x,*(x,S(y))) *(0(),y) -> 0() +Full(0(),y) -> y f(x) -> *(x,x) goal(xs) -> map(xs) map(Cons(x,xs)) -> Cons(f(x),map(xs)) map(Nil()) -> Nil() Signature: {*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1} Obligation: Innermost basic terms: {*,+Full,f,goal,map}/{+,0,Cons,Nil,S} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(+) = {2}, uargs(Cons) = {1,2} Following symbols are considered usable: {} TcT has computed the following interpretation: p(*) = [8] x1 + [9] p(+) = [1] x2 + [0] p(+Full) = [6] x1 + [4] x2 + [1] p(0) = [2] p(Cons) = [1] x1 + [1] x2 + [2] p(Nil) = [0] p(S) = [1] x1 + [4] p(f) = [8] x1 + [9] p(goal) = [10] x1 + [8] p(map) = [8] x1 + [6] Following rules are strictly oriented: +Full(S(x),y) = [6] x + [4] y + [25] > [6] x + [4] y + [17] = +Full(x,S(y)) Following rules are (at-least) weakly oriented: *(x,0()) = [8] x + [9] >= [2] = 0() *(x,S(0())) = [8] x + [9] >= [1] x + [0] = x *(x,S(S(y))) = [8] x + [9] >= [8] x + [9] = +(x,*(x,S(y))) *(0(),y) = [25] >= [2] = 0() +Full(0(),y) = [4] y + [13] >= [1] y + [0] = y f(x) = [8] x + [9] >= [8] x + [9] = *(x,x) goal(xs) = [10] xs + [8] >= [8] xs + [6] = map(xs) map(Cons(x,xs)) = [8] x + [8] xs + [22] >= [8] x + [8] xs + [17] = Cons(f(x),map(xs)) map(Nil()) = [6] >= [0] = Nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: Weak TRS Rules: *(x,0()) -> 0() *(x,S(0())) -> x *(x,S(S(y))) -> +(x,*(x,S(y))) *(0(),y) -> 0() +Full(0(),y) -> y +Full(S(x),y) -> +Full(x,S(y)) f(x) -> *(x,x) goal(xs) -> map(xs) map(Cons(x,xs)) -> Cons(f(x),map(xs)) map(Nil()) -> Nil() Signature: {*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1} Obligation: Innermost basic terms: {*,+Full,f,goal,map}/{+,0,Cons,Nil,S} Applied Processor: EmptyProcessor Proof: The problem is already closed. The intended complexity is O(1).