MAYBE * Step 1: WeightGap MAYBE + Considered Problem: - Strict TRS: a__head(X) -> head(X) a__head(cons(X,XS)) -> mark(X) a__incr(X) -> incr(X) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__nats() -> cons(0(),incr(nats())) a__nats() -> nats() a__odds() -> a__incr(a__pairs()) a__odds() -> odds() a__pairs() -> cons(0(),incr(odds())) a__pairs() -> pairs() a__tail(X) -> tail(X) a__tail(cons(X,XS)) -> mark(XS) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(head(X)) -> a__head(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(odds()) -> a__odds() mark(pairs()) -> a__pairs() mark(s(X)) -> s(mark(X)) mark(tail(X)) -> a__tail(mark(X)) - Signature: {a__head/1,a__incr/1,a__nats/0,a__odds/0,a__pairs/0,a__tail/1,mark/1} / {0/0,cons/2,head/1,incr/1,nats/0 ,odds/0,pairs/0,s/1,tail/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__head,a__incr,a__nats,a__odds,a__pairs,a__tail ,mark} and constructors {0,cons,head,incr,nats,odds,pairs,s,tail} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: 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(a__head) = {1}, uargs(a__incr) = {1}, uargs(a__tail) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__head) = [1] x1 + [0] p(a__incr) = [1] x1 + [0] p(a__nats) = [0] p(a__odds) = [0] p(a__pairs) = [0] p(a__tail) = [1] x1 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(head) = [1] x1 + [0] p(incr) = [1] x1 + [8] p(mark) = [1] x1 + [8] p(nats) = [0] p(odds) = [8] p(pairs) = [4] p(s) = [1] x1 + [4] p(tail) = [1] x1 + [1] Following rules are strictly oriented: mark(0()) = [8] > [0] = 0() mark(incr(X)) = [1] X + [16] > [1] X + [8] = a__incr(mark(X)) mark(nats()) = [8] > [0] = a__nats() mark(odds()) = [16] > [0] = a__odds() mark(pairs()) = [12] > [0] = a__pairs() mark(tail(X)) = [1] X + [9] > [1] X + [8] = a__tail(mark(X)) Following rules are (at-least) weakly oriented: a__head(X) = [1] X + [0] >= [1] X + [0] = head(X) a__head(cons(X,XS)) = [1] X + [1] XS + [0] >= [1] X + [8] = mark(X) a__incr(X) = [1] X + [0] >= [1] X + [8] = incr(X) a__incr(cons(X,XS)) = [1] X + [1] XS + [0] >= [1] X + [1] XS + [20] = cons(s(mark(X)),incr(XS)) a__nats() = [0] >= [8] = cons(0(),incr(nats())) a__nats() = [0] >= [0] = nats() a__odds() = [0] >= [0] = a__incr(a__pairs()) a__odds() = [0] >= [8] = odds() a__pairs() = [0] >= [16] = cons(0(),incr(odds())) a__pairs() = [0] >= [4] = pairs() a__tail(X) = [1] X + [0] >= [1] X + [1] = tail(X) a__tail(cons(X,XS)) = [1] X + [1] XS + [0] >= [1] XS + [8] = mark(XS) mark(cons(X1,X2)) = [1] X1 + [1] X2 + [8] >= [1] X1 + [1] X2 + [8] = cons(mark(X1),X2) mark(head(X)) = [1] X + [8] >= [1] X + [8] = a__head(mark(X)) mark(s(X)) = [1] X + [12] >= [1] X + [12] = s(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap MAYBE + Considered Problem: - Strict TRS: a__head(X) -> head(X) a__head(cons(X,XS)) -> mark(X) a__incr(X) -> incr(X) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__nats() -> cons(0(),incr(nats())) a__nats() -> nats() a__odds() -> a__incr(a__pairs()) a__odds() -> odds() a__pairs() -> cons(0(),incr(odds())) a__pairs() -> pairs() a__tail(X) -> tail(X) a__tail(cons(X,XS)) -> mark(XS) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(head(X)) -> a__head(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: mark(0()) -> 0() mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(odds()) -> a__odds() mark(pairs()) -> a__pairs() mark(tail(X)) -> a__tail(mark(X)) - Signature: {a__head/1,a__incr/1,a__nats/0,a__odds/0,a__pairs/0,a__tail/1,mark/1} / {0/0,cons/2,head/1,incr/1,nats/0 ,odds/0,pairs/0,s/1,tail/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__head,a__incr,a__nats,a__odds,a__pairs,a__tail ,mark} and constructors {0,cons,head,incr,nats,odds,pairs,s,tail} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: 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(a__head) = {1}, uargs(a__incr) = {1}, uargs(a__tail) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [5] p(a__head) = [1] x1 + [1] p(a__incr) = [1] x1 + [1] p(a__nats) = [0] p(a__odds) = [0] p(a__pairs) = [0] p(a__tail) = [1] x1 + [0] p(cons) = [1] x1 + [1] x2 + [12] p(head) = [1] x1 + [4] p(incr) = [1] x1 + [1] p(mark) = [1] x1 + [0] p(nats) = [0] p(odds) = [0] p(pairs) = [4] p(s) = [1] x1 + [8] p(tail) = [1] x1 + [0] Following rules are strictly oriented: a__head(cons(X,XS)) = [1] X + [1] XS + [13] > [1] X + [0] = mark(X) a__tail(cons(X,XS)) = [1] X + [1] XS + [12] > [1] XS + [0] = mark(XS) mark(head(X)) = [1] X + [4] > [1] X + [1] = a__head(mark(X)) Following rules are (at-least) weakly oriented: a__head(X) = [1] X + [1] >= [1] X + [4] = head(X) a__incr(X) = [1] X + [1] >= [1] X + [1] = incr(X) a__incr(cons(X,XS)) = [1] X + [1] XS + [13] >= [1] X + [1] XS + [21] = cons(s(mark(X)),incr(XS)) a__nats() = [0] >= [18] = cons(0(),incr(nats())) a__nats() = [0] >= [0] = nats() a__odds() = [0] >= [1] = a__incr(a__pairs()) a__odds() = [0] >= [0] = odds() a__pairs() = [0] >= [18] = cons(0(),incr(odds())) a__pairs() = [0] >= [4] = pairs() a__tail(X) = [1] X + [0] >= [1] X + [0] = tail(X) mark(0()) = [5] >= [5] = 0() mark(cons(X1,X2)) = [1] X1 + [1] X2 + [12] >= [1] X1 + [1] X2 + [12] = cons(mark(X1),X2) mark(incr(X)) = [1] X + [1] >= [1] X + [1] = a__incr(mark(X)) mark(nats()) = [0] >= [0] = a__nats() mark(odds()) = [0] >= [0] = a__odds() mark(pairs()) = [4] >= [0] = a__pairs() mark(s(X)) = [1] X + [8] >= [1] X + [8] = s(mark(X)) mark(tail(X)) = [1] X + [0] >= [1] X + [0] = a__tail(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap MAYBE + Considered Problem: - Strict TRS: a__head(X) -> head(X) a__incr(X) -> incr(X) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__nats() -> cons(0(),incr(nats())) a__nats() -> nats() a__odds() -> a__incr(a__pairs()) a__odds() -> odds() a__pairs() -> cons(0(),incr(odds())) a__pairs() -> pairs() a__tail(X) -> tail(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__head(cons(X,XS)) -> mark(X) a__tail(cons(X,XS)) -> mark(XS) mark(0()) -> 0() mark(head(X)) -> a__head(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(odds()) -> a__odds() mark(pairs()) -> a__pairs() mark(tail(X)) -> a__tail(mark(X)) - Signature: {a__head/1,a__incr/1,a__nats/0,a__odds/0,a__pairs/0,a__tail/1,mark/1} / {0/0,cons/2,head/1,incr/1,nats/0 ,odds/0,pairs/0,s/1,tail/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__head,a__incr,a__nats,a__odds,a__pairs,a__tail ,mark} and constructors {0,cons,head,incr,nats,odds,pairs,s,tail} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: 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(a__head) = {1}, uargs(a__incr) = {1}, uargs(a__tail) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__head) = [1] x1 + [0] p(a__incr) = [1] x1 + [0] p(a__nats) = [0] p(a__odds) = [1] p(a__pairs) = [0] p(a__tail) = [1] x1 + [0] p(cons) = [1] x1 + [1] p(head) = [1] p(incr) = [1] x1 + [1] p(mark) = [1] p(nats) = [15] p(odds) = [0] p(pairs) = [1] p(s) = [1] x1 + [1] p(tail) = [0] Following rules are strictly oriented: a__odds() = [1] > [0] = a__incr(a__pairs()) a__odds() = [1] > [0] = odds() Following rules are (at-least) weakly oriented: a__head(X) = [1] X + [0] >= [1] = head(X) a__head(cons(X,XS)) = [1] X + [1] >= [1] = mark(X) a__incr(X) = [1] X + [0] >= [1] X + [1] = incr(X) a__incr(cons(X,XS)) = [1] X + [1] >= [3] = cons(s(mark(X)),incr(XS)) a__nats() = [0] >= [1] = cons(0(),incr(nats())) a__nats() = [0] >= [15] = nats() a__pairs() = [0] >= [1] = cons(0(),incr(odds())) a__pairs() = [0] >= [1] = pairs() a__tail(X) = [1] X + [0] >= [0] = tail(X) a__tail(cons(X,XS)) = [1] X + [1] >= [1] = mark(XS) mark(0()) = [1] >= [0] = 0() mark(cons(X1,X2)) = [1] >= [2] = cons(mark(X1),X2) mark(head(X)) = [1] >= [1] = a__head(mark(X)) mark(incr(X)) = [1] >= [1] = a__incr(mark(X)) mark(nats()) = [1] >= [0] = a__nats() mark(odds()) = [1] >= [1] = a__odds() mark(pairs()) = [1] >= [0] = a__pairs() mark(s(X)) = [1] >= [2] = s(mark(X)) mark(tail(X)) = [1] >= [1] = a__tail(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap MAYBE + Considered Problem: - Strict TRS: a__head(X) -> head(X) a__incr(X) -> incr(X) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__nats() -> cons(0(),incr(nats())) a__nats() -> nats() a__pairs() -> cons(0(),incr(odds())) a__pairs() -> pairs() a__tail(X) -> tail(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__head(cons(X,XS)) -> mark(X) a__odds() -> a__incr(a__pairs()) a__odds() -> odds() a__tail(cons(X,XS)) -> mark(XS) mark(0()) -> 0() mark(head(X)) -> a__head(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(odds()) -> a__odds() mark(pairs()) -> a__pairs() mark(tail(X)) -> a__tail(mark(X)) - Signature: {a__head/1,a__incr/1,a__nats/0,a__odds/0,a__pairs/0,a__tail/1,mark/1} / {0/0,cons/2,head/1,incr/1,nats/0 ,odds/0,pairs/0,s/1,tail/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__head,a__incr,a__nats,a__odds,a__pairs,a__tail ,mark} and constructors {0,cons,head,incr,nats,odds,pairs,s,tail} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: 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(a__head) = {1}, uargs(a__incr) = {1}, uargs(a__tail) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(a__head) = [1] x1 + [0] p(a__incr) = [1] x1 + [0] p(a__nats) = [0] p(a__odds) = [4] p(a__pairs) = [4] p(a__tail) = [1] x1 + [0] p(cons) = [1] x1 + [8] p(head) = [0] p(incr) = [0] p(mark) = [4] p(nats) = [0] p(odds) = [4] p(pairs) = [0] p(s) = [1] x1 + [0] p(tail) = [1] x1 + [0] Following rules are strictly oriented: a__pairs() = [4] > [0] = pairs() Following rules are (at-least) weakly oriented: a__head(X) = [1] X + [0] >= [0] = head(X) a__head(cons(X,XS)) = [1] X + [8] >= [4] = mark(X) a__incr(X) = [1] X + [0] >= [0] = incr(X) a__incr(cons(X,XS)) = [1] X + [8] >= [12] = cons(s(mark(X)),incr(XS)) a__nats() = [0] >= [10] = cons(0(),incr(nats())) a__nats() = [0] >= [0] = nats() a__odds() = [4] >= [4] = a__incr(a__pairs()) a__odds() = [4] >= [4] = odds() a__pairs() = [4] >= [10] = cons(0(),incr(odds())) a__tail(X) = [1] X + [0] >= [1] X + [0] = tail(X) a__tail(cons(X,XS)) = [1] X + [8] >= [4] = mark(XS) mark(0()) = [4] >= [2] = 0() mark(cons(X1,X2)) = [4] >= [12] = cons(mark(X1),X2) mark(head(X)) = [4] >= [4] = a__head(mark(X)) mark(incr(X)) = [4] >= [4] = a__incr(mark(X)) mark(nats()) = [4] >= [0] = a__nats() mark(odds()) = [4] >= [4] = a__odds() mark(pairs()) = [4] >= [4] = a__pairs() mark(s(X)) = [4] >= [4] = s(mark(X)) mark(tail(X)) = [4] >= [4] = a__tail(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap MAYBE + Considered Problem: - Strict TRS: a__head(X) -> head(X) a__incr(X) -> incr(X) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__nats() -> cons(0(),incr(nats())) a__nats() -> nats() a__pairs() -> cons(0(),incr(odds())) a__tail(X) -> tail(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__head(cons(X,XS)) -> mark(X) a__odds() -> a__incr(a__pairs()) a__odds() -> odds() a__pairs() -> pairs() a__tail(cons(X,XS)) -> mark(XS) mark(0()) -> 0() mark(head(X)) -> a__head(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(odds()) -> a__odds() mark(pairs()) -> a__pairs() mark(tail(X)) -> a__tail(mark(X)) - Signature: {a__head/1,a__incr/1,a__nats/0,a__odds/0,a__pairs/0,a__tail/1,mark/1} / {0/0,cons/2,head/1,incr/1,nats/0 ,odds/0,pairs/0,s/1,tail/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__head,a__incr,a__nats,a__odds,a__pairs,a__tail ,mark} and constructors {0,cons,head,incr,nats,odds,pairs,s,tail} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: 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(a__head) = {1}, uargs(a__incr) = {1}, uargs(a__tail) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__head) = [1] x1 + [0] p(a__incr) = [1] x1 + [0] p(a__nats) = [1] p(a__odds) = [1] p(a__pairs) = [0] p(a__tail) = [1] x1 + [0] p(cons) = [1] x1 + [2] p(head) = [1] x1 + [0] p(incr) = [14] p(mark) = [1] p(nats) = [0] p(odds) = [0] p(pairs) = [0] p(s) = [1] x1 + [0] p(tail) = [1] Following rules are strictly oriented: a__nats() = [1] > [0] = nats() Following rules are (at-least) weakly oriented: a__head(X) = [1] X + [0] >= [1] X + [0] = head(X) a__head(cons(X,XS)) = [1] X + [2] >= [1] = mark(X) a__incr(X) = [1] X + [0] >= [14] = incr(X) a__incr(cons(X,XS)) = [1] X + [2] >= [3] = cons(s(mark(X)),incr(XS)) a__nats() = [1] >= [2] = cons(0(),incr(nats())) a__odds() = [1] >= [0] = a__incr(a__pairs()) a__odds() = [1] >= [0] = odds() a__pairs() = [0] >= [2] = cons(0(),incr(odds())) a__pairs() = [0] >= [0] = pairs() a__tail(X) = [1] X + [0] >= [1] = tail(X) a__tail(cons(X,XS)) = [1] X + [2] >= [1] = mark(XS) mark(0()) = [1] >= [0] = 0() mark(cons(X1,X2)) = [1] >= [3] = cons(mark(X1),X2) mark(head(X)) = [1] >= [1] = a__head(mark(X)) mark(incr(X)) = [1] >= [1] = a__incr(mark(X)) mark(nats()) = [1] >= [1] = a__nats() mark(odds()) = [1] >= [1] = a__odds() mark(pairs()) = [1] >= [0] = a__pairs() mark(s(X)) = [1] >= [1] = s(mark(X)) mark(tail(X)) = [1] >= [1] = a__tail(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: WeightGap MAYBE + Considered Problem: - Strict TRS: a__head(X) -> head(X) a__incr(X) -> incr(X) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__tail(X) -> tail(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__head(cons(X,XS)) -> mark(X) a__nats() -> nats() a__odds() -> a__incr(a__pairs()) a__odds() -> odds() a__pairs() -> pairs() a__tail(cons(X,XS)) -> mark(XS) mark(0()) -> 0() mark(head(X)) -> a__head(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(odds()) -> a__odds() mark(pairs()) -> a__pairs() mark(tail(X)) -> a__tail(mark(X)) - Signature: {a__head/1,a__incr/1,a__nats/0,a__odds/0,a__pairs/0,a__tail/1,mark/1} / {0/0,cons/2,head/1,incr/1,nats/0 ,odds/0,pairs/0,s/1,tail/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__head,a__incr,a__nats,a__odds,a__pairs,a__tail ,mark} and constructors {0,cons,head,incr,nats,odds,pairs,s,tail} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: 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(a__head) = {1}, uargs(a__incr) = {1}, uargs(a__tail) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__head) = [1] x1 + [2] p(a__incr) = [1] x1 + [0] p(a__nats) = [3] p(a__odds) = [0] p(a__pairs) = [0] p(a__tail) = [1] x1 + [8] p(cons) = [1] x1 + [1] x2 + [2] p(head) = [1] x1 + [4] p(incr) = [1] x1 + [0] p(mark) = [1] x1 + [4] p(nats) = [0] p(odds) = [0] p(pairs) = [0] p(s) = [1] x1 + [1] p(tail) = [1] x1 + [8] Following rules are strictly oriented: a__nats() = [3] > [2] = cons(0(),incr(nats())) Following rules are (at-least) weakly oriented: a__head(X) = [1] X + [2] >= [1] X + [4] = head(X) a__head(cons(X,XS)) = [1] X + [1] XS + [4] >= [1] X + [4] = mark(X) a__incr(X) = [1] X + [0] >= [1] X + [0] = incr(X) a__incr(cons(X,XS)) = [1] X + [1] XS + [2] >= [1] X + [1] XS + [7] = cons(s(mark(X)),incr(XS)) a__nats() = [3] >= [0] = nats() a__odds() = [0] >= [0] = a__incr(a__pairs()) a__odds() = [0] >= [0] = odds() a__pairs() = [0] >= [2] = cons(0(),incr(odds())) a__pairs() = [0] >= [0] = pairs() a__tail(X) = [1] X + [8] >= [1] X + [8] = tail(X) a__tail(cons(X,XS)) = [1] X + [1] XS + [10] >= [1] XS + [4] = mark(XS) mark(0()) = [4] >= [0] = 0() mark(cons(X1,X2)) = [1] X1 + [1] X2 + [6] >= [1] X1 + [1] X2 + [6] = cons(mark(X1),X2) mark(head(X)) = [1] X + [8] >= [1] X + [6] = a__head(mark(X)) mark(incr(X)) = [1] X + [4] >= [1] X + [4] = a__incr(mark(X)) mark(nats()) = [4] >= [3] = a__nats() mark(odds()) = [4] >= [0] = a__odds() mark(pairs()) = [4] >= [0] = a__pairs() mark(s(X)) = [1] X + [5] >= [1] X + [5] = s(mark(X)) mark(tail(X)) = [1] X + [12] >= [1] X + [12] = a__tail(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: WeightGap MAYBE + Considered Problem: - Strict TRS: a__head(X) -> head(X) a__incr(X) -> incr(X) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__pairs() -> cons(0(),incr(odds())) a__tail(X) -> tail(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__head(cons(X,XS)) -> mark(X) a__nats() -> cons(0(),incr(nats())) a__nats() -> nats() a__odds() -> a__incr(a__pairs()) a__odds() -> odds() a__pairs() -> pairs() a__tail(cons(X,XS)) -> mark(XS) mark(0()) -> 0() mark(head(X)) -> a__head(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(odds()) -> a__odds() mark(pairs()) -> a__pairs() mark(tail(X)) -> a__tail(mark(X)) - Signature: {a__head/1,a__incr/1,a__nats/0,a__odds/0,a__pairs/0,a__tail/1,mark/1} / {0/0,cons/2,head/1,incr/1,nats/0 ,odds/0,pairs/0,s/1,tail/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__head,a__incr,a__nats,a__odds,a__pairs,a__tail ,mark} and constructors {0,cons,head,incr,nats,odds,pairs,s,tail} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: 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(a__head) = {1}, uargs(a__incr) = {1}, uargs(a__tail) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__head) = [1] x1 + [2] p(a__incr) = [1] x1 + [0] p(a__nats) = [0] p(a__odds) = [1] p(a__pairs) = [1] p(a__tail) = [1] x1 + [4] p(cons) = [1] x1 + [1] x2 + [0] p(head) = [1] x1 + [2] p(incr) = [1] x1 + [0] p(mark) = [1] x1 + [1] p(nats) = [0] p(odds) = [0] p(pairs) = [0] p(s) = [1] x1 + [7] p(tail) = [1] x1 + [4] Following rules are strictly oriented: a__pairs() = [1] > [0] = cons(0(),incr(odds())) Following rules are (at-least) weakly oriented: a__head(X) = [1] X + [2] >= [1] X + [2] = head(X) a__head(cons(X,XS)) = [1] X + [1] XS + [2] >= [1] X + [1] = mark(X) a__incr(X) = [1] X + [0] >= [1] X + [0] = incr(X) a__incr(cons(X,XS)) = [1] X + [1] XS + [0] >= [1] X + [1] XS + [8] = cons(s(mark(X)),incr(XS)) a__nats() = [0] >= [0] = cons(0(),incr(nats())) a__nats() = [0] >= [0] = nats() a__odds() = [1] >= [1] = a__incr(a__pairs()) a__odds() = [1] >= [0] = odds() a__pairs() = [1] >= [0] = pairs() a__tail(X) = [1] X + [4] >= [1] X + [4] = tail(X) a__tail(cons(X,XS)) = [1] X + [1] XS + [4] >= [1] XS + [1] = mark(XS) mark(0()) = [1] >= [0] = 0() mark(cons(X1,X2)) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = cons(mark(X1),X2) mark(head(X)) = [1] X + [3] >= [1] X + [3] = a__head(mark(X)) mark(incr(X)) = [1] X + [1] >= [1] X + [1] = a__incr(mark(X)) mark(nats()) = [1] >= [0] = a__nats() mark(odds()) = [1] >= [1] = a__odds() mark(pairs()) = [1] >= [1] = a__pairs() mark(s(X)) = [1] X + [8] >= [1] X + [8] = s(mark(X)) mark(tail(X)) = [1] X + [5] >= [1] X + [5] = a__tail(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: NaturalMI MAYBE + Considered Problem: - Strict TRS: a__head(X) -> head(X) a__incr(X) -> incr(X) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__tail(X) -> tail(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__head(cons(X,XS)) -> mark(X) a__nats() -> cons(0(),incr(nats())) a__nats() -> nats() a__odds() -> a__incr(a__pairs()) a__odds() -> odds() a__pairs() -> cons(0(),incr(odds())) a__pairs() -> pairs() a__tail(cons(X,XS)) -> mark(XS) mark(0()) -> 0() mark(head(X)) -> a__head(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(odds()) -> a__odds() mark(pairs()) -> a__pairs() mark(tail(X)) -> a__tail(mark(X)) - Signature: {a__head/1,a__incr/1,a__nats/0,a__odds/0,a__pairs/0,a__tail/1,mark/1} / {0/0,cons/2,head/1,incr/1,nats/0 ,odds/0,pairs/0,s/1,tail/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__head,a__incr,a__nats,a__odds,a__pairs,a__tail ,mark} and constructors {0,cons,head,incr,nats,odds,pairs,s,tail} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__head) = {1}, uargs(a__incr) = {1}, uargs(a__tail) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__head,a__incr,a__nats,a__odds,a__pairs,a__tail,mark} TcT has computed the following interpretation: p(0) = [0] [0] p(a__head) = [1 0] x1 + [0] [0 1] [0] p(a__incr) = [1 4] x1 + [0] [0 1] [0] p(a__nats) = [0] [0] p(a__odds) = [0] [0] p(a__pairs) = [0] [0] p(a__tail) = [1 2] x1 + [2] [0 1] [2] p(cons) = [1 4] x1 + [1 2] x2 + [0] [0 1] [0 1] [0] p(head) = [1 0] x1 + [0] [0 1] [0] p(incr) = [1 4] x1 + [0] [0 1] [0] p(mark) = [1 4] x1 + [0] [0 1] [0] p(nats) = [0] [0] p(odds) = [0] [0] p(pairs) = [0] [0] p(s) = [1 0] x1 + [0] [0 1] [0] p(tail) = [1 2] x1 + [0] [0 1] [2] Following rules are strictly oriented: a__tail(X) = [1 2] X + [2] [0 1] [2] > [1 2] X + [0] [0 1] [2] = tail(X) Following rules are (at-least) weakly oriented: a__head(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = head(X) a__head(cons(X,XS)) = [1 4] X + [1 2] XS + [0] [0 1] [0 1] [0] >= [1 4] X + [0] [0 1] [0] = mark(X) a__incr(X) = [1 4] X + [0] [0 1] [0] >= [1 4] X + [0] [0 1] [0] = incr(X) a__incr(cons(X,XS)) = [1 8] X + [1 6] XS + [0] [0 1] [0 1] [0] >= [1 8] X + [1 6] XS + [0] [0 1] [0 1] [0] = cons(s(mark(X)),incr(XS)) a__nats() = [0] [0] >= [0] [0] = cons(0(),incr(nats())) a__nats() = [0] [0] >= [0] [0] = nats() a__odds() = [0] [0] >= [0] [0] = a__incr(a__pairs()) a__odds() = [0] [0] >= [0] [0] = odds() a__pairs() = [0] [0] >= [0] [0] = cons(0(),incr(odds())) a__pairs() = [0] [0] >= [0] [0] = pairs() a__tail(cons(X,XS)) = [1 6] X + [1 4] XS + [2] [0 1] [0 1] [2] >= [1 4] XS + [0] [0 1] [0] = mark(XS) mark(0()) = [0] [0] >= [0] [0] = 0() mark(cons(X1,X2)) = [1 8] X1 + [1 6] X2 + [0] [0 1] [0 1] [0] >= [1 8] X1 + [1 2] X2 + [0] [0 1] [0 1] [0] = cons(mark(X1),X2) mark(head(X)) = [1 4] X + [0] [0 1] [0] >= [1 4] X + [0] [0 1] [0] = a__head(mark(X)) mark(incr(X)) = [1 8] X + [0] [0 1] [0] >= [1 8] X + [0] [0 1] [0] = a__incr(mark(X)) mark(nats()) = [0] [0] >= [0] [0] = a__nats() mark(odds()) = [0] [0] >= [0] [0] = a__odds() mark(pairs()) = [0] [0] >= [0] [0] = a__pairs() mark(s(X)) = [1 4] X + [0] [0 1] [0] >= [1 4] X + [0] [0 1] [0] = s(mark(X)) mark(tail(X)) = [1 6] X + [8] [0 1] [2] >= [1 6] X + [2] [0 1] [2] = a__tail(mark(X)) * Step 9: NaturalMI MAYBE + Considered Problem: - Strict TRS: a__head(X) -> head(X) a__incr(X) -> incr(X) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__head(cons(X,XS)) -> mark(X) a__nats() -> cons(0(),incr(nats())) a__nats() -> nats() a__odds() -> a__incr(a__pairs()) a__odds() -> odds() a__pairs() -> cons(0(),incr(odds())) a__pairs() -> pairs() a__tail(X) -> tail(X) a__tail(cons(X,XS)) -> mark(XS) mark(0()) -> 0() mark(head(X)) -> a__head(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(odds()) -> a__odds() mark(pairs()) -> a__pairs() mark(tail(X)) -> a__tail(mark(X)) - Signature: {a__head/1,a__incr/1,a__nats/0,a__odds/0,a__pairs/0,a__tail/1,mark/1} / {0/0,cons/2,head/1,incr/1,nats/0 ,odds/0,pairs/0,s/1,tail/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__head,a__incr,a__nats,a__odds,a__pairs,a__tail ,mark} and constructors {0,cons,head,incr,nats,odds,pairs,s,tail} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__head) = {1}, uargs(a__incr) = {1}, uargs(a__tail) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__head,a__incr,a__nats,a__odds,a__pairs,a__tail,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__head) = [1 1 2] [3] [0 1 3] x1 + [3] [0 0 0] [0] p(a__incr) = [1 2 3] [0] [0 1 3] x1 + [0] [0 0 1] [0] p(a__nats) = [0] [0] [0] p(a__odds) = [0] [0] [0] p(a__pairs) = [0] [0] [0] p(a__tail) = [1 1 1] [0] [0 1 3] x1 + [0] [0 0 0] [0] p(cons) = [1 0 0] [1 0 0] [0] [0 1 3] x1 + [0 1 3] x2 + [0] [0 0 0] [0 0 0] [0] p(head) = [1 1 0] [0] [0 1 3] x1 + [3] [0 0 0] [0] p(incr) = [1 2 3] [0] [0 1 3] x1 + [0] [0 0 0] [0] p(mark) = [1 1 0] [0] [0 1 3] x1 + [0] [0 0 0] [0] p(nats) = [0] [0] [0] p(odds) = [0] [0] [0] p(pairs) = [0] [0] [0] p(s) = [1 0 1] [0] [0 1 3] x1 + [0] [0 0 0] [0] p(tail) = [1 1 0] [0] [0 1 3] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: a__head(X) = [1 1 2] [3] [0 1 3] X + [3] [0 0 0] [0] > [1 1 0] [0] [0 1 3] X + [3] [0 0 0] [0] = head(X) Following rules are (at-least) weakly oriented: a__head(cons(X,XS)) = [1 1 3] [1 1 3] [3] [0 1 3] X + [0 1 3] XS + [3] [0 0 0] [0 0 0] [0] >= [1 1 0] [0] [0 1 3] X + [0] [0 0 0] [0] = mark(X) a__incr(X) = [1 2 3] [0] [0 1 3] X + [0] [0 0 1] [0] >= [1 2 3] [0] [0 1 3] X + [0] [0 0 0] [0] = incr(X) a__incr(cons(X,XS)) = [1 2 6] [1 2 6] [0] [0 1 3] X + [0 1 3] XS + [0] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 2 3] [0] [0 1 3] X + [0 1 3] XS + [0] [0 0 0] [0 0 0] [0] = cons(s(mark(X)),incr(XS)) a__nats() = [0] [0] [0] >= [0] [0] [0] = cons(0(),incr(nats())) a__nats() = [0] [0] [0] >= [0] [0] [0] = nats() a__odds() = [0] [0] [0] >= [0] [0] [0] = a__incr(a__pairs()) a__odds() = [0] [0] [0] >= [0] [0] [0] = odds() a__pairs() = [0] [0] [0] >= [0] [0] [0] = cons(0(),incr(odds())) a__pairs() = [0] [0] [0] >= [0] [0] [0] = pairs() a__tail(X) = [1 1 1] [0] [0 1 3] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 3] X + [0] [0 0 0] [0] = tail(X) a__tail(cons(X,XS)) = [1 1 3] [1 1 3] [0] [0 1 3] X + [0 1 3] XS + [0] [0 0 0] [0 0 0] [0] >= [1 1 0] [0] [0 1 3] XS + [0] [0 0 0] [0] = mark(XS) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(cons(X1,X2)) = [1 1 3] [1 1 3] [0] [0 1 3] X1 + [0 1 3] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 0 0] [0] [0 1 3] X1 + [0 1 3] X2 + [0] [0 0 0] [0 0 0] [0] = cons(mark(X1),X2) mark(head(X)) = [1 2 3] [3] [0 1 3] X + [3] [0 0 0] [0] >= [1 2 3] [3] [0 1 3] X + [3] [0 0 0] [0] = a__head(mark(X)) mark(incr(X)) = [1 3 6] [0] [0 1 3] X + [0] [0 0 0] [0] >= [1 3 6] [0] [0 1 3] X + [0] [0 0 0] [0] = a__incr(mark(X)) mark(nats()) = [0] [0] [0] >= [0] [0] [0] = a__nats() mark(odds()) = [0] [0] [0] >= [0] [0] [0] = a__odds() mark(pairs()) = [0] [0] [0] >= [0] [0] [0] = a__pairs() mark(s(X)) = [1 1 4] [0] [0 1 3] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 3] X + [0] [0 0 0] [0] = s(mark(X)) mark(tail(X)) = [1 2 3] [0] [0 1 3] X + [0] [0 0 0] [0] >= [1 2 3] [0] [0 1 3] X + [0] [0 0 0] [0] = a__tail(mark(X)) * Step 10: Failure MAYBE + Considered Problem: - Strict TRS: a__incr(X) -> incr(X) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__head(X) -> head(X) a__head(cons(X,XS)) -> mark(X) a__nats() -> cons(0(),incr(nats())) a__nats() -> nats() a__odds() -> a__incr(a__pairs()) a__odds() -> odds() a__pairs() -> cons(0(),incr(odds())) a__pairs() -> pairs() a__tail(X) -> tail(X) a__tail(cons(X,XS)) -> mark(XS) mark(0()) -> 0() mark(head(X)) -> a__head(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(odds()) -> a__odds() mark(pairs()) -> a__pairs() mark(tail(X)) -> a__tail(mark(X)) - Signature: {a__head/1,a__incr/1,a__nats/0,a__odds/0,a__pairs/0,a__tail/1,mark/1} / {0/0,cons/2,head/1,incr/1,nats/0 ,odds/0,pairs/0,s/1,tail/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__head,a__incr,a__nats,a__odds,a__pairs,a__tail ,mark} and constructors {0,cons,head,incr,nats,odds,pairs,s,tail} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE