WORST_CASE(?,O(n^2)) * Step 1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(0(),0()) -> true() eq(0(),s(X)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifrm(false(),N,add(M,X)) -> add(M,rm(N,X)) ifrm(true(),N,add(M,X)) -> rm(N,X) purge(add(N,X)) -> add(N,purge(rm(N,X))) purge(nil()) -> nil() rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X)) rm(N,nil()) -> nil() - Signature: {eq/2,ifrm/3,purge/1,rm/2} / {0/0,add/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s ,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, 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(add) = {2}, uargs(ifrm) = {1}, uargs(purge) = {1} Following symbols are considered usable: {eq,ifrm,purge,rm} TcT has computed the following interpretation: p(0) = [2] p(add) = [1] x2 + [0] p(eq) = [0] p(false) = [0] p(ifrm) = [1] x1 + [0] p(nil) = [0] p(purge) = [1] x1 + [1] p(rm) = [0] p(s) = [1] p(true) = [0] Following rules are strictly oriented: purge(nil()) = [1] > [0] = nil() Following rules are (at-least) weakly oriented: eq(0(),0()) = [0] >= [0] = true() eq(0(),s(X)) = [0] >= [0] = false() eq(s(X),0()) = [0] >= [0] = false() eq(s(X),s(Y)) = [0] >= [0] = eq(X,Y) ifrm(false(),N,add(M,X)) = [0] >= [0] = add(M,rm(N,X)) ifrm(true(),N,add(M,X)) = [0] >= [0] = rm(N,X) purge(add(N,X)) = [1] X + [1] >= [1] = add(N,purge(rm(N,X))) rm(N,add(M,X)) = [0] >= [0] = ifrm(eq(N,M),N,add(M,X)) rm(N,nil()) = [0] >= [0] = nil() * Step 2: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(0(),0()) -> true() eq(0(),s(X)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifrm(false(),N,add(M,X)) -> add(M,rm(N,X)) ifrm(true(),N,add(M,X)) -> rm(N,X) purge(add(N,X)) -> add(N,purge(rm(N,X))) rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X)) rm(N,nil()) -> nil() - Weak TRS: purge(nil()) -> nil() - Signature: {eq/2,ifrm/3,purge/1,rm/2} / {0/0,add/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s ,true} + 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(add) = {2}, uargs(ifrm) = {1}, uargs(purge) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(add) = [1] x1 + [1] x2 + [0] p(eq) = [12] p(false) = [0] p(ifrm) = [1] x1 + [1] x3 + [0] p(nil) = [6] p(purge) = [1] x1 + [0] p(rm) = [1] x2 + [0] p(s) = [1] p(true) = [0] Following rules are strictly oriented: eq(0(),0()) = [12] > [0] = true() eq(0(),s(X)) = [12] > [0] = false() eq(s(X),0()) = [12] > [0] = false() Following rules are (at-least) weakly oriented: eq(s(X),s(Y)) = [12] >= [12] = eq(X,Y) ifrm(false(),N,add(M,X)) = [1] M + [1] X + [0] >= [1] M + [1] X + [0] = add(M,rm(N,X)) ifrm(true(),N,add(M,X)) = [1] M + [1] X + [0] >= [1] X + [0] = rm(N,X) purge(add(N,X)) = [1] N + [1] X + [0] >= [1] N + [1] X + [0] = add(N,purge(rm(N,X))) purge(nil()) = [6] >= [6] = nil() rm(N,add(M,X)) = [1] M + [1] X + [0] >= [1] M + [1] X + [12] = ifrm(eq(N,M),N,add(M,X)) rm(N,nil()) = [6] >= [6] = nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(s(X),s(Y)) -> eq(X,Y) ifrm(false(),N,add(M,X)) -> add(M,rm(N,X)) ifrm(true(),N,add(M,X)) -> rm(N,X) purge(add(N,X)) -> add(N,purge(rm(N,X))) rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X)) rm(N,nil()) -> nil() - Weak TRS: eq(0(),0()) -> true() eq(0(),s(X)) -> false() eq(s(X),0()) -> false() purge(nil()) -> nil() - Signature: {eq/2,ifrm/3,purge/1,rm/2} / {0/0,add/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s ,true} + 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(add) = {2}, uargs(ifrm) = {1}, uargs(purge) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [4] p(add) = [1] x1 + [1] x2 + [10] p(eq) = [8] p(false) = [4] p(ifrm) = [1] x1 + [1] x3 + [0] p(nil) = [1] p(purge) = [1] x1 + [0] p(rm) = [1] x2 + [5] p(s) = [1] x1 + [8] p(true) = [5] Following rules are strictly oriented: ifrm(true(),N,add(M,X)) = [1] M + [1] X + [15] > [1] X + [5] = rm(N,X) rm(N,nil()) = [6] > [1] = nil() Following rules are (at-least) weakly oriented: eq(0(),0()) = [8] >= [5] = true() eq(0(),s(X)) = [8] >= [4] = false() eq(s(X),0()) = [8] >= [4] = false() eq(s(X),s(Y)) = [8] >= [8] = eq(X,Y) ifrm(false(),N,add(M,X)) = [1] M + [1] X + [14] >= [1] M + [1] X + [15] = add(M,rm(N,X)) purge(add(N,X)) = [1] N + [1] X + [10] >= [1] N + [1] X + [15] = add(N,purge(rm(N,X))) purge(nil()) = [1] >= [1] = nil() rm(N,add(M,X)) = [1] M + [1] X + [15] >= [1] M + [1] X + [18] = ifrm(eq(N,M),N,add(M,X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(s(X),s(Y)) -> eq(X,Y) ifrm(false(),N,add(M,X)) -> add(M,rm(N,X)) purge(add(N,X)) -> add(N,purge(rm(N,X))) rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(X)) -> false() eq(s(X),0()) -> false() ifrm(true(),N,add(M,X)) -> rm(N,X) purge(nil()) -> nil() rm(N,nil()) -> nil() - Signature: {eq/2,ifrm/3,purge/1,rm/2} / {0/0,add/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s ,true} + 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(add) = {2}, uargs(ifrm) = {1}, uargs(purge) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(add) = [1] x2 + [8] p(eq) = [4] p(false) = [2] p(ifrm) = [1] x1 + [12] p(nil) = [0] p(purge) = [1] x1 + [2] p(rm) = [2] p(s) = [0] p(true) = [4] Following rules are strictly oriented: ifrm(false(),N,add(M,X)) = [14] > [10] = add(M,rm(N,X)) Following rules are (at-least) weakly oriented: eq(0(),0()) = [4] >= [4] = true() eq(0(),s(X)) = [4] >= [2] = false() eq(s(X),0()) = [4] >= [2] = false() eq(s(X),s(Y)) = [4] >= [4] = eq(X,Y) ifrm(true(),N,add(M,X)) = [16] >= [2] = rm(N,X) purge(add(N,X)) = [1] X + [10] >= [12] = add(N,purge(rm(N,X))) purge(nil()) = [2] >= [0] = nil() rm(N,add(M,X)) = [2] >= [16] = ifrm(eq(N,M),N,add(M,X)) rm(N,nil()) = [2] >= [0] = nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(s(X),s(Y)) -> eq(X,Y) purge(add(N,X)) -> add(N,purge(rm(N,X))) rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(X)) -> false() eq(s(X),0()) -> false() ifrm(false(),N,add(M,X)) -> add(M,rm(N,X)) ifrm(true(),N,add(M,X)) -> rm(N,X) purge(nil()) -> nil() rm(N,nil()) -> nil() - Signature: {eq/2,ifrm/3,purge/1,rm/2} / {0/0,add/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s ,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, 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(add) = {2}, uargs(ifrm) = {1}, uargs(purge) = {1} Following symbols are considered usable: {eq,ifrm,purge,rm} TcT has computed the following interpretation: p(0) = [0] p(add) = [1] x2 + [2] p(eq) = [0] p(false) = [0] p(ifrm) = [8] x1 + [1] x3 + [1] p(nil) = [1] p(purge) = [8] x1 + [8] p(rm) = [1] x2 + [1] p(s) = [0] p(true) = [0] Following rules are strictly oriented: purge(add(N,X)) = [8] X + [24] > [8] X + [18] = add(N,purge(rm(N,X))) Following rules are (at-least) weakly oriented: eq(0(),0()) = [0] >= [0] = true() eq(0(),s(X)) = [0] >= [0] = false() eq(s(X),0()) = [0] >= [0] = false() eq(s(X),s(Y)) = [0] >= [0] = eq(X,Y) ifrm(false(),N,add(M,X)) = [1] X + [3] >= [1] X + [3] = add(M,rm(N,X)) ifrm(true(),N,add(M,X)) = [1] X + [3] >= [1] X + [1] = rm(N,X) purge(nil()) = [16] >= [1] = nil() rm(N,add(M,X)) = [1] X + [3] >= [1] X + [3] = ifrm(eq(N,M),N,add(M,X)) rm(N,nil()) = [2] >= [1] = nil() * Step 6: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(s(X),s(Y)) -> eq(X,Y) rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(X)) -> false() eq(s(X),0()) -> false() ifrm(false(),N,add(M,X)) -> add(M,rm(N,X)) ifrm(true(),N,add(M,X)) -> rm(N,X) purge(add(N,X)) -> add(N,purge(rm(N,X))) purge(nil()) -> nil() rm(N,nil()) -> nil() - Signature: {eq/2,ifrm/3,purge/1,rm/2} / {0/0,add/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s ,true} + 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(add) = {2}, uargs(ifrm) = {1}, uargs(purge) = {1} Following symbols are considered usable: {eq,ifrm,purge,rm} TcT has computed the following interpretation: p(0) = [0] [1] p(add) = [1 4] x2 + [0] [0 1] [1] p(eq) = [0 0] x2 + [0] [0 1] [0] p(false) = [0] [0] p(ifrm) = [4 0] x1 + [1 1] x3 + [1] [0 0] [0 1] [0] p(nil) = [4] [1] p(purge) = [2 4] x1 + [0] [0 1] [0] p(rm) = [1 1] x2 + [2] [0 1] [0] p(s) = [0 0] x1 + [0] [0 1] [0] p(true) = [0] [0] Following rules are strictly oriented: rm(N,add(M,X)) = [1 5] X + [3] [0 1] [1] > [1 5] X + [2] [0 1] [1] = ifrm(eq(N,M),N,add(M,X)) Following rules are (at-least) weakly oriented: eq(0(),0()) = [0] [1] >= [0] [0] = true() eq(0(),s(X)) = [0 0] X + [0] [0 1] [0] >= [0] [0] = false() eq(s(X),0()) = [0] [1] >= [0] [0] = false() eq(s(X),s(Y)) = [0 0] Y + [0] [0 1] [0] >= [0 0] Y + [0] [0 1] [0] = eq(X,Y) ifrm(false(),N,add(M,X)) = [1 5] X + [2] [0 1] [1] >= [1 5] X + [2] [0 1] [1] = add(M,rm(N,X)) ifrm(true(),N,add(M,X)) = [1 5] X + [2] [0 1] [1] >= [1 1] X + [2] [0 1] [0] = rm(N,X) purge(add(N,X)) = [2 12] X + [4] [0 1] [1] >= [2 10] X + [4] [0 1] [1] = add(N,purge(rm(N,X))) purge(nil()) = [12] [1] >= [4] [1] = nil() rm(N,nil()) = [7] [1] >= [4] [1] = nil() * Step 7: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(s(X),s(Y)) -> eq(X,Y) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(X)) -> false() eq(s(X),0()) -> false() ifrm(false(),N,add(M,X)) -> add(M,rm(N,X)) ifrm(true(),N,add(M,X)) -> rm(N,X) purge(add(N,X)) -> add(N,purge(rm(N,X))) purge(nil()) -> nil() rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X)) rm(N,nil()) -> nil() - Signature: {eq/2,ifrm/3,purge/1,rm/2} / {0/0,add/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s ,true} + 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(add) = {2}, uargs(ifrm) = {1}, uargs(purge) = {1} Following symbols are considered usable: {eq,ifrm,purge,rm} TcT has computed the following interpretation: p(0) = [0] [0] [3] p(add) = [0 0 0] [1 0 2] [0] [0 0 0] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [2] p(eq) = [0 0 0] [0 0 1] [1] [0 0 0] x1 + [0 0 0] x2 + [3] [2 0 0] [0 0 0] [0] p(false) = [2] [1] [0] p(ifrm) = [1 1 0] [0 0 0] [1 1 0] [0] [0 1 0] x1 + [0 2 0] x2 + [0 1 1] x3 + [0] [0 0 0] [0 0 0] [0 0 1] [0] p(nil) = [2] [0] [0] p(purge) = [2 0 2] [0] [0 0 2] x1 + [3] [0 0 1] [0] p(rm) = [0 0 0] [1 0 1] [2] [0 2 0] x1 + [0 0 2] x2 + [1] [0 0 0] [0 0 1] [0] p(s) = [1 1 3] [0] [0 0 1] x1 + [0] [0 0 1] [1] p(true) = [0] [2] [0] Following rules are strictly oriented: eq(s(X),s(Y)) = [0 0 0] [0 0 1] [2] [0 0 0] X + [0 0 0] Y + [3] [2 2 6] [0 0 0] [0] > [0 0 0] [0 0 1] [1] [0 0 0] X + [0 0 0] Y + [3] [2 0 0] [0 0 0] [0] = eq(X,Y) Following rules are (at-least) weakly oriented: eq(0(),0()) = [4] [3] [0] >= [0] [2] [0] = true() eq(0(),s(X)) = [0 0 1] [2] [0 0 0] X + [3] [0 0 0] [0] >= [2] [1] [0] = false() eq(s(X),0()) = [0 0 0] [4] [0 0 0] X + [3] [2 2 6] [0] >= [2] [1] [0] = false() ifrm(false(),N,add(M,X)) = [0 0 0] [0 0 0] [1 0 3] [3] [0 0 1] M + [0 2 0] N + [0 0 2] X + [3] [0 0 1] [0 0 0] [0 0 1] [2] >= [0 0 0] [1 0 3] [2] [0 0 0] M + [0 0 1] X + [0] [0 0 1] [0 0 1] [2] = add(M,rm(N,X)) ifrm(true(),N,add(M,X)) = [0 0 0] [0 0 0] [1 0 3] [2] [0 0 1] M + [0 2 0] N + [0 0 2] X + [4] [0 0 1] [0 0 0] [0 0 1] [2] >= [0 0 0] [1 0 1] [2] [0 2 0] N + [0 0 2] X + [1] [0 0 0] [0 0 1] [0] = rm(N,X) purge(add(N,X)) = [0 0 2] [2 0 6] [4] [0 0 2] N + [0 0 2] X + [7] [0 0 1] [0 0 1] [2] >= [0 0 0] [2 0 6] [4] [0 0 0] N + [0 0 1] X + [0] [0 0 1] [0 0 1] [2] = add(N,purge(rm(N,X))) purge(nil()) = [4] [3] [0] >= [2] [0] [0] = nil() rm(N,add(M,X)) = [0 0 1] [0 0 0] [1 0 3] [4] [0 0 2] M + [0 2 0] N + [0 0 2] X + [5] [0 0 1] [0 0 0] [0 0 1] [2] >= [0 0 1] [0 0 0] [1 0 3] [4] [0 0 1] M + [0 2 0] N + [0 0 2] X + [5] [0 0 1] [0 0 0] [0 0 1] [2] = ifrm(eq(N,M),N,add(M,X)) rm(N,nil()) = [0 0 0] [4] [0 2 0] N + [1] [0 0 0] [0] >= [2] [0] [0] = nil() * Step 8: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: eq(0(),0()) -> true() eq(0(),s(X)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifrm(false(),N,add(M,X)) -> add(M,rm(N,X)) ifrm(true(),N,add(M,X)) -> rm(N,X) purge(add(N,X)) -> add(N,purge(rm(N,X))) purge(nil()) -> nil() rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X)) rm(N,nil()) -> nil() - Signature: {eq/2,ifrm/3,purge/1,rm/2} / {0/0,add/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s ,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))