* Step 1: WeightGap WORST_CASE(?,O(n^3)) + 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: 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) = {1,3}, uargs(purge) = {1}, uargs(rm) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(add) = [1] x1 + [1] x2 + [0] p(eq) = [5] p(false) = [0] p(ifrm) = [1] x1 + [1] x3 + [5] p(nil) = [0] p(purge) = [1] x1 + [5] p(rm) = [1] x2 + [3] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: eq(0(),0()) = [5] > [0] = true() eq(0(),s(X)) = [5] > [0] = false() eq(s(X),0()) = [5] > [0] = false() ifrm(false(),N,add(M,X)) = [1] M + [1] X + [5] > [1] M + [1] X + [3] = add(M,rm(N,X)) ifrm(true(),N,add(M,X)) = [1] M + [1] X + [5] > [1] X + [3] = rm(N,X) purge(nil()) = [5] > [0] = nil() rm(N,nil()) = [3] > [0] = nil() Following rules are (at-least) weakly oriented: eq(s(X),s(Y)) = [5] >= [5] = eq(X,Y) purge(add(N,X)) = [1] N + [1] X + [5] >= [1] N + [1] X + [8] = add(N,purge(rm(N,X))) rm(N,add(M,X)) = [1] M + [1] X + [3] >= [1] M + [1] X + [10] = 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 2: MI WORST_CASE(?,O(n^3)) + 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: runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s,true} + Applied Processor: MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)): The following argument positions are considered usable: uargs(add) = {2}, uargs(ifrm) = {1,3}, uargs(purge) = {1}, uargs(rm) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(add) = [1] x_1 + [1] x_2 + [3] p(eq) = [0] p(false) = [0] p(ifrm) = [8] x_1 + [1] x_3 + [1] p(nil) = [0] p(purge) = [2] x_1 + [0] p(rm) = [1] x_2 + [1] p(s) = [4] p(true) = [0] Following rules are strictly oriented: purge(add(N,X)) = [2] N + [2] X + [6] > [1] N + [2] X + [5] = 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] M + [1] X + [4] >= [1] M + [1] X + [4] = add(M,rm(N,X)) ifrm(true(),N,add(M,X)) = [1] M + [1] X + [4] >= [1] X + [1] = rm(N,X) purge(nil()) = [0] >= [0] = nil() rm(N,add(M,X)) = [1] M + [1] X + [4] >= [1] M + [1] X + [4] = ifrm(eq(N,M),N,add(M,X)) rm(N,nil()) = [1] >= [0] = nil() * Step 3: MI WORST_CASE(?,O(n^3)) + 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: runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s,true} + Applied Processor: MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)): The following argument positions are considered usable: uargs(add) = {2}, uargs(ifrm) = {1,3}, uargs(purge) = {1}, uargs(rm) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] [0] p(add) = [1 4] x_2 + [0] [0 1] [2] p(eq) = [0] [2] p(false) = [0] [2] p(ifrm) = [2 1] x_1 + [1 2] x_3 + [0] [0 0] [0 1] [0] p(nil) = [0] [0] p(purge) = [2 4] x_1 + [4] [0 1] [0] p(rm) = [1 2] x_2 + [3] [0 1] [0] p(s) = [1 0] x_1 + [0] [0 1] [0] p(true) = [0] [2] Following rules are strictly oriented: rm(N,add(M,X)) = [1 6] X + [7] [0 1] [2] > [1 6] X + [6] [0 1] [2] = ifrm(eq(N,M),N,add(M,X)) Following rules are (at-least) weakly oriented: eq(0(),0()) = [0] [2] >= [0] [2] = true() eq(0(),s(X)) = [0] [2] >= [0] [2] = false() eq(s(X),0()) = [0] [2] >= [0] [2] = false() eq(s(X),s(Y)) = [0] [2] >= [0] [2] = eq(X,Y) ifrm(false(),N,add(M,X)) = [1 6] X + [6] [0 1] [2] >= [1 6] X + [3] [0 1] [2] = add(M,rm(N,X)) ifrm(true(),N,add(M,X)) = [1 6] X + [6] [0 1] [2] >= [1 2] X + [3] [0 1] [0] = rm(N,X) purge(add(N,X)) = [2 12] X + [12] [0 1] [2] >= [2 12] X + [10] [0 1] [2] = add(N,purge(rm(N,X))) purge(nil()) = [4] [0] >= [0] [0] = nil() rm(N,nil()) = [3] [0] >= [0] [0] = nil() * Step 4: MI WORST_CASE(?,O(n^3)) + 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: runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s,true} + Applied Processor: MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 3))), miDimension = 4, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 3))): The following argument positions are considered usable: uargs(add) = {2}, uargs(ifrm) = {1,3}, uargs(purge) = {1}, uargs(rm) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] [1] [0] [1] p(add) = [0 0 0 0] [1 0 2 0] [0] [0 0 0 0] x_1 + [0 0 1 0] x_2 + [0] [0 0 0 1] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0] p(eq) = [0 0 0 0] [0 0 0 1] [1] [1 1 1 1] x_1 + [0 0 0 1] x_2 + [0] [0 0 0 0] [0 0 0 0] [0] [1 1 0 2] [0 0 0 0] [2] p(false) = [0] [0] [0] [0] p(ifrm) = [1 0 0 0] [0 0 0 0] [1 1 0 0] [0] [0 0 0 0] x_1 + [0 0 2 0] x_2 + [1 0 0 0] x_3 + [2] [0 0 0 0] [0 0 0 0] [0 0 1 0] [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [2] p(nil) = [2] [0] [0] [2] p(purge) = [2 0 0 0] [0] [0 1 0 0] x_1 + [0] [0 0 1 0] [0] [2 0 0 0] [0] p(rm) = [0 0 0 0] [1 0 1 0] [0] [0 0 2 0] x_1 + [1 0 0 0] x_2 + [2] [0 0 0 0] [0 0 1 0] [0] [0 0 0 0] [0 0 0 0] [2] p(s) = [1 1 2 0] [0] [0 0 0 0] x_1 + [2] [0 0 0 0] [3] [0 0 0 1] [1] p(true) = [0] [3] [0] [1] Following rules are strictly oriented: eq(s(X),s(Y)) = [0 0 0 0] [0 0 0 1] [2] [1 1 2 1] X + [0 0 0 1] Y + [7] [0 0 0 0] [0 0 0 0] [0] [1 1 2 2] [0 0 0 0] [6] > [0 0 0 0] [0 0 0 1] [1] [1 1 1 1] X + [0 0 0 1] Y + [0] [0 0 0 0] [0 0 0 0] [0] [1 1 0 2] [0 0 0 0] [2] = eq(X,Y) Following rules are (at-least) weakly oriented: eq(0(),0()) = [2] [3] [0] [5] >= [0] [3] [0] [1] = true() eq(0(),s(X)) = [0 0 0 1] [2] [0 0 0 1] X + [3] [0 0 0 0] [0] [0 0 0 0] [5] >= [0] [0] [0] [0] = false() eq(s(X),0()) = [0 0 0 0] [2] [1 1 2 1] X + [7] [0 0 0 0] [0] [1 1 2 2] [6] >= [0] [0] [0] [0] = false() ifrm(false(),N,add(M,X)) = [0 0 0 0] [0 0 0 0] [1 0 3 0] [0] [0 0 0 0] M + [0 0 2 0] N + [1 0 2 0] X + [2] [0 0 0 1] [0 0 0 0] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [2] >= [0 0 0 0] [1 0 3 0] [0] [0 0 0 0] M + [0 0 1 0] X + [0] [0 0 0 1] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0] = add(M,rm(N,X)) ifrm(true(),N,add(M,X)) = [0 0 0 0] [0 0 0 0] [1 0 3 0] [0] [0 0 0 0] M + [0 0 2 0] N + [1 0 2 0] X + [2] [0 0 0 1] [0 0 0 0] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [2] >= [0 0 0 0] [1 0 1 0] [0] [0 0 2 0] N + [1 0 0 0] X + [2] [0 0 0 0] [0 0 1 0] [0] [0 0 0 0] [0 0 0 0] [2] = rm(N,X) purge(add(N,X)) = [0 0 0 0] [2 0 4 0] [0] [0 0 0 0] N + [0 0 1 0] X + [0] [0 0 0 1] [0 0 1 0] [1] [0 0 0 0] [2 0 4 0] [0] >= [0 0 0 0] [2 0 4 0] [0] [0 0 0 0] N + [0 0 1 0] X + [0] [0 0 0 1] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0] = add(N,purge(rm(N,X))) purge(nil()) = [4] [0] [0] [4] >= [2] [0] [0] [2] = nil() rm(N,add(M,X)) = [0 0 0 1] [0 0 0 0] [1 0 3 0] [1] [0 0 0 0] M + [0 0 2 0] N + [1 0 2 0] X + [2] [0 0 0 1] [0 0 0 0] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [2] >= [0 0 0 1] [0 0 0 0] [1 0 3 0] [1] [0 0 0 0] M + [0 0 2 0] N + [1 0 2 0] X + [2] [0 0 0 1] [0 0 0 0] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [2] = ifrm(eq(N,M),N,add(M,X)) rm(N,nil()) = [0 0 0 0] [2] [0 0 2 0] N + [4] [0 0 0 0] [0] [0 0 0 0] [2] >= [2] [0] [0] [2] = nil() * Step 5: 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: 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^3))