/export/starexec/sandbox2/solver/bin/starexec_run_tct_rc /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),O(n^3)) * Step 1: Sum. WORST_CASE(Omega(n^1),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: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + 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: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + 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: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: eq(x,y){x -> s(x),y -> s(y)} = eq(s(x),s(y)) ->^+ eq(x,y) = C[eq(x,y) = eq(x,y){}] ** Step 1.b:1: NaturalMI. 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: 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) = {1,3}, uargs(purge) = {1}, uargs(rm) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(add) = [1] x2 + [0] p(eq) = [0] p(false) = [0] p(ifrm) = [4] x1 + [1] x3 + [0] p(nil) = [1] p(purge) = [8] x1 + [2] p(rm) = [1] x2 + [0] p(s) = [1] x1 + [1] p(true) = [0] Following rules are strictly oriented: purge(nil()) = [10] > [1] = 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)) = [1] X + [0] >= [1] X + [0] = add(M,rm(N,X)) ifrm(true(),N,add(M,X)) = [1] X + [0] >= [1] X + [0] = rm(N,X) purge(add(N,X)) = [8] X + [2] >= [8] X + [2] = add(N,purge(rm(N,X))) rm(N,add(M,X)) = [1] X + [0] >= [1] X + [0] = ifrm(eq(N,M),N,add(M,X)) rm(N,nil()) = [1] >= [1] = nil() ** Step 1.b:2: NaturalMI. 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))) 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: 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) = {1,3}, 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 + [3] p(eq) = [0] p(false) = [0] p(ifrm) = [1] x1 + [1] x3 + [0] p(nil) = [0] p(purge) = [1] x1 + [3] p(rm) = [1] x2 + [0] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: ifrm(true(),N,add(M,X)) = [1] M + [1] X + [3] > [1] X + [0] = 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 + [3] >= [1] M + [1] X + [3] = add(M,rm(N,X)) purge(add(N,X)) = [1] N + [1] X + [6] >= [1] N + [1] X + [6] = add(N,purge(rm(N,X))) purge(nil()) = [3] >= [0] = nil() rm(N,add(M,X)) = [1] M + [1] X + [3] >= [1] M + [1] X + [3] = ifrm(eq(N,M),N,add(M,X)) rm(N,nil()) = [0] >= [0] = nil() ** Step 1.b:3: NaturalMI. 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)) 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: ifrm(true(),N,add(M,X)) -> rm(N,X) 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: 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) = {1,3}, uargs(purge) = {1}, uargs(rm) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [4] p(add) = [1] x2 + [2] p(eq) = [0] p(false) = [0] p(ifrm) = [8] x1 + [1] x3 + [0] p(nil) = [5] p(purge) = [4] x1 + [4] p(rm) = [1] x2 + [0] p(s) = [0] p(true) = [0] Following rules are strictly oriented: purge(add(N,X)) = [4] X + [12] > [4] X + [6] = 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 + [2] >= [1] X + [2] = add(M,rm(N,X)) ifrm(true(),N,add(M,X)) = [1] X + [2] >= [1] X + [0] = rm(N,X) purge(nil()) = [24] >= [5] = nil() rm(N,add(M,X)) = [1] X + [2] >= [1] X + [2] = ifrm(eq(N,M),N,add(M,X)) rm(N,nil()) = [5] >= [5] = nil() ** Step 1.b:4: NaturalMI. 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)) rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X)) rm(N,nil()) -> nil() - Weak TRS: ifrm(true(),N,add(M,X)) -> rm(N,X) purge(add(N,X)) -> add(N,purge(rm(N,X))) 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: 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) = {1,3}, uargs(purge) = {1}, uargs(rm) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(add) = [1] x2 + [2] p(eq) = [0] p(false) = [0] p(ifrm) = [4] x1 + [1] x3 + [1] p(nil) = [0] p(purge) = [8] x1 + [0] p(rm) = [1] x2 + [1] p(s) = [8] p(true) = [0] Following rules are strictly oriented: rm(N,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)) = [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(add(N,X)) = [8] X + [16] >= [8] X + [10] = add(N,purge(rm(N,X))) purge(nil()) = [0] >= [0] = nil() rm(N,add(M,X)) = [1] X + [3] >= [1] X + [3] = ifrm(eq(N,M),N,add(M,X)) ** Step 1.b:5: NaturalMI. 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)) rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X)) - Weak TRS: 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: 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) = {1,3}, uargs(purge) = {1}, uargs(rm) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(add) = [1] x2 + [4] p(eq) = [1] p(false) = [1] p(ifrm) = [2] x1 + [1] x3 + [0] p(nil) = [6] p(purge) = [2] x1 + [4] p(rm) = [1] x2 + [2] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: eq(0(),0()) = [1] > [0] = true() Following rules are (at-least) weakly oriented: eq(0(),s(X)) = [1] >= [1] = false() eq(s(X),0()) = [1] >= [1] = false() eq(s(X),s(Y)) = [1] >= [1] = eq(X,Y) ifrm(false(),N,add(M,X)) = [1] X + [6] >= [1] X + [6] = add(M,rm(N,X)) ifrm(true(),N,add(M,X)) = [1] X + [4] >= [1] X + [2] = rm(N,X) purge(add(N,X)) = [2] X + [12] >= [2] X + [12] = add(N,purge(rm(N,X))) purge(nil()) = [16] >= [6] = nil() rm(N,add(M,X)) = [1] X + [6] >= [1] X + [6] = ifrm(eq(N,M),N,add(M,X)) rm(N,nil()) = [8] >= [6] = nil() ** Step 1.b:6: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: 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)) rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X)) - Weak TRS: eq(0(),0()) -> true() 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: NaturalMI {miDimension = 3, miDegree = 3, 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) = {1,3}, uargs(purge) = {1}, uargs(rm) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] [1] [1] p(add) = [1 0 0] [1 2 0] [0] [0 0 0] x1 + [0 1 3] x2 + [0] [0 0 1] [0 0 1] [1] p(eq) = [0 0 1] [1] [0 0 0] x2 + [0] [0 0 0] [0] p(false) = [2] [0] [0] p(ifrm) = [3 0 0] [1 1 0] [0] [0 0 0] x1 + [0 1 0] x3 + [0] [0 0 0] [0 0 1] [0] p(nil) = [0] [1] [2] p(purge) = [2 2 0] [1] [0 1 0] x1 + [0] [0 0 1] [0] p(rm) = [1 1 3] [0] [0 1 0] x2 + [0] [0 0 1] [0] p(s) = [0 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [1] p(true) = [0] [0] [0] Following rules are strictly oriented: eq(s(X),s(Y)) = [0 0 1] [2] [0 0 0] Y + [0] [0 0 0] [0] > [0 0 1] [1] [0 0 0] Y + [0] [0 0 0] [0] = eq(X,Y) ifrm(false(),N,add(M,X)) = [1 0 0] [1 3 3] [6] [0 0 0] M + [0 1 3] X + [0] [0 0 1] [0 0 1] [1] > [1 0 0] [1 3 3] [0] [0 0 0] M + [0 1 3] X + [0] [0 0 1] [0 0 1] [1] = add(M,rm(N,X)) Following rules are (at-least) weakly oriented: eq(0(),0()) = [2] [0] [0] >= [0] [0] [0] = true() eq(0(),s(X)) = [0 0 1] [2] [0 0 0] X + [0] [0 0 0] [0] >= [2] [0] [0] = false() eq(s(X),0()) = [2] [0] [0] >= [2] [0] [0] = false() ifrm(true(),N,add(M,X)) = [1 0 0] [1 3 3] [0] [0 0 0] M + [0 1 3] X + [0] [0 0 1] [0 0 1] [1] >= [1 1 3] [0] [0 1 0] X + [0] [0 0 1] [0] = rm(N,X) purge(add(N,X)) = [2 0 0] [2 6 6] [1] [0 0 0] N + [0 1 3] X + [0] [0 0 1] [0 0 1] [1] >= [1 0 0] [2 6 6] [1] [0 0 0] N + [0 1 3] X + [0] [0 0 1] [0 0 1] [1] = add(N,purge(rm(N,X))) purge(nil()) = [3] [1] [2] >= [0] [1] [2] = nil() rm(N,add(M,X)) = [1 0 3] [1 3 6] [3] [0 0 0] M + [0 1 3] X + [0] [0 0 1] [0 0 1] [1] >= [1 0 3] [1 3 3] [3] [0 0 0] M + [0 1 3] X + [0] [0 0 1] [0 0 1] [1] = ifrm(eq(N,M),N,add(M,X)) rm(N,nil()) = [7] [1] [2] >= [0] [1] [2] = nil() ** Step 1.b:7: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: eq(0(),s(X)) -> false() eq(s(X),0()) -> false() rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X)) - Weak TRS: eq(0(),0()) -> true() 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,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: NaturalMI {miDimension = 3, miDegree = 3, 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) = {1,3}, uargs(purge) = {1}, uargs(rm) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] [0] [2] p(add) = [0 0 0] [1 0 2] [0] [0 0 0] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(eq) = [0 0 1] [0] [0 0 0] x2 + [0] [0 0 0] [0] p(false) = [0] [0] [0] p(ifrm) = [1 0 0] [0 0 0] [1 1 0] [0] [0 0 0] x1 + [0 0 2] x2 + [0 0 1] x3 + [0] [0 0 0] [0 0 0] [0 0 1] [0] p(nil) = [2] [0] [3] p(purge) = [2 0 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(rm) = [0 0 0] [1 0 1] [0] [0 0 2] x1 + [0 0 1] x2 + [0] [0 0 0] [0 0 1] [0] p(s) = [0 0 1] [0] [0 1 0] x1 + [1] [0 0 1] [0] p(true) = [0] [0] [0] Following rules are strictly oriented: eq(s(X),0()) = [2] [0] [0] > [0] [0] [0] = false() Following rules are (at-least) weakly oriented: eq(0(),0()) = [2] [0] [0] >= [0] [0] [0] = true() eq(0(),s(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] >= [0] [0] [0] = false() eq(s(X),s(Y)) = [0 0 1] [0] [0 0 0] Y + [0] [0 0 0] [0] >= [0 0 1] [0] [0 0 0] Y + [0] [0 0 0] [0] = eq(X,Y) ifrm(false(),N,add(M,X)) = [0 0 0] [0 0 0] [1 0 3] [0] [0 0 1] M + [0 0 2] N + [0 0 1] X + [0] [0 0 1] [0 0 0] [0 0 1] [0] >= [0 0 0] [1 0 3] [0] [0 0 0] M + [0 0 1] X + [0] [0 0 1] [0 0 1] [0] = add(M,rm(N,X)) ifrm(true(),N,add(M,X)) = [0 0 0] [0 0 0] [1 0 3] [0] [0 0 1] M + [0 0 2] N + [0 0 1] X + [0] [0 0 1] [0 0 0] [0 0 1] [0] >= [0 0 0] [1 0 1] [0] [0 0 2] N + [0 0 1] X + [0] [0 0 0] [0 0 1] [0] = rm(N,X) purge(add(N,X)) = [0 0 0] [2 0 4] [0] [0 0 1] N + [0 0 1] X + [0] [0 0 1] [0 0 1] [0] >= [0 0 0] [2 0 4] [0] [0 0 0] N + [0 0 1] X + [0] [0 0 1] [0 0 1] [0] = add(N,purge(rm(N,X))) purge(nil()) = [4] [3] [3] >= [2] [0] [3] = nil() rm(N,add(M,X)) = [0 0 1] [0 0 0] [1 0 3] [0] [0 0 1] M + [0 0 2] N + [0 0 1] X + [0] [0 0 1] [0 0 0] [0 0 1] [0] >= [0 0 1] [0 0 0] [1 0 3] [0] [0 0 1] M + [0 0 2] N + [0 0 1] X + [0] [0 0 1] [0 0 0] [0 0 1] [0] = ifrm(eq(N,M),N,add(M,X)) rm(N,nil()) = [0 0 0] [5] [0 0 2] N + [3] [0 0 0] [3] >= [2] [0] [3] = nil() ** Step 1.b:8: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: eq(0(),s(X)) -> false() rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X)) - Weak TRS: eq(0(),0()) -> true() 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,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: NaturalMI {miDimension = 3, miDegree = 3, 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) = {1,3}, uargs(purge) = {1}, uargs(rm) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] [0] [2] p(add) = [0 0 0] [1 2 0] [0] [0 1 3] x1 + [0 1 0] x2 + [2] [0 0 0] [0 0 0] [0] p(eq) = [0 0 0] [1] [0 1 0] x2 + [1] [0 0 3] [0] p(false) = [0] [0] [0] p(ifrm) = [2 0 0] [0 0 0] [1 1 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0 1 0] x3 + [0] [2 2 2] [3 0 0] [1 0 0] [0] p(nil) = [0] [0] [0] p(purge) = [2 2 0] [2] [0 1 0] x1 + [0] [1 0 0] [2] p(rm) = [0 0 0] [1 1 0] [2] [0 0 0] x1 + [0 1 0] x2 + [0] [3 0 0] [1 2 0] [0] p(s) = [0 0 1] [0] [0 1 1] x1 + [0] [0 0 1] [0] p(true) = [0] [1] [2] Following rules are strictly oriented: eq(0(),s(X)) = [0 0 0] [1] [0 1 1] X + [1] [0 0 3] [0] > [0] [0] [0] = false() Following rules are (at-least) weakly oriented: eq(0(),0()) = [1] [1] [6] >= [0] [1] [2] = true() eq(s(X),0()) = [1] [1] [6] >= [0] [0] [0] = false() eq(s(X),s(Y)) = [0 0 0] [1] [0 1 1] Y + [1] [0 0 3] [0] >= [0 0 0] [1] [0 1 0] Y + [1] [0 0 3] [0] = eq(X,Y) ifrm(false(),N,add(M,X)) = [0 1 3] [0 0 0] [1 3 0] [2] [0 1 3] M + [0 0 0] N + [0 1 0] X + [2] [0 0 0] [3 0 0] [1 2 0] [0] >= [0 0 0] [1 3 0] [2] [0 1 3] M + [0 1 0] X + [2] [0 0 0] [0 0 0] [0] = add(M,rm(N,X)) ifrm(true(),N,add(M,X)) = [0 1 3] [0 0 0] [1 3 0] [2] [0 1 3] M + [0 0 0] N + [0 1 0] X + [2] [0 0 0] [3 0 0] [1 2 0] [6] >= [0 0 0] [1 1 0] [2] [0 0 0] N + [0 1 0] X + [0] [3 0 0] [1 2 0] [0] = rm(N,X) purge(add(N,X)) = [0 2 6] [2 6 0] [6] [0 1 3] N + [0 1 0] X + [2] [0 0 0] [1 2 0] [2] >= [0 0 0] [2 6 0] [6] [0 1 3] N + [0 1 0] X + [2] [0 0 0] [0 0 0] [0] = add(N,purge(rm(N,X))) purge(nil()) = [2] [0] [2] >= [0] [0] [0] = nil() rm(N,add(M,X)) = [0 1 3] [0 0 0] [1 3 0] [4] [0 1 3] M + [0 0 0] N + [0 1 0] X + [2] [0 2 6] [3 0 0] [1 4 0] [4] >= [0 1 3] [0 0 0] [1 3 0] [4] [0 1 3] M + [0 0 0] N + [0 1 0] X + [2] [0 2 6] [3 0 0] [1 2 0] [4] = ifrm(eq(N,M),N,add(M,X)) rm(N,nil()) = [0 0 0] [2] [0 0 0] N + [0] [3 0 0] [0] >= [0] [0] [0] = nil() ** Step 1.b:9: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 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() 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,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: NaturalMI {miDimension = 3, miDegree = 3, 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) = {1,3}, uargs(purge) = {1}, uargs(rm) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] [0] [0] p(add) = [0 0 2] [1 2 0] [0] [0 1 1] x1 + [0 1 0] x2 + [2] [0 0 0] [0 0 0] [0] p(eq) = [0 0 0] [0] [0 1 1] x2 + [1] [0 0 0] [0] p(false) = [0] [1] [0] p(ifrm) = [2 0 0] [1 1 0] [0] [0 0 0] x1 + [0 1 0] x3 + [0] [0 2 0] [2 0 0] [1] p(nil) = [0] [0] [0] p(purge) = [2 1 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(rm) = [1 1 0] [1] [0 1 0] x2 + [0] [2 2 0] [2] p(s) = [1 1 0] [0] [0 1 2] x1 + [0] [0 0 0] [0] p(true) = [0] [1] [0] Following rules are strictly oriented: rm(N,add(M,X)) = [0 1 3] [1 3 0] [3] [0 1 1] M + [0 1 0] X + [2] [0 2 6] [2 6 0] [6] > [0 1 3] [1 3 0] [2] [0 1 1] M + [0 1 0] X + [2] [0 2 6] [2 4 0] [3] = ifrm(eq(N,M),N,add(M,X)) Following rules are (at-least) weakly oriented: eq(0(),0()) = [0] [1] [0] >= [0] [1] [0] = true() eq(0(),s(X)) = [0 0 0] [0] [0 1 2] X + [1] [0 0 0] [0] >= [0] [1] [0] = false() eq(s(X),0()) = [0] [1] [0] >= [0] [1] [0] = false() eq(s(X),s(Y)) = [0 0 0] [0] [0 1 2] Y + [1] [0 0 0] [0] >= [0 0 0] [0] [0 1 1] Y + [1] [0 0 0] [0] = eq(X,Y) ifrm(false(),N,add(M,X)) = [0 1 3] [1 3 0] [2] [0 1 1] M + [0 1 0] X + [2] [0 0 4] [2 4 0] [3] >= [0 0 2] [1 3 0] [1] [0 1 1] M + [0 1 0] X + [2] [0 0 0] [0 0 0] [0] = add(M,rm(N,X)) ifrm(true(),N,add(M,X)) = [0 1 3] [1 3 0] [2] [0 1 1] M + [0 1 0] X + [2] [0 0 4] [2 4 0] [3] >= [1 1 0] [1] [0 1 0] X + [0] [2 2 0] [2] = rm(N,X) purge(add(N,X)) = [0 1 5] [2 5 0] [2] [0 1 1] N + [0 1 0] X + [2] [0 0 0] [0 0 0] [0] >= [0 0 2] [2 5 0] [2] [0 1 1] N + [0 1 0] X + [2] [0 0 0] [0 0 0] [0] = add(N,purge(rm(N,X))) purge(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() rm(N,nil()) = [1] [0] [2] >= [0] [0] [0] = nil() ** Step 1.b:10: 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(Omega(n^1),O(n^3))