YES(O(1),O(n^2)) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict Trs: { eq(0(), 0()) -> true() , eq(0(), s(X)) -> false() , eq(s(X), 0()) -> false() , eq(s(X), s(Y)) -> eq(X, Y) , rm(N, nil()) -> nil() , rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X)) , ifrm(true(), N, add(M, X)) -> rm(N, X) , ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) , purge(nil()) -> nil() , purge(add(N, X)) -> add(N, purge(rm(N, X))) } Obligation: runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(rm) = {2}, Uargs(add) = {2}, Uargs(ifrm) = {1, 3}, Uargs(purge) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [eq](x1, x2) = [4] [0] = [7] [true] = [0] [s](x1) = [1] x1 + [7] [false] = [0] [rm](x1, x2) = [1] x2 + [0] [nil] = [4] [add](x1, x2) = [1] x1 + [1] x2 + [0] [ifrm](x1, x2, x3) = [1] x1 + [1] x3 + [0] [purge](x1) = [1] x1 + [0] The following symbols are considered usable {eq, rm, ifrm, purge} The order satisfies the following ordering constraints: [eq(0(), 0())] = [4] > [0] = [true()] [eq(0(), s(X))] = [4] > [0] = [false()] [eq(s(X), 0())] = [4] > [0] = [false()] [eq(s(X), s(Y))] = [4] >= [4] = [eq(X, Y)] [rm(N, nil())] = [4] >= [4] = [nil()] [rm(N, add(M, X))] = [1] X + [1] M + [0] ? [1] X + [1] M + [4] = [ifrm(eq(N, M), N, add(M, X))] [ifrm(true(), N, add(M, X))] = [1] X + [1] M + [0] >= [1] X + [0] = [rm(N, X)] [ifrm(false(), N, add(M, X))] = [1] X + [1] M + [0] >= [1] X + [1] M + [0] = [add(M, rm(N, X))] [purge(nil())] = [4] >= [4] = [nil()] [purge(add(N, X))] = [1] X + [1] N + [0] >= [1] X + [1] N + [0] = [add(N, purge(rm(N, X)))] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict Trs: { eq(s(X), s(Y)) -> eq(X, Y) , rm(N, nil()) -> nil() , rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X)) , ifrm(true(), N, add(M, X)) -> rm(N, X) , ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) , purge(nil()) -> nil() , purge(add(N, X)) -> add(N, purge(rm(N, X))) } Weak Trs: { eq(0(), 0()) -> true() , eq(0(), s(X)) -> false() , eq(s(X), 0()) -> false() } Obligation: runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(rm) = {2}, Uargs(add) = {2}, Uargs(ifrm) = {1, 3}, Uargs(purge) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [eq](x1, x2) = [1] [0] = [7] [true] = [1] [s](x1) = [1] x1 + [7] [false] = [0] [rm](x1, x2) = [1] x2 + [0] [nil] = [4] [add](x1, x2) = [1] x1 + [1] x2 + [0] [ifrm](x1, x2, x3) = [1] x1 + [1] x3 + [0] [purge](x1) = [1] x1 + [0] The following symbols are considered usable {eq, rm, ifrm, purge} The order satisfies the following ordering constraints: [eq(0(), 0())] = [1] >= [1] = [true()] [eq(0(), s(X))] = [1] > [0] = [false()] [eq(s(X), 0())] = [1] > [0] = [false()] [eq(s(X), s(Y))] = [1] >= [1] = [eq(X, Y)] [rm(N, nil())] = [4] >= [4] = [nil()] [rm(N, add(M, X))] = [1] X + [1] M + [0] ? [1] X + [1] M + [1] = [ifrm(eq(N, M), N, add(M, X))] [ifrm(true(), N, add(M, X))] = [1] X + [1] M + [1] > [1] X + [0] = [rm(N, X)] [ifrm(false(), N, add(M, X))] = [1] X + [1] M + [0] >= [1] X + [1] M + [0] = [add(M, rm(N, X))] [purge(nil())] = [4] >= [4] = [nil()] [purge(add(N, X))] = [1] X + [1] N + [0] >= [1] X + [1] N + [0] = [add(N, purge(rm(N, X)))] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict Trs: { eq(s(X), s(Y)) -> eq(X, Y) , rm(N, nil()) -> nil() , rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X)) , ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) , purge(nil()) -> nil() , purge(add(N, X)) -> add(N, purge(rm(N, 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) } Obligation: runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(rm) = {2}, Uargs(add) = {2}, Uargs(ifrm) = {1, 3}, Uargs(purge) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [eq](x1, x2) = [1] [0] = [7] [true] = [0] [s](x1) = [1] x1 + [7] [false] = [1] [rm](x1, x2) = [1] x2 + [0] [nil] = [4] [add](x1, x2) = [1] x1 + [1] x2 + [0] [ifrm](x1, x2, x3) = [1] x1 + [1] x3 + [0] [purge](x1) = [1] x1 + [0] The following symbols are considered usable {eq, rm, ifrm, purge} The order satisfies the following ordering constraints: [eq(0(), 0())] = [1] > [0] = [true()] [eq(0(), s(X))] = [1] >= [1] = [false()] [eq(s(X), 0())] = [1] >= [1] = [false()] [eq(s(X), s(Y))] = [1] >= [1] = [eq(X, Y)] [rm(N, nil())] = [4] >= [4] = [nil()] [rm(N, add(M, X))] = [1] X + [1] M + [0] ? [1] X + [1] M + [1] = [ifrm(eq(N, M), N, add(M, X))] [ifrm(true(), N, add(M, X))] = [1] X + [1] M + [0] >= [1] X + [0] = [rm(N, X)] [ifrm(false(), N, add(M, X))] = [1] X + [1] M + [1] > [1] X + [1] M + [0] = [add(M, rm(N, X))] [purge(nil())] = [4] >= [4] = [nil()] [purge(add(N, X))] = [1] X + [1] N + [0] >= [1] X + [1] N + [0] = [add(N, purge(rm(N, X)))] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict Trs: { eq(s(X), s(Y)) -> eq(X, Y) , rm(N, nil()) -> nil() , rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X)) , purge(nil()) -> nil() , purge(add(N, X)) -> add(N, purge(rm(N, 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) , ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) } Obligation: runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(rm) = {2}, Uargs(add) = {2}, Uargs(ifrm) = {1, 3}, Uargs(purge) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [eq](x1, x2) = [4] [0] = [7] [true] = [4] [s](x1) = [1] x1 + [7] [false] = [0] [rm](x1, x2) = [1] x2 + [0] [nil] = [4] [add](x1, x2) = [1] x2 + [0] [ifrm](x1, x2, x3) = [1] x1 + [1] x3 + [0] [purge](x1) = [1] x1 + [4] The following symbols are considered usable {eq, rm, ifrm, purge} The order satisfies the following ordering constraints: [eq(0(), 0())] = [4] >= [4] = [true()] [eq(0(), s(X))] = [4] > [0] = [false()] [eq(s(X), 0())] = [4] > [0] = [false()] [eq(s(X), s(Y))] = [4] >= [4] = [eq(X, Y)] [rm(N, nil())] = [4] >= [4] = [nil()] [rm(N, add(M, X))] = [1] X + [0] ? [1] X + [4] = [ifrm(eq(N, M), N, add(M, X))] [ifrm(true(), N, add(M, X))] = [1] X + [4] > [1] X + [0] = [rm(N, X)] [ifrm(false(), N, add(M, X))] = [1] X + [0] >= [1] X + [0] = [add(M, rm(N, X))] [purge(nil())] = [8] > [4] = [nil()] [purge(add(N, X))] = [1] X + [4] >= [1] X + [4] = [add(N, purge(rm(N, X)))] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict Trs: { eq(s(X), s(Y)) -> eq(X, Y) , rm(N, nil()) -> nil() , rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X)) , purge(add(N, X)) -> add(N, purge(rm(N, 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) , ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) , purge(nil()) -> nil() } Obligation: runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(rm) = {2}, Uargs(add) = {2}, Uargs(ifrm) = {1, 3}, Uargs(purge) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [eq](x1, x2) = [1] [0] = [7] [true] = [1] [s](x1) = [1] x1 + [7] [false] = [1] [rm](x1, x2) = [1] x2 + [1] [nil] = [7] [add](x1, x2) = [1] x1 + [1] x2 + [0] [ifrm](x1, x2, x3) = [1] x1 + [1] x3 + [0] [purge](x1) = [1] x1 + [0] The following symbols are considered usable {eq, rm, ifrm, purge} The order satisfies the following ordering constraints: [eq(0(), 0())] = [1] >= [1] = [true()] [eq(0(), s(X))] = [1] >= [1] = [false()] [eq(s(X), 0())] = [1] >= [1] = [false()] [eq(s(X), s(Y))] = [1] >= [1] = [eq(X, Y)] [rm(N, nil())] = [8] > [7] = [nil()] [rm(N, add(M, X))] = [1] X + [1] M + [1] >= [1] X + [1] M + [1] = [ifrm(eq(N, M), N, add(M, X))] [ifrm(true(), N, add(M, X))] = [1] X + [1] M + [1] >= [1] X + [1] = [rm(N, X)] [ifrm(false(), N, add(M, X))] = [1] X + [1] M + [1] >= [1] X + [1] M + [1] = [add(M, rm(N, X))] [purge(nil())] = [7] >= [7] = [nil()] [purge(add(N, X))] = [1] X + [1] N + [0] ? [1] X + [1] N + [1] = [add(N, purge(rm(N, X)))] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict Trs: { eq(s(X), s(Y)) -> eq(X, Y) , rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X)) , purge(add(N, X)) -> add(N, purge(rm(N, X))) } Weak Trs: { eq(0(), 0()) -> true() , eq(0(), s(X)) -> false() , eq(s(X), 0()) -> false() , rm(N, nil()) -> nil() , ifrm(true(), N, add(M, X)) -> rm(N, X) , ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) , purge(nil()) -> nil() } Obligation: runtime complexity Answer: YES(O(1),O(n^2)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. Trs: { purge(add(N, X)) -> add(N, purge(rm(N, X))) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^1)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(rm) = {2}, Uargs(add) = {2}, Uargs(ifrm) = {1, 3}, Uargs(purge) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [eq](x1, x2) = [0] [0] = [0] [true] = [0] [s](x1) = [1] x1 + [0] [false] = [0] [rm](x1, x2) = [1] x2 + [0] [nil] = [0] [add](x1, x2) = [1] x2 + [1] [ifrm](x1, x2, x3) = [4] x1 + [1] x3 + [0] [purge](x1) = [4] x1 + [0] The following symbols are considered usable {eq, rm, ifrm, purge} The order satisfies the following ordering constraints: [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)] [rm(N, nil())] = [0] >= [0] = [nil()] [rm(N, add(M, X))] = [1] X + [1] >= [1] X + [1] = [ifrm(eq(N, M), N, add(M, X))] [ifrm(true(), N, add(M, X))] = [1] X + [1] > [1] X + [0] = [rm(N, X)] [ifrm(false(), N, add(M, X))] = [1] X + [1] >= [1] X + [1] = [add(M, rm(N, X))] [purge(nil())] = [0] >= [0] = [nil()] [purge(add(N, X))] = [4] X + [4] > [4] X + [1] = [add(N, purge(rm(N, X)))] We return to the main proof. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). 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() , rm(N, nil()) -> nil() , ifrm(true(), N, add(M, X)) -> rm(N, X) , ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) , purge(nil()) -> nil() , purge(add(N, X)) -> add(N, purge(rm(N, X))) } Obligation: runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(rm) = {2}, Uargs(add) = {2}, Uargs(ifrm) = {1, 3}, Uargs(purge) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [eq](x1, x2) = [0 1] x2 + [0] [0 0] [4] [0] = [0] [4] [true] = [0] [0] [s](x1) = [0 0] x1 + [0] [0 1] [4] [false] = [0] [0] [rm](x1, x2) = [1 4] x2 + [0] [0 0] [0] [nil] = [0] [0] [add](x1, x2) = [0 0] x1 + [1 4] x2 + [0] [0 1] [0 0] [0] [ifrm](x1, x2, x3) = [1 0] x1 + [1 0] x3 + [0] [0 0] [0 1] [4] [purge](x1) = [1 0] x1 + [0] [0 1] [0] The following symbols are considered usable {eq, rm, ifrm, purge} The order satisfies the following ordering constraints: [eq(0(), 0())] = [4] [4] > [0] [0] = [true()] [eq(0(), s(X))] = [0 1] X + [4] [0 0] [4] > [0] [0] = [false()] [eq(s(X), 0())] = [4] [4] > [0] [0] = [false()] [eq(s(X), s(Y))] = [0 1] Y + [4] [0 0] [4] > [0 1] Y + [0] [0 0] [4] = [eq(X, Y)] [rm(N, nil())] = [0] [0] >= [0] [0] = [nil()] [rm(N, add(M, X))] = [1 4] X + [0 4] M + [0] [0 0] [0 0] [0] ? [1 4] X + [0 1] M + [0] [0 0] [0 1] [4] = [ifrm(eq(N, M), N, add(M, X))] [ifrm(true(), N, add(M, X))] = [1 4] X + [0 0] M + [0] [0 0] [0 1] [4] >= [1 4] X + [0] [0 0] [0] = [rm(N, X)] [ifrm(false(), N, add(M, X))] = [1 4] X + [0 0] M + [0] [0 0] [0 1] [4] >= [1 4] X + [0 0] M + [0] [0 0] [0 1] [0] = [add(M, rm(N, X))] [purge(nil())] = [0] [0] >= [0] [0] = [nil()] [purge(add(N, X))] = [1 4] X + [0 0] N + [0] [0 0] [0 1] [0] >= [1 4] X + [0 0] N + [0] [0 0] [0 1] [0] = [add(N, purge(rm(N, X)))] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). 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) , rm(N, nil()) -> nil() , ifrm(true(), N, add(M, X)) -> rm(N, X) , ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) , purge(nil()) -> nil() , purge(add(N, X)) -> add(N, purge(rm(N, X))) } Obligation: runtime complexity Answer: YES(O(1),O(n^2)) We use the processor 'matrix interpretation of dimension 2' to orient following rules strictly. Trs: { rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X)) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^2)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(rm) = {2}, Uargs(add) = {2}, Uargs(ifrm) = {1, 3}, Uargs(purge) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [eq](x1, x2) = [0] [0] [0] = [0] [0] [true] = [0] [0] [s](x1) = [0] [0] [false] = [0] [0] [rm](x1, x2) = [1 2] x2 + [1] [0 1] [0] [nil] = [0] [0] [add](x1, x2) = [1 4] x2 + [0] [0 1] [2] [ifrm](x1, x2, x3) = [1 0] x1 + [1 2] x3 + [0] [0 0] [0 1] [0] [purge](x1) = [2 2] x1 + [4] [0 1] [0] The following symbols are considered usable {eq, rm, ifrm, purge} The order satisfies the following ordering constraints: [eq(0(), 0())] = [0] [0] >= [0] [0] = [true()] [eq(0(), s(X))] = [0] [0] >= [0] [0] = [false()] [eq(s(X), 0())] = [0] [0] >= [0] [0] = [false()] [eq(s(X), s(Y))] = [0] [0] >= [0] [0] = [eq(X, Y)] [rm(N, nil())] = [1] [0] > [0] [0] = [nil()] [rm(N, add(M, X))] = [1 6] X + [5] [0 1] [2] > [1 6] X + [4] [0 1] [2] = [ifrm(eq(N, M), N, add(M, X))] [ifrm(true(), N, add(M, X))] = [1 6] X + [4] [0 1] [2] > [1 2] X + [1] [0 1] [0] = [rm(N, X)] [ifrm(false(), N, add(M, X))] = [1 6] X + [4] [0 1] [2] > [1 6] X + [1] [0 1] [2] = [add(M, rm(N, X))] [purge(nil())] = [4] [0] > [0] [0] = [nil()] [purge(add(N, X))] = [2 10] X + [8] [0 1] [2] > [2 10] X + [6] [0 1] [2] = [add(N, purge(rm(N, X)))] We return to the main proof. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak Trs: { eq(0(), 0()) -> true() , eq(0(), s(X)) -> false() , eq(s(X), 0()) -> false() , eq(s(X), s(Y)) -> eq(X, Y) , rm(N, nil()) -> nil() , rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X)) , ifrm(true(), N, add(M, X)) -> rm(N, X) , ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) , purge(nil()) -> nil() , purge(add(N, X)) -> add(N, purge(rm(N, X))) } Obligation: runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Hurray, we answered YES(O(1),O(n^2))