/export/starexec/sandbox/solver/bin/starexec_run_tct_rc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {if_mod,le,minus,mod} and constructors {0,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {if_mod,le,minus,mod} and constructors {0,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {if_mod,le,minus,mod} and constructors {0,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: le(x,y){x -> s(x),y -> s(y)} = le(s(x),s(y)) ->^+ le(x,y) = C[le(x,y) = le(x,y){}] ** Step 1.b:1: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {if_mod,le,minus,mod} and constructors {0,false,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(if_mod) = {1}, uargs(mod) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(if_mod) = [1] x1 + [8] x2 + [1] x3 + [2] p(le) = [0] p(minus) = [1] x1 + [0] p(mod) = [8] x1 + [1] x2 + [7] p(s) = [1] x1 + [1] p(true) = [0] Following rules are strictly oriented: if_mod(false(),s(x),s(y)) = [8] x + [1] y + [11] > [1] x + [1] = s(x) if_mod(true(),s(x),s(y)) = [8] x + [1] y + [11] > [8] x + [1] y + [8] = mod(minus(x,y),s(y)) minus(s(x),s(y)) = [1] x + [1] > [1] x + [0] = minus(x,y) mod(0(),y) = [1] y + [7] > [0] = 0() mod(s(x),0()) = [8] x + [15] > [0] = 0() mod(s(x),s(y)) = [8] x + [1] y + [16] > [8] x + [1] y + [11] = if_mod(le(y,x),s(x),s(y)) Following rules are (at-least) weakly oriented: le(0(),y) = [0] >= [0] = true() le(s(x),0()) = [0] >= [0] = false() le(s(x),s(y)) = [0] >= [0] = le(x,y) minus(x,0()) = [1] x + [0] >= [1] x + [0] = x ** Step 1.b:2: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x - Weak TRS: if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y)) minus(s(x),s(y)) -> minus(x,y) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {if_mod,le,minus,mod} and constructors {0,false,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(if_mod) = {1}, uargs(mod) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(false) = [0] p(if_mod) = [2] x1 + [4] x2 + [5] p(le) = [1] p(minus) = [1] x1 + [0] p(mod) = [4] x1 + [7] p(s) = [1] x1 + [0] p(true) = [1] Following rules are strictly oriented: le(s(x),0()) = [1] > [0] = false() Following rules are (at-least) weakly oriented: if_mod(false(),s(x),s(y)) = [4] x + [5] >= [1] x + [0] = s(x) if_mod(true(),s(x),s(y)) = [4] x + [7] >= [4] x + [7] = mod(minus(x,y),s(y)) le(0(),y) = [1] >= [1] = true() le(s(x),s(y)) = [1] >= [1] = le(x,y) minus(x,0()) = [1] x + [0] >= [1] x + [0] = x minus(s(x),s(y)) = [1] x + [0] >= [1] x + [0] = minus(x,y) mod(0(),y) = [15] >= [2] = 0() mod(s(x),0()) = [4] x + [7] >= [2] = 0() mod(s(x),s(y)) = [4] x + [7] >= [4] x + [7] = if_mod(le(y,x),s(x),s(y)) ** Step 1.b:3: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: le(0(),y) -> true() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x - Weak TRS: if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y)) le(s(x),0()) -> false() minus(s(x),s(y)) -> minus(x,y) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {if_mod,le,minus,mod} and constructors {0,false,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(if_mod) = {1}, uargs(mod) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [10] p(false) = [1] p(if_mod) = [2] x1 + [1] x2 + [0] p(le) = [1] p(minus) = [1] x1 + [1] p(mod) = [1] x1 + [2] p(s) = [1] x1 + [8] p(true) = [0] Following rules are strictly oriented: le(0(),y) = [1] > [0] = true() minus(x,0()) = [1] x + [1] > [1] x + [0] = x Following rules are (at-least) weakly oriented: if_mod(false(),s(x),s(y)) = [1] x + [10] >= [1] x + [8] = s(x) if_mod(true(),s(x),s(y)) = [1] x + [8] >= [1] x + [3] = mod(minus(x,y),s(y)) le(s(x),0()) = [1] >= [1] = false() le(s(x),s(y)) = [1] >= [1] = le(x,y) minus(s(x),s(y)) = [1] x + [9] >= [1] x + [1] = minus(x,y) mod(0(),y) = [12] >= [10] = 0() mod(s(x),0()) = [1] x + [10] >= [10] = 0() mod(s(x),s(y)) = [1] x + [10] >= [1] x + [10] = if_mod(le(y,x),s(x),s(y)) ** Step 1.b:4: NaturalPI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: le(s(x),s(y)) -> le(x,y) - Weak TRS: if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {if_mod,le,minus,mod} and constructors {0,false,s,true} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(if_mod) = {1}, uargs(mod) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = 0 p(false) = 0 p(if_mod) = 3 + x1 + 2*x2^2 + x3 p(le) = 2*x2 p(minus) = x1 p(mod) = 2*x1 + 2*x1^2 + x2 p(s) = 2 + x1 p(true) = 0 Following rules are strictly oriented: le(s(x),s(y)) = 4 + 2*y > 2*y = le(x,y) Following rules are (at-least) weakly oriented: if_mod(false(),s(x),s(y)) = 13 + 8*x + 2*x^2 + y >= 2 + x = s(x) if_mod(true(),s(x),s(y)) = 13 + 8*x + 2*x^2 + y >= 2 + 2*x + 2*x^2 + y = mod(minus(x,y),s(y)) le(0(),y) = 2*y >= 0 = true() le(s(x),0()) = 0 >= 0 = false() minus(x,0()) = x >= x = x minus(s(x),s(y)) = 2 + x >= x = minus(x,y) mod(0(),y) = y >= 0 = 0() mod(s(x),0()) = 12 + 10*x + 2*x^2 >= 0 = 0() mod(s(x),s(y)) = 14 + 10*x + 2*x^2 + y >= 13 + 10*x + 2*x^2 + y = if_mod(le(y,x),s(x),s(y)) ** Step 1.b:5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {if_mod,le,minus,mod} and constructors {0,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^2))