/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^1)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: divp(x,y) -> =(rem(x,y),0()) prime(0()) -> false() prime(s(0())) -> false() prime(s(s(x))) -> prime1(s(s(x)),s(x)) prime1(x,0()) -> false() prime1(x,s(0())) -> true() prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y))) - Signature: {divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {divp,prime,prime1} and constructors {0,=,and,false,not,rem,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: divp(x,y) -> =(rem(x,y),0()) prime(0()) -> false() prime(s(0())) -> false() prime(s(s(x))) -> prime1(s(s(x)),s(x)) prime1(x,0()) -> false() prime1(x,s(0())) -> true() prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y))) - Signature: {divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {divp,prime,prime1} and constructors {0,=,and,false,not,rem,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: prime1(x,s(y)){y -> s(y)} = prime1(x,s(s(y))) ->^+ and(not(divp(s(s(y)),x)),prime1(x,s(y))) = C[prime1(x,s(y)) = prime1(x,s(y)){}] ** Step 1.b:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: divp(x,y) -> =(rem(x,y),0()) prime(0()) -> false() prime(s(0())) -> false() prime(s(s(x))) -> prime1(s(s(x)),s(x)) prime1(x,0()) -> false() prime1(x,s(0())) -> true() prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y))) - Signature: {divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {divp,prime,prime1} and constructors {0,=,and,false,not,rem,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(and) = {1,2}, uargs(not) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(=) = [1] x1 + [0] p(and) = [1] x1 + [1] x2 + [0] p(divp) = [9] x1 + [0] p(false) = [1] p(not) = [1] x1 + [0] p(prime) = [11] p(prime1) = [2] p(rem) = [0] p(s) = [0] p(true) = [1] Following rules are strictly oriented: prime(0()) = [11] > [1] = false() prime(s(0())) = [11] > [1] = false() prime(s(s(x))) = [11] > [2] = prime1(s(s(x)),s(x)) prime1(x,0()) = [2] > [1] = false() prime1(x,s(0())) = [2] > [1] = true() Following rules are (at-least) weakly oriented: divp(x,y) = [9] x + [0] >= [0] = =(rem(x,y),0()) prime1(x,s(s(y))) = [2] >= [2] = and(not(divp(s(s(y)),x)),prime1(x,s(y))) ** Step 1.b:2: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: divp(x,y) -> =(rem(x,y),0()) prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y))) - Weak TRS: prime(0()) -> false() prime(s(0())) -> false() prime(s(s(x))) -> prime1(s(s(x)),s(x)) prime1(x,0()) -> false() prime1(x,s(0())) -> true() - Signature: {divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {divp,prime,prime1} and constructors {0,=,and,false,not,rem,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(and) = {1,2}, uargs(not) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(=) = [1] x1 + [0] p(and) = [1] x1 + [1] x2 + [0] p(divp) = [1] p(false) = [1] p(not) = [1] x1 + [0] p(prime) = [8] x1 + [0] p(prime1) = [8] x2 + [6] p(rem) = [1] p(s) = [1] x1 + [1] p(true) = [0] Following rules are strictly oriented: prime1(x,s(s(y))) = [8] y + [22] > [8] y + [15] = and(not(divp(s(s(y)),x)),prime1(x,s(y))) Following rules are (at-least) weakly oriented: divp(x,y) = [1] >= [1] = =(rem(x,y),0()) prime(0()) = [8] >= [1] = false() prime(s(0())) = [16] >= [1] = false() prime(s(s(x))) = [8] x + [16] >= [8] x + [14] = prime1(s(s(x)),s(x)) prime1(x,0()) = [14] >= [1] = false() prime1(x,s(0())) = [22] >= [0] = true() ** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: divp(x,y) -> =(rem(x,y),0()) - Weak TRS: prime(0()) -> false() prime(s(0())) -> false() prime(s(s(x))) -> prime1(s(s(x)),s(x)) prime1(x,0()) -> false() prime1(x,s(0())) -> true() prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y))) - Signature: {divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {divp,prime,prime1} and constructors {0,=,and,false,not,rem,s,true} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(and) = {1,2}, uargs(not) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = 8 p(=) = 3 p(and) = x1 + x2 p(divp) = 4 p(false) = 3 p(not) = 4 + x1 p(prime) = 8 + x1 p(prime1) = 9 + x2 p(rem) = 2 + x2 p(s) = 8 + x1 p(true) = 8 Following rules are strictly oriented: divp(x,y) = 4 > 3 = =(rem(x,y),0()) Following rules are (at-least) weakly oriented: prime(0()) = 16 >= 3 = false() prime(s(0())) = 24 >= 3 = false() prime(s(s(x))) = 24 + x >= 17 + x = prime1(s(s(x)),s(x)) prime1(x,0()) = 17 >= 3 = false() prime1(x,s(0())) = 25 >= 8 = true() prime1(x,s(s(y))) = 25 + y >= 25 + y = and(not(divp(s(s(y)),x)),prime1(x,s(y))) ** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: divp(x,y) -> =(rem(x,y),0()) prime(0()) -> false() prime(s(0())) -> false() prime(s(s(x))) -> prime1(s(s(x)),s(x)) prime1(x,0()) -> false() prime1(x,s(0())) -> true() prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y))) - Signature: {divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {divp,prime,prime1} and constructors {0,=,and,false,not,rem,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))