* Step 1: DependencyPairs WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1} / {0/0,p/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {*,+,minus} and constructors {0,p,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs *#(0(),y) -> c_1() *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)) *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) +#(0(),y) -> c_4() +#(p(x),y) -> c_5(+#(x,y)) +#(s(x),y) -> c_6(+#(x,y)) minus#(0()) -> c_7() minus#(p(x)) -> c_8(minus#(x)) minus#(s(x)) -> c_9(minus#(x)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: *#(0(),y) -> c_1() *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)) *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) +#(0(),y) -> c_4() +#(p(x),y) -> c_5(+#(x,y)) +#(s(x),y) -> c_6(+#(x,y)) minus#(0()) -> c_7() minus#(p(x)) -> c_8(minus#(x)) minus#(s(x)) -> c_9(minus#(x)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4,7} by application of Pre({1,4,7}) = {2,3,5,6,8,9}. Here rules are labelled as follows: 1: *#(0(),y) -> c_1() 2: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)) 3: *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) 4: +#(0(),y) -> c_4() 5: +#(p(x),y) -> c_5(+#(x,y)) 6: +#(s(x),y) -> c_6(+#(x,y)) 7: minus#(0()) -> c_7() 8: minus#(p(x)) -> c_8(minus#(x)) 9: minus#(s(x)) -> c_9(minus#(x)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)) *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) +#(p(x),y) -> c_5(+#(x,y)) +#(s(x),y) -> c_6(+#(x,y)) minus#(p(x)) -> c_8(minus#(x)) minus#(s(x)) -> c_9(minus#(x)) - Weak DPs: *#(0(),y) -> c_1() +#(0(),y) -> c_4() minus#(0()) -> c_7() - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)) -->_3 minus#(s(x)) -> c_9(minus#(x)):6 -->_3 minus#(p(x)) -> c_8(minus#(x)):5 -->_1 +#(s(x),y) -> c_6(+#(x,y)):4 -->_1 +#(p(x),y) -> c_5(+#(x,y)):3 -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):2 -->_3 minus#(0()) -> c_7():9 -->_1 +#(0(),y) -> c_4():8 -->_2 *#(0(),y) -> c_1():7 -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):1 2:S:*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) -->_1 +#(s(x),y) -> c_6(+#(x,y)):4 -->_1 +#(p(x),y) -> c_5(+#(x,y)):3 -->_1 +#(0(),y) -> c_4():8 -->_2 *#(0(),y) -> c_1():7 -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):2 -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):1 3:S:+#(p(x),y) -> c_5(+#(x,y)) -->_1 +#(s(x),y) -> c_6(+#(x,y)):4 -->_1 +#(0(),y) -> c_4():8 -->_1 +#(p(x),y) -> c_5(+#(x,y)):3 4:S:+#(s(x),y) -> c_6(+#(x,y)) -->_1 +#(0(),y) -> c_4():8 -->_1 +#(s(x),y) -> c_6(+#(x,y)):4 -->_1 +#(p(x),y) -> c_5(+#(x,y)):3 5:S:minus#(p(x)) -> c_8(minus#(x)) -->_1 minus#(s(x)) -> c_9(minus#(x)):6 -->_1 minus#(0()) -> c_7():9 -->_1 minus#(p(x)) -> c_8(minus#(x)):5 6:S:minus#(s(x)) -> c_9(minus#(x)) -->_1 minus#(0()) -> c_7():9 -->_1 minus#(s(x)) -> c_9(minus#(x)):6 -->_1 minus#(p(x)) -> c_8(minus#(x)):5 7:W:*#(0(),y) -> c_1() 8:W:+#(0(),y) -> c_4() 9:W:minus#(0()) -> c_7() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: *#(0(),y) -> c_1() 8: +#(0(),y) -> c_4() 9: minus#(0()) -> c_7() * Step 4: PredecessorEstimationCP WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)) *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) +#(p(x),y) -> c_5(+#(x,y)) +#(s(x),y) -> c_6(+#(x,y)) minus#(p(x)) -> c_8(minus#(x)) minus#(s(x)) -> c_9(minus#(x)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)) 5: minus#(p(x)) -> c_8(minus#(x)) The strictly oriented rules are moved into the weak component. ** Step 4.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)) *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) +#(p(x),y) -> c_5(+#(x,y)) +#(s(x),y) -> c_6(+#(x,y)) minus#(p(x)) -> c_8(minus#(x)) minus#(s(x)) -> c_9(minus#(x)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_2) = {1,2,3}, uargs(c_3) = {1,2}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1} Following symbols are considered usable: {*#,+#,minus#} TcT has computed the following interpretation: p(*) = 1 + 2*x2^2 p(+) = 1 + x1 + x1^2 + x2 p(0) = 0 p(minus) = 1 p(p) = 1 + x1 p(s) = x1 p(*#) = 3*x1 + 2*x1*x2 + x1^2 p(+#) = 0 p(minus#) = 2*x1 p(c_1) = 0 p(c_2) = x1 + x2 + x3 p(c_3) = x1 + x2 p(c_4) = 0 p(c_5) = x1 p(c_6) = x1 p(c_7) = 0 p(c_8) = x1 p(c_9) = x1 Following rules are strictly oriented: *#(p(x),y) = 4 + 5*x + 2*x*y + x^2 + 2*y > 3*x + 2*x*y + x^2 + 2*y = c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)) minus#(p(x)) = 2 + 2*x > 2*x = c_8(minus#(x)) Following rules are (at-least) weakly oriented: *#(s(x),y) = 3*x + 2*x*y + x^2 >= 3*x + 2*x*y + x^2 = c_3(+#(*(x,y),y),*#(x,y)) +#(p(x),y) = 0 >= 0 = c_5(+#(x,y)) +#(s(x),y) = 0 >= 0 = c_6(+#(x,y)) minus#(s(x)) = 2*x >= 2*x = c_9(minus#(x)) ** Step 4.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) +#(p(x),y) -> c_5(+#(x,y)) +#(s(x),y) -> c_6(+#(x,y)) minus#(s(x)) -> c_9(minus#(x)) - Weak DPs: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)) minus#(p(x)) -> c_8(minus#(x)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ** Step 4.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) +#(p(x),y) -> c_5(+#(x,y)) +#(s(x),y) -> c_6(+#(x,y)) minus#(s(x)) -> c_9(minus#(x)) - Weak DPs: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)) minus#(p(x)) -> c_8(minus#(x)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) 4: minus#(s(x)) -> c_9(minus#(x)) The strictly oriented rules are moved into the weak component. *** Step 4.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) +#(p(x),y) -> c_5(+#(x,y)) +#(s(x),y) -> c_6(+#(x,y)) minus#(s(x)) -> c_9(minus#(x)) - Weak DPs: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)) minus#(p(x)) -> c_8(minus#(x)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_2) = {1,2,3}, uargs(c_3) = {1,2}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1} Following symbols are considered usable: {*#,+#,minus#} TcT has computed the following interpretation: p(*) = 1 + 2*x1 p(+) = x1*x2 + 3*x2 + 3*x2^2 p(0) = 0 p(minus) = 0 p(p) = 1 + x1 p(s) = 1 + x1 p(*#) = 2 + 2*x1 + x1*x2 + 2*x1^2 + 3*x2 + 2*x2^2 p(+#) = 0 p(minus#) = 2 + x1 p(c_1) = 1 p(c_2) = 1 + x1 + x2 + x3 p(c_3) = x1 + x2 p(c_4) = 0 p(c_5) = x1 p(c_6) = x1 p(c_7) = 0 p(c_8) = x1 p(c_9) = x1 Following rules are strictly oriented: *#(s(x),y) = 6 + 6*x + x*y + 2*x^2 + 4*y + 2*y^2 > 2 + 2*x + x*y + 2*x^2 + 3*y + 2*y^2 = c_3(+#(*(x,y),y),*#(x,y)) minus#(s(x)) = 3 + x > 2 + x = c_9(minus#(x)) Following rules are (at-least) weakly oriented: *#(p(x),y) = 6 + 6*x + x*y + 2*x^2 + 4*y + 2*y^2 >= 5 + 2*x + x*y + 2*x^2 + 4*y + 2*y^2 = c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)) +#(p(x),y) = 0 >= 0 = c_5(+#(x,y)) +#(s(x),y) = 0 >= 0 = c_6(+#(x,y)) minus#(p(x)) = 3 + x >= 2 + x = c_8(minus#(x)) *** Step 4.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: +#(p(x),y) -> c_5(+#(x,y)) +#(s(x),y) -> c_6(+#(x,y)) - Weak DPs: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)) *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) minus#(p(x)) -> c_8(minus#(x)) minus#(s(x)) -> c_9(minus#(x)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 4.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: +#(p(x),y) -> c_5(+#(x,y)) +#(s(x),y) -> c_6(+#(x,y)) - Weak DPs: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)) *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) minus#(p(x)) -> c_8(minus#(x)) minus#(s(x)) -> c_9(minus#(x)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:+#(p(x),y) -> c_5(+#(x,y)) -->_1 +#(s(x),y) -> c_6(+#(x,y)):2 -->_1 +#(p(x),y) -> c_5(+#(x,y)):1 2:S:+#(s(x),y) -> c_6(+#(x,y)) -->_1 +#(s(x),y) -> c_6(+#(x,y)):2 -->_1 +#(p(x),y) -> c_5(+#(x,y)):1 3:W:*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)) -->_3 minus#(s(x)) -> c_9(minus#(x)):6 -->_3 minus#(p(x)) -> c_8(minus#(x)):5 -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):4 -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):3 -->_1 +#(s(x),y) -> c_6(+#(x,y)):2 -->_1 +#(p(x),y) -> c_5(+#(x,y)):1 4:W:*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):4 -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):3 -->_1 +#(s(x),y) -> c_6(+#(x,y)):2 -->_1 +#(p(x),y) -> c_5(+#(x,y)):1 5:W:minus#(p(x)) -> c_8(minus#(x)) -->_1 minus#(s(x)) -> c_9(minus#(x)):6 -->_1 minus#(p(x)) -> c_8(minus#(x)):5 6:W:minus#(s(x)) -> c_9(minus#(x)) -->_1 minus#(s(x)) -> c_9(minus#(x)):6 -->_1 minus#(p(x)) -> c_8(minus#(x)):5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: minus#(s(x)) -> c_9(minus#(x)) 5: minus#(p(x)) -> c_8(minus#(x)) *** Step 4.b:1.b:2: SimplifyRHS WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: +#(p(x),y) -> c_5(+#(x,y)) +#(s(x),y) -> c_6(+#(x,y)) - Weak DPs: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)) *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:+#(p(x),y) -> c_5(+#(x,y)) -->_1 +#(s(x),y) -> c_6(+#(x,y)):2 -->_1 +#(p(x),y) -> c_5(+#(x,y)):1 2:S:+#(s(x),y) -> c_6(+#(x,y)) -->_1 +#(s(x),y) -> c_6(+#(x,y)):2 -->_1 +#(p(x),y) -> c_5(+#(x,y)):1 3:W:*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)) -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):4 -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):3 -->_1 +#(s(x),y) -> c_6(+#(x,y)):2 -->_1 +#(p(x),y) -> c_5(+#(x,y)):1 4:W:*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):4 -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):3 -->_1 +#(s(x),y) -> c_6(+#(x,y)):2 -->_1 +#(p(x),y) -> c_5(+#(x,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)) *** Step 4.b:1.b:3: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: +#(p(x),y) -> c_5(+#(x,y)) +#(s(x),y) -> c_6(+#(x,y)) - Weak DPs: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)) *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)) *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) and a lower component +#(p(x),y) -> c_5(+#(x,y)) +#(s(x),y) -> c_6(+#(x,y)) Further, following extension rules are added to the lower component. *#(p(x),y) -> *#(x,y) *#(p(x),y) -> +#(*(x,y),minus(y)) *#(s(x),y) -> *#(x,y) *#(s(x),y) -> +#(*(x,y),y) **** Step 4.b:1.b:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)) *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)) The strictly oriented rules are moved into the weak component. ***** Step 4.b:1.b:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)) *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1,2}, uargs(c_3) = {1,2} Following symbols are considered usable: {*#,+#,minus#} TcT has computed the following interpretation: p(*) = [4] x1 + [0] p(+) = [3] x1 + [2] x2 + [8] p(0) = [1] p(minus) = [8] p(p) = [1] x1 + [1] p(s) = [1] x1 + [0] p(*#) = [8] x1 + [0] p(+#) = [0] p(minus#) = [0] p(c_1) = [2] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [2] x1 + [1] x2 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] Following rules are strictly oriented: *#(p(x),y) = [8] x + [8] > [8] x + [0] = c_2(+#(*(x,y),minus(y)),*#(x,y)) Following rules are (at-least) weakly oriented: *#(s(x),y) = [8] x + [0] >= [8] x + [0] = c_3(+#(*(x,y),y),*#(x,y)) ***** Step 4.b:1.b:3.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) - Weak DPs: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 4.b:1.b:3.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) - Weak DPs: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) The strictly oriented rules are moved into the weak component. ****** Step 4.b:1.b:3.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) - Weak DPs: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1,2}, uargs(c_3) = {1,2} Following symbols are considered usable: {*#,+#,minus#} TcT has computed the following interpretation: p(*) = [12] x1 + [2] x2 + [0] p(+) = [2] x1 + [1] p(0) = [0] p(minus) = [10] x1 + [9] p(p) = [1] x1 + [1] p(s) = [1] x1 + [1] p(*#) = [4] x1 + [2] x2 + [0] p(+#) = [0] p(minus#) = [0] p(c_1) = [0] p(c_2) = [4] x1 + [1] x2 + [2] p(c_3) = [4] x1 + [1] x2 + [0] p(c_4) = [8] p(c_5) = [1] x1 + [0] p(c_6) = [2] x1 + [1] p(c_7) = [0] p(c_8) = [2] x1 + [0] p(c_9) = [1] Following rules are strictly oriented: *#(s(x),y) = [4] x + [2] y + [4] > [4] x + [2] y + [0] = c_3(+#(*(x,y),y),*#(x,y)) Following rules are (at-least) weakly oriented: *#(p(x),y) = [4] x + [2] y + [4] >= [4] x + [2] y + [2] = c_2(+#(*(x,y),minus(y)),*#(x,y)) ****** Step 4.b:1.b:3.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)) *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 4.b:1.b:3.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)) *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)) -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):2 -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)):1 2:W:*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):2 -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)) 2: *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)) ****** Step 4.b:1.b:3.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 4.b:1.b:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: +#(p(x),y) -> c_5(+#(x,y)) +#(s(x),y) -> c_6(+#(x,y)) - Weak DPs: *#(p(x),y) -> *#(x,y) *#(p(x),y) -> +#(*(x,y),minus(y)) *#(s(x),y) -> *#(x,y) *#(s(x),y) -> +#(*(x,y),y) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: +#(s(x),y) -> c_6(+#(x,y)) The strictly oriented rules are moved into the weak component. ***** Step 4.b:1.b:3.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: +#(p(x),y) -> c_5(+#(x,y)) +#(s(x),y) -> c_6(+#(x,y)) - Weak DPs: *#(p(x),y) -> *#(x,y) *#(p(x),y) -> +#(*(x,y),minus(y)) *#(s(x),y) -> *#(x,y) *#(s(x),y) -> +#(*(x,y),y) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_5) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {*,+,minus,*#,+#,minus#} TcT has computed the following interpretation: p(*) = 2*x1*x2 p(+) = x1 + 2*x2 p(0) = 0 p(minus) = x1 p(p) = 1 + x1 p(s) = 1 + x1 p(*#) = 3 + 2*x1*x2 p(+#) = 3 + x1 p(minus#) = 2 + x1 + 2*x1^2 p(c_1) = 1 p(c_2) = x2 p(c_3) = 1 p(c_4) = 0 p(c_5) = 1 + x1 p(c_6) = x1 p(c_7) = 0 p(c_8) = 1 + x1 p(c_9) = 1 Following rules are strictly oriented: +#(s(x),y) = 4 + x > 3 + x = c_6(+#(x,y)) Following rules are (at-least) weakly oriented: *#(p(x),y) = 3 + 2*x*y + 2*y >= 3 + 2*x*y = *#(x,y) *#(p(x),y) = 3 + 2*x*y + 2*y >= 3 + 2*x*y = +#(*(x,y),minus(y)) *#(s(x),y) = 3 + 2*x*y + 2*y >= 3 + 2*x*y = *#(x,y) *#(s(x),y) = 3 + 2*x*y + 2*y >= 3 + 2*x*y = +#(*(x,y),y) +#(p(x),y) = 4 + x >= 4 + x = c_5(+#(x,y)) *(0(),y) = 0 >= 0 = 0() *(p(x),y) = 2*x*y + 2*y >= 2*x*y + 2*y = +(*(x,y),minus(y)) *(s(x),y) = 2*x*y + 2*y >= 2*x*y + 2*y = +(*(x,y),y) +(0(),y) = 2*y >= y = y +(p(x),y) = 1 + x + 2*y >= 1 + x + 2*y = p(+(x,y)) +(s(x),y) = 1 + x + 2*y >= 1 + x + 2*y = s(+(x,y)) minus(0()) = 0 >= 0 = 0() minus(p(x)) = 1 + x >= 1 + x = s(minus(x)) minus(s(x)) = 1 + x >= 1 + x = p(minus(x)) ***** Step 4.b:1.b:3.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: +#(p(x),y) -> c_5(+#(x,y)) - Weak DPs: *#(p(x),y) -> *#(x,y) *#(p(x),y) -> +#(*(x,y),minus(y)) *#(s(x),y) -> *#(x,y) *#(s(x),y) -> +#(*(x,y),y) +#(s(x),y) -> c_6(+#(x,y)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 4.b:1.b:3.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: +#(p(x),y) -> c_5(+#(x,y)) - Weak DPs: *#(p(x),y) -> *#(x,y) *#(p(x),y) -> +#(*(x,y),minus(y)) *#(s(x),y) -> *#(x,y) *#(s(x),y) -> +#(*(x,y),y) +#(s(x),y) -> c_6(+#(x,y)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: +#(p(x),y) -> c_5(+#(x,y)) The strictly oriented rules are moved into the weak component. ****** Step 4.b:1.b:3.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: +#(p(x),y) -> c_5(+#(x,y)) - Weak DPs: *#(p(x),y) -> *#(x,y) *#(p(x),y) -> +#(*(x,y),minus(y)) *#(s(x),y) -> *#(x,y) *#(s(x),y) -> +#(*(x,y),y) +#(s(x),y) -> c_6(+#(x,y)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_5) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {*,+,minus,*#,+#,minus#} TcT has computed the following interpretation: p(*) = x1*x2 + 2*x1^2 p(+) = 1 + x1 + x2 p(0) = 0 p(minus) = x1 p(p) = 1 + x1 p(s) = 1 + x1 p(*#) = x1*x2 + 2*x1^2 + 3*x2 + 3*x2^2 p(+#) = 2 + x1 + x2 + 2*x2^2 p(minus#) = 2*x1 + x1^2 p(c_1) = 0 p(c_2) = x1 p(c_3) = 0 p(c_4) = 0 p(c_5) = x1 p(c_6) = x1 p(c_7) = 0 p(c_8) = 0 p(c_9) = 1 Following rules are strictly oriented: +#(p(x),y) = 3 + x + y + 2*y^2 > 2 + x + y + 2*y^2 = c_5(+#(x,y)) Following rules are (at-least) weakly oriented: *#(p(x),y) = 2 + 4*x + x*y + 2*x^2 + 4*y + 3*y^2 >= x*y + 2*x^2 + 3*y + 3*y^2 = *#(x,y) *#(p(x),y) = 2 + 4*x + x*y + 2*x^2 + 4*y + 3*y^2 >= 2 + x*y + 2*x^2 + y + 2*y^2 = +#(*(x,y),minus(y)) *#(s(x),y) = 2 + 4*x + x*y + 2*x^2 + 4*y + 3*y^2 >= x*y + 2*x^2 + 3*y + 3*y^2 = *#(x,y) *#(s(x),y) = 2 + 4*x + x*y + 2*x^2 + 4*y + 3*y^2 >= 2 + x*y + 2*x^2 + y + 2*y^2 = +#(*(x,y),y) +#(s(x),y) = 3 + x + y + 2*y^2 >= 2 + x + y + 2*y^2 = c_6(+#(x,y)) *(0(),y) = 0 >= 0 = 0() *(p(x),y) = 2 + 4*x + x*y + 2*x^2 + y >= 1 + x*y + 2*x^2 + y = +(*(x,y),minus(y)) *(s(x),y) = 2 + 4*x + x*y + 2*x^2 + y >= 1 + x*y + 2*x^2 + y = +(*(x,y),y) +(0(),y) = 1 + y >= y = y +(p(x),y) = 2 + x + y >= 2 + x + y = p(+(x,y)) +(s(x),y) = 2 + x + y >= 2 + x + y = s(+(x,y)) minus(0()) = 0 >= 0 = 0() minus(p(x)) = 1 + x >= 1 + x = s(minus(x)) minus(s(x)) = 1 + x >= 1 + x = p(minus(x)) ****** Step 4.b:1.b:3.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: *#(p(x),y) -> *#(x,y) *#(p(x),y) -> +#(*(x,y),minus(y)) *#(s(x),y) -> *#(x,y) *#(s(x),y) -> +#(*(x,y),y) +#(p(x),y) -> c_5(+#(x,y)) +#(s(x),y) -> c_6(+#(x,y)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 4.b:1.b:3.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: *#(p(x),y) -> *#(x,y) *#(p(x),y) -> +#(*(x,y),minus(y)) *#(s(x),y) -> *#(x,y) *#(s(x),y) -> +#(*(x,y),y) +#(p(x),y) -> c_5(+#(x,y)) +#(s(x),y) -> c_6(+#(x,y)) - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:*#(p(x),y) -> *#(x,y) -->_1 *#(s(x),y) -> +#(*(x,y),y):4 -->_1 *#(s(x),y) -> *#(x,y):3 -->_1 *#(p(x),y) -> +#(*(x,y),minus(y)):2 -->_1 *#(p(x),y) -> *#(x,y):1 2:W:*#(p(x),y) -> +#(*(x,y),minus(y)) -->_1 +#(s(x),y) -> c_6(+#(x,y)):6 -->_1 +#(p(x),y) -> c_5(+#(x,y)):5 3:W:*#(s(x),y) -> *#(x,y) -->_1 *#(s(x),y) -> +#(*(x,y),y):4 -->_1 *#(s(x),y) -> *#(x,y):3 -->_1 *#(p(x),y) -> +#(*(x,y),minus(y)):2 -->_1 *#(p(x),y) -> *#(x,y):1 4:W:*#(s(x),y) -> +#(*(x,y),y) -->_1 +#(s(x),y) -> c_6(+#(x,y)):6 -->_1 +#(p(x),y) -> c_5(+#(x,y)):5 5:W:+#(p(x),y) -> c_5(+#(x,y)) -->_1 +#(s(x),y) -> c_6(+#(x,y)):6 -->_1 +#(p(x),y) -> c_5(+#(x,y)):5 6:W:+#(s(x),y) -> c_6(+#(x,y)) -->_1 +#(s(x),y) -> c_6(+#(x,y)):6 -->_1 +#(p(x),y) -> c_5(+#(x,y)):5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: *#(p(x),y) -> *#(x,y) 3: *#(s(x),y) -> *#(x,y) 2: *#(p(x),y) -> +#(*(x,y),minus(y)) 4: *#(s(x),y) -> +#(*(x,y),y) 6: +#(s(x),y) -> c_6(+#(x,y)) 5: +#(p(x),y) -> c_5(+#(x,y)) ****** Step 4.b:1.b:3.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: *(0(),y) -> 0() *(p(x),y) -> +(*(x,y),minus(y)) *(s(x),y) -> +(*(x,y),y) +(0(),y) -> y +(p(x),y) -> p(+(x,y)) +(s(x),y) -> s(+(x,y)) minus(0()) -> 0() minus(p(x)) -> s(minus(x)) minus(s(x)) -> p(minus(x)) - Signature: {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))