/export/starexec/sandbox/solver/bin/starexec_run_tct_dci /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?,O(n^2)) * Step 1: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(f(x)) -> f(active(x)) active(f(x)) -> mark(x) check(x) -> start(match(f(X()),x)) check(f(x)) -> f(check(x)) f(found(x)) -> found(f(x)) f(mark(x)) -> mark(f(x)) f(ok(x)) -> ok(f(x)) match(X(),x) -> proper(x) match(f(x),f(y)) -> f(match(x,y)) proper(c()) -> ok(c()) proper(f(x)) -> f(proper(x)) start(ok(x)) -> found(x) top(active(c())) -> top(mark(c())) top(found(x)) -> top(active(x)) top(mark(x)) -> top(check(x)) - Signature: {active/1,check/1,f/1,match/2,proper/1,start/1,top/1} / {X/0,c/0,found/1,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {X,active,c,check,f,found,mark,match,ok,proper,start,top} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(X) = [0] p(active) = [1] x1 + [0] p(c) = [0] p(check) = [1] x1 + [0] p(f) = [1] x1 + [0] p(found) = [1] x1 + [0] p(mark) = [1] x1 + [5] p(match) = [1] x1 + [1] x2 + [0] p(ok) = [1] x1 + [0] p(proper) = [1] x1 + [0] p(start) = [1] x1 + [0] p(top) = [1] x1 + [0] Following rules are strictly oriented: top(mark(x)) = [1] x + [5] > [1] x + [0] = top(check(x)) Following rules are (at-least) weakly oriented: active(f(x)) = [1] x + [0] >= [1] x + [0] = f(active(x)) active(f(x)) = [1] x + [0] >= [1] x + [5] = mark(x) check(x) = [1] x + [0] >= [1] x + [0] = start(match(f(X()),x)) check(f(x)) = [1] x + [0] >= [1] x + [0] = f(check(x)) f(found(x)) = [1] x + [0] >= [1] x + [0] = found(f(x)) f(mark(x)) = [1] x + [5] >= [1] x + [5] = mark(f(x)) f(ok(x)) = [1] x + [0] >= [1] x + [0] = ok(f(x)) match(X(),x) = [1] x + [0] >= [1] x + [0] = proper(x) match(f(x),f(y)) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = f(match(x,y)) proper(c()) = [0] >= [0] = ok(c()) proper(f(x)) = [1] x + [0] >= [1] x + [0] = f(proper(x)) start(ok(x)) = [1] x + [0] >= [1] x + [0] = found(x) top(active(c())) = [0] >= [5] = top(mark(c())) top(found(x)) = [1] x + [0] >= [1] x + [0] = top(active(x)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(f(x)) -> f(active(x)) active(f(x)) -> mark(x) check(x) -> start(match(f(X()),x)) check(f(x)) -> f(check(x)) f(found(x)) -> found(f(x)) f(mark(x)) -> mark(f(x)) f(ok(x)) -> ok(f(x)) match(X(),x) -> proper(x) match(f(x),f(y)) -> f(match(x,y)) proper(c()) -> ok(c()) proper(f(x)) -> f(proper(x)) start(ok(x)) -> found(x) top(active(c())) -> top(mark(c())) top(found(x)) -> top(active(x)) - Weak TRS: top(mark(x)) -> top(check(x)) - Signature: {active/1,check/1,f/1,match/2,proper/1,start/1,top/1} / {X/0,c/0,found/1,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {X,active,c,check,f,found,mark,match,ok,proper,start,top} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(X) = [0] p(active) = [1] x1 + [0] p(c) = [0] p(check) = [1] x1 + [0] p(f) = [1] x1 + [0] p(found) = [1] x1 + [11] p(mark) = [1] x1 + [0] p(match) = [1] x1 + [1] x2 + [0] p(ok) = [1] x1 + [0] p(proper) = [1] x1 + [0] p(start) = [1] x1 + [0] p(top) = [1] x1 + [0] Following rules are strictly oriented: top(found(x)) = [1] x + [11] > [1] x + [0] = top(active(x)) Following rules are (at-least) weakly oriented: active(f(x)) = [1] x + [0] >= [1] x + [0] = f(active(x)) active(f(x)) = [1] x + [0] >= [1] x + [0] = mark(x) check(x) = [1] x + [0] >= [1] x + [0] = start(match(f(X()),x)) check(f(x)) = [1] x + [0] >= [1] x + [0] = f(check(x)) f(found(x)) = [1] x + [11] >= [1] x + [11] = found(f(x)) f(mark(x)) = [1] x + [0] >= [1] x + [0] = mark(f(x)) f(ok(x)) = [1] x + [0] >= [1] x + [0] = ok(f(x)) match(X(),x) = [1] x + [0] >= [1] x + [0] = proper(x) match(f(x),f(y)) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = f(match(x,y)) proper(c()) = [0] >= [0] = ok(c()) proper(f(x)) = [1] x + [0] >= [1] x + [0] = f(proper(x)) start(ok(x)) = [1] x + [0] >= [1] x + [11] = found(x) top(active(c())) = [0] >= [0] = top(mark(c())) top(mark(x)) = [1] x + [0] >= [1] x + [0] = top(check(x)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(f(x)) -> f(active(x)) active(f(x)) -> mark(x) check(x) -> start(match(f(X()),x)) check(f(x)) -> f(check(x)) f(found(x)) -> found(f(x)) f(mark(x)) -> mark(f(x)) f(ok(x)) -> ok(f(x)) match(X(),x) -> proper(x) match(f(x),f(y)) -> f(match(x,y)) proper(c()) -> ok(c()) proper(f(x)) -> f(proper(x)) start(ok(x)) -> found(x) top(active(c())) -> top(mark(c())) - Weak TRS: top(found(x)) -> top(active(x)) top(mark(x)) -> top(check(x)) - Signature: {active/1,check/1,f/1,match/2,proper/1,start/1,top/1} / {X/0,c/0,found/1,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {X,active,c,check,f,found,mark,match,ok,proper,start,top} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(X) = [0] p(active) = [1] x1 + [11] p(c) = [0] p(check) = [1] x1 + [0] p(f) = [1] x1 + [0] p(found) = [1] x1 + [11] p(mark) = [1] x1 + [0] p(match) = [1] x1 + [1] x2 + [0] p(ok) = [1] x1 + [0] p(proper) = [1] x1 + [0] p(start) = [1] x1 + [0] p(top) = [1] x1 + [0] Following rules are strictly oriented: active(f(x)) = [1] x + [11] > [1] x + [0] = mark(x) top(active(c())) = [11] > [0] = top(mark(c())) Following rules are (at-least) weakly oriented: active(f(x)) = [1] x + [11] >= [1] x + [11] = f(active(x)) check(x) = [1] x + [0] >= [1] x + [0] = start(match(f(X()),x)) check(f(x)) = [1] x + [0] >= [1] x + [0] = f(check(x)) f(found(x)) = [1] x + [11] >= [1] x + [11] = found(f(x)) f(mark(x)) = [1] x + [0] >= [1] x + [0] = mark(f(x)) f(ok(x)) = [1] x + [0] >= [1] x + [0] = ok(f(x)) match(X(),x) = [1] x + [0] >= [1] x + [0] = proper(x) match(f(x),f(y)) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = f(match(x,y)) proper(c()) = [0] >= [0] = ok(c()) proper(f(x)) = [1] x + [0] >= [1] x + [0] = f(proper(x)) start(ok(x)) = [1] x + [0] >= [1] x + [11] = found(x) top(found(x)) = [1] x + [11] >= [1] x + [11] = top(active(x)) top(mark(x)) = [1] x + [0] >= [1] x + [0] = top(check(x)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(f(x)) -> f(active(x)) check(x) -> start(match(f(X()),x)) check(f(x)) -> f(check(x)) f(found(x)) -> found(f(x)) f(mark(x)) -> mark(f(x)) f(ok(x)) -> ok(f(x)) match(X(),x) -> proper(x) match(f(x),f(y)) -> f(match(x,y)) proper(c()) -> ok(c()) proper(f(x)) -> f(proper(x)) start(ok(x)) -> found(x) - Weak TRS: active(f(x)) -> mark(x) top(active(c())) -> top(mark(c())) top(found(x)) -> top(active(x)) top(mark(x)) -> top(check(x)) - Signature: {active/1,check/1,f/1,match/2,proper/1,start/1,top/1} / {X/0,c/0,found/1,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {X,active,c,check,f,found,mark,match,ok,proper,start,top} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(X) = [0] p(active) = [1] x1 + [4] p(c) = [8] p(check) = [1] x1 + [4] p(f) = [1] x1 + [0] p(found) = [1] x1 + [4] p(mark) = [1] x1 + [4] p(match) = [1] x1 + [1] x2 + [14] p(ok) = [1] x1 + [4] p(proper) = [1] x1 + [2] p(start) = [1] x1 + [3] p(top) = [1] x1 + [0] Following rules are strictly oriented: match(X(),x) = [1] x + [14] > [1] x + [2] = proper(x) start(ok(x)) = [1] x + [7] > [1] x + [4] = found(x) Following rules are (at-least) weakly oriented: active(f(x)) = [1] x + [4] >= [1] x + [4] = f(active(x)) active(f(x)) = [1] x + [4] >= [1] x + [4] = mark(x) check(x) = [1] x + [4] >= [1] x + [17] = start(match(f(X()),x)) check(f(x)) = [1] x + [4] >= [1] x + [4] = f(check(x)) f(found(x)) = [1] x + [4] >= [1] x + [4] = found(f(x)) f(mark(x)) = [1] x + [4] >= [1] x + [4] = mark(f(x)) f(ok(x)) = [1] x + [4] >= [1] x + [4] = ok(f(x)) match(f(x),f(y)) = [1] x + [1] y + [14] >= [1] x + [1] y + [14] = f(match(x,y)) proper(c()) = [10] >= [12] = ok(c()) proper(f(x)) = [1] x + [2] >= [1] x + [2] = f(proper(x)) top(active(c())) = [12] >= [12] = top(mark(c())) top(found(x)) = [1] x + [4] >= [1] x + [4] = top(active(x)) top(mark(x)) = [1] x + [4] >= [1] x + [4] = top(check(x)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(f(x)) -> f(active(x)) check(x) -> start(match(f(X()),x)) check(f(x)) -> f(check(x)) f(found(x)) -> found(f(x)) f(mark(x)) -> mark(f(x)) f(ok(x)) -> ok(f(x)) match(f(x),f(y)) -> f(match(x,y)) proper(c()) -> ok(c()) proper(f(x)) -> f(proper(x)) - Weak TRS: active(f(x)) -> mark(x) match(X(),x) -> proper(x) start(ok(x)) -> found(x) top(active(c())) -> top(mark(c())) top(found(x)) -> top(active(x)) top(mark(x)) -> top(check(x)) - Signature: {active/1,check/1,f/1,match/2,proper/1,start/1,top/1} / {X/0,c/0,found/1,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {X,active,c,check,f,found,mark,match,ok,proper,start,top} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(X) = [3] p(active) = [1] x1 + [5] p(c) = [0] p(check) = [1] x1 + [5] p(f) = [1] x1 + [0] p(found) = [1] x1 + [5] p(mark) = [1] x1 + [5] p(match) = [1] x1 + [1] x2 + [2] p(ok) = [1] x1 + [3] p(proper) = [1] x1 + [5] p(start) = [1] x1 + [2] p(top) = [1] x1 + [0] Following rules are strictly oriented: proper(c()) = [5] > [3] = ok(c()) Following rules are (at-least) weakly oriented: active(f(x)) = [1] x + [5] >= [1] x + [5] = f(active(x)) active(f(x)) = [1] x + [5] >= [1] x + [5] = mark(x) check(x) = [1] x + [5] >= [1] x + [7] = start(match(f(X()),x)) check(f(x)) = [1] x + [5] >= [1] x + [5] = f(check(x)) f(found(x)) = [1] x + [5] >= [1] x + [5] = found(f(x)) f(mark(x)) = [1] x + [5] >= [1] x + [5] = mark(f(x)) f(ok(x)) = [1] x + [3] >= [1] x + [3] = ok(f(x)) match(X(),x) = [1] x + [5] >= [1] x + [5] = proper(x) match(f(x),f(y)) = [1] x + [1] y + [2] >= [1] x + [1] y + [2] = f(match(x,y)) proper(f(x)) = [1] x + [5] >= [1] x + [5] = f(proper(x)) start(ok(x)) = [1] x + [5] >= [1] x + [5] = found(x) top(active(c())) = [5] >= [5] = top(mark(c())) top(found(x)) = [1] x + [5] >= [1] x + [5] = top(active(x)) top(mark(x)) = [1] x + [5] >= [1] x + [5] = top(check(x)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(f(x)) -> f(active(x)) check(x) -> start(match(f(X()),x)) check(f(x)) -> f(check(x)) f(found(x)) -> found(f(x)) f(mark(x)) -> mark(f(x)) f(ok(x)) -> ok(f(x)) match(f(x),f(y)) -> f(match(x,y)) proper(f(x)) -> f(proper(x)) - Weak TRS: active(f(x)) -> mark(x) match(X(),x) -> proper(x) proper(c()) -> ok(c()) start(ok(x)) -> found(x) top(active(c())) -> top(mark(c())) top(found(x)) -> top(active(x)) top(mark(x)) -> top(check(x)) - Signature: {active/1,check/1,f/1,match/2,proper/1,start/1,top/1} / {X/0,c/0,found/1,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {X,active,c,check,f,found,mark,match,ok,proper,start,top} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(X) = [7] p(active) = [1] x1 + [0] p(c) = [0] p(check) = [1] x1 + [0] p(f) = [1] x1 + [4] p(found) = [1] x1 + [0] p(mark) = [1] x1 + [0] p(match) = [1] x1 + [1] x2 + [1] p(ok) = [1] x1 + [0] p(proper) = [1] x1 + [0] p(start) = [1] x1 + [2] p(top) = [1] x1 + [0] Following rules are strictly oriented: match(f(x),f(y)) = [1] x + [1] y + [9] > [1] x + [1] y + [5] = f(match(x,y)) Following rules are (at-least) weakly oriented: active(f(x)) = [1] x + [4] >= [1] x + [4] = f(active(x)) active(f(x)) = [1] x + [4] >= [1] x + [0] = mark(x) check(x) = [1] x + [0] >= [1] x + [14] = start(match(f(X()),x)) check(f(x)) = [1] x + [4] >= [1] x + [4] = f(check(x)) f(found(x)) = [1] x + [4] >= [1] x + [4] = found(f(x)) f(mark(x)) = [1] x + [4] >= [1] x + [4] = mark(f(x)) f(ok(x)) = [1] x + [4] >= [1] x + [4] = ok(f(x)) match(X(),x) = [1] x + [8] >= [1] x + [0] = proper(x) proper(c()) = [0] >= [0] = ok(c()) proper(f(x)) = [1] x + [4] >= [1] x + [4] = f(proper(x)) start(ok(x)) = [1] x + [2] >= [1] x + [0] = found(x) top(active(c())) = [0] >= [0] = top(mark(c())) top(found(x)) = [1] x + [0] >= [1] x + [0] = top(active(x)) top(mark(x)) = [1] x + [0] >= [1] x + [0] = top(check(x)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(f(x)) -> f(active(x)) check(x) -> start(match(f(X()),x)) check(f(x)) -> f(check(x)) f(found(x)) -> found(f(x)) f(mark(x)) -> mark(f(x)) f(ok(x)) -> ok(f(x)) proper(f(x)) -> f(proper(x)) - Weak TRS: active(f(x)) -> mark(x) match(X(),x) -> proper(x) match(f(x),f(y)) -> f(match(x,y)) proper(c()) -> ok(c()) start(ok(x)) -> found(x) top(active(c())) -> top(mark(c())) top(found(x)) -> top(active(x)) top(mark(x)) -> top(check(x)) - Signature: {active/1,check/1,f/1,match/2,proper/1,start/1,top/1} / {X/0,c/0,found/1,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {X,active,c,check,f,found,mark,match,ok,proper,start,top} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 2, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): Following symbols are considered usable: all TcT has computed the following interpretation: p(X) = [0] [0] [0] [0] p(active) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(c) = [1] [0] [0] [0] p(check) = [1 0 0 1] [0] [0 0 1 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [0] p(f) = [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [1] p(found) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(mark) = [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [1] p(match) = [1 0 0 0] [1 0 0 0] [0] [0 0 0 0] x1 + [0 0 0 0] x2 + [1] [0 0 0 0] [0 0 0 1] [1] [0 0 0 0] [0 0 0 1] [0] p(ok) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(proper) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 1] [0] [0 0 0 1] [0] p(start) = [1 0 0 0] [0] [0 0 0 1] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(top) = [1 0 0 0] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [1] [0 0 0 0] [1] Following rules are strictly oriented: check(f(x)) = [1 0 0 2] [1] [0 0 0 1] x + [2] [0 0 0 1] [2] [0 0 0 1] [1] > [1 0 0 2] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] = f(check(x)) Following rules are (at-least) weakly oriented: active(f(x)) = [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] >= [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] = f(active(x)) active(f(x)) = [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] >= [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] = mark(x) check(x) = [1 0 0 1] [0] [0 0 1 1] x + [1] [0 0 0 1] [1] [0 0 0 1] [0] >= [1 0 0 0] [0] [0 0 0 1] x + [0] [0 0 0 0] [0] [0 0 0 1] [0] = start(match(f(X()),x)) f(found(x)) = [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] >= [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] = found(f(x)) f(mark(x)) = [1 0 0 2] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [2] >= [1 0 0 2] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [2] = mark(f(x)) f(ok(x)) = [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] >= [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] = ok(f(x)) match(X(),x) = [1 0 0 0] [0] [0 0 0 0] x + [1] [0 0 0 1] [1] [0 0 0 1] [0] >= [1 0 0 0] [0] [0 0 0 0] x + [0] [0 0 0 1] [0] [0 0 0 1] [0] = proper(x) match(f(x),f(y)) = [1 0 0 1] [1 0 0 1] [0] [0 0 0 0] x + [0 0 0 0] y + [1] [0 0 0 0] [0 0 0 1] [2] [0 0 0 0] [0 0 0 1] [1] >= [1 0 0 0] [1 0 0 1] [0] [0 0 0 0] x + [0 0 0 0] y + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 1] [1] = f(match(x,y)) proper(c()) = [1] [0] [0] [0] >= [1] [0] [0] [0] = ok(c()) proper(f(x)) = [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 1] [1] [0 0 0 1] [1] >= [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] = f(proper(x)) start(ok(x)) = [1 0 0 0] [0] [0 0 0 1] x + [0] [0 0 0 0] [0] [0 0 0 1] [0] >= [1 0 0 0] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [0] = found(x) top(active(c())) = [1] [1] [1] [1] >= [1] [1] [1] [1] = top(mark(c())) top(found(x)) = [1 0 0 0] [0] [0 0 0 0] x + [1] [0 0 0 0] [1] [0 0 0 0] [1] >= [1 0 0 0] [0] [0 0 0 0] x + [1] [0 0 0 0] [1] [0 0 0 0] [1] = top(active(x)) top(mark(x)) = [1 0 0 1] [0] [0 0 0 0] x + [1] [0 0 0 0] [1] [0 0 0 0] [1] >= [1 0 0 1] [0] [0 0 0 0] x + [1] [0 0 0 0] [1] [0 0 0 0] [1] = top(check(x)) * Step 8: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(f(x)) -> f(active(x)) check(x) -> start(match(f(X()),x)) f(found(x)) -> found(f(x)) f(mark(x)) -> mark(f(x)) f(ok(x)) -> ok(f(x)) proper(f(x)) -> f(proper(x)) - Weak TRS: active(f(x)) -> mark(x) check(f(x)) -> f(check(x)) match(X(),x) -> proper(x) match(f(x),f(y)) -> f(match(x,y)) proper(c()) -> ok(c()) start(ok(x)) -> found(x) top(active(c())) -> top(mark(c())) top(found(x)) -> top(active(x)) top(mark(x)) -> top(check(x)) - Signature: {active/1,check/1,f/1,match/2,proper/1,start/1,top/1} / {X/0,c/0,found/1,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {X,active,c,check,f,found,mark,match,ok,proper,start,top} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 2, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): Following symbols are considered usable: all TcT has computed the following interpretation: p(X) = [0] [0] [0] [0] p(active) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(c) = [0] [0] [0] [0] p(check) = [1 1 0 0] [0] [0 1 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(f) = [1 1 0 0] [0] [0 1 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(found) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(mark) = [1 1 0 0] [0] [0 1 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(match) = [1 0 0 0] [1 1 0 0] [0] [0 0 0 0] x1 + [0 1 0 0] x2 + [1] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] p(ok) = [1 0 0 0] [0] [0 1 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(proper) = [1 1 0 0] [0] [0 1 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(start) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(top) = [1 0 0 0] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [1] [0 0 0 0] [0] Following rules are strictly oriented: f(ok(x)) = [1 1 0 0] [1] [0 1 0 0] x + [2] [0 0 0 0] [0] [0 0 0 0] [0] > [1 1 0 0] [0] [0 1 0 0] x + [2] [0 0 0 0] [0] [0 0 0 0] [0] = ok(f(x)) Following rules are (at-least) weakly oriented: active(f(x)) = [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = f(active(x)) active(f(x)) = [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = mark(x) check(x) = [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = start(match(f(X()),x)) check(f(x)) = [1 2 0 0] [1] [0 1 0 0] x + [2] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 2 0 0] [1] [0 1 0 0] x + [2] [0 0 0 0] [0] [0 0 0 0] [0] = f(check(x)) f(found(x)) = [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = found(f(x)) f(mark(x)) = [1 2 0 0] [1] [0 1 0 0] x + [2] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 2 0 0] [1] [0 1 0 0] x + [2] [0 0 0 0] [0] [0 0 0 0] [0] = mark(f(x)) match(X(),x) = [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = proper(x) match(f(x),f(y)) = [1 1 0 0] [1 2 0 0] [1] [0 0 0 0] x + [0 1 0 0] y + [2] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 0] [1 2 0 0] [1] [0 0 0 0] x + [0 1 0 0] y + [2] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] = f(match(x,y)) proper(c()) = [0] [1] [0] [0] >= [0] [1] [0] [0] = ok(c()) proper(f(x)) = [1 2 0 0] [1] [0 1 0 0] x + [2] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 2 0 0] [1] [0 1 0 0] x + [2] [0 0 0 0] [0] [0 0 0 0] [0] = f(proper(x)) start(ok(x)) = [1 0 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [0] [0 1 0 0] x + [0] [0 0 0 0] [0] [0 0 0 0] [0] = found(x) top(active(c())) = [0] [1] [1] [0] >= [0] [1] [1] [0] = top(mark(c())) top(found(x)) = [1 0 0 0] [0] [0 0 0 0] x + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 0 0] [0] [0 0 0 0] x + [1] [0 0 0 0] [1] [0 0 0 0] [0] = top(active(x)) top(mark(x)) = [1 1 0 0] [0] [0 0 0 0] x + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 1 0 0] [0] [0 0 0 0] x + [1] [0 0 0 0] [1] [0 0 0 0] [0] = top(check(x)) * Step 9: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(f(x)) -> f(active(x)) check(x) -> start(match(f(X()),x)) f(found(x)) -> found(f(x)) f(mark(x)) -> mark(f(x)) proper(f(x)) -> f(proper(x)) - Weak TRS: active(f(x)) -> mark(x) check(f(x)) -> f(check(x)) f(ok(x)) -> ok(f(x)) match(X(),x) -> proper(x) match(f(x),f(y)) -> f(match(x,y)) proper(c()) -> ok(c()) start(ok(x)) -> found(x) top(active(c())) -> top(mark(c())) top(found(x)) -> top(active(x)) top(mark(x)) -> top(check(x)) - Signature: {active/1,check/1,f/1,match/2,proper/1,start/1,top/1} / {X/0,c/0,found/1,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {X,active,c,check,f,found,mark,match,ok,proper,start,top} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 2, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): Following symbols are considered usable: all TcT has computed the following interpretation: p(X) = [1] [0] [0] [1] p(active) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 1] [0] [0 0 0 0] [0] p(c) = [0] [0] [0] [1] p(check) = [1 0 0 0] [1] [0 1 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(f) = [1 1 0 0] [0] [0 1 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(found) = [1 0 0 1] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(mark) = [1 0 0 0] [0] [0 1 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(match) = [1 0 0 0] [1 0 0 0] [0] [0 0 0 0] x1 + [0 1 0 0] x2 + [0] [0 0 0 1] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [1] p(ok) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 1] [0] [0 0 0 0] [0] p(proper) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(start) = [1 0 1 0] [0] [0 1 1 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(top) = [1 1 1 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] Following rules are strictly oriented: f(mark(x)) = [1 1 0 0] [1] [0 1 0 0] x + [2] [0 0 0 0] [0] [0 0 0 0] [0] > [1 1 0 0] [0] [0 1 0 0] x + [2] [0 0 0 0] [0] [0 0 0 0] [0] = mark(f(x)) Following rules are (at-least) weakly oriented: active(f(x)) = [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = f(active(x)) active(f(x)) = [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = mark(x) check(x) = [1 0 0 0] [1] [0 1 0 0] x + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [1] [0 1 0 0] x + [0] [0 0 0 0] [0] [0 0 0 0] [0] = start(match(f(X()),x)) check(f(x)) = [1 1 0 0] [1] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = f(check(x)) f(found(x)) = [1 1 0 1] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = found(f(x)) f(ok(x)) = [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = ok(f(x)) match(X(),x) = [1 0 0 0] [1] [0 1 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [1] >= [1 0 0 0] [0] [0 1 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [0] = proper(x) match(f(x),f(y)) = [1 1 0 0] [1 1 0 0] [0] [0 0 0 0] x + [0 1 0 0] y + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [1] >= [1 0 0 0] [1 1 0 0] [0] [0 0 0 0] x + [0 1 0 0] y + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] = f(match(x,y)) proper(c()) = [0] [0] [1] [0] >= [0] [0] [1] [0] = ok(c()) proper(f(x)) = [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = f(proper(x)) start(ok(x)) = [1 0 0 1] [0] [0 1 0 1] x + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 1] [0] [0 1 0 0] x + [0] [0 0 0 0] [0] [0 0 0 0] [0] = found(x) top(active(c())) = [1] [0] [0] [0] >= [1] [0] [0] [0] = top(mark(c())) top(found(x)) = [1 1 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 0] [0] = top(active(x)) top(mark(x)) = [1 1 0 0] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 0] [0] = top(check(x)) * Step 10: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(f(x)) -> f(active(x)) check(x) -> start(match(f(X()),x)) f(found(x)) -> found(f(x)) proper(f(x)) -> f(proper(x)) - Weak TRS: active(f(x)) -> mark(x) check(f(x)) -> f(check(x)) f(mark(x)) -> mark(f(x)) f(ok(x)) -> ok(f(x)) match(X(),x) -> proper(x) match(f(x),f(y)) -> f(match(x,y)) proper(c()) -> ok(c()) start(ok(x)) -> found(x) top(active(c())) -> top(mark(c())) top(found(x)) -> top(active(x)) top(mark(x)) -> top(check(x)) - Signature: {active/1,check/1,f/1,match/2,proper/1,start/1,top/1} / {X/0,c/0,found/1,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {X,active,c,check,f,found,mark,match,ok,proper,start,top} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 2, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): Following symbols are considered usable: all TcT has computed the following interpretation: p(X) = [0] [0] [0] [0] p(active) = [1 0 0 0] [1] [0 1 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(c) = [0] [0] [0] [0] p(check) = [1 1 0 0] [1] [0 1 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(f) = [1 1 0 0] [0] [0 1 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(found) = [1 0 0 1] [1] [0 1 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(mark) = [1 1 0 0] [1] [0 1 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(match) = [1 0 0 0] [1 0 0 0] [0] [0 0 0 0] x1 + [0 1 0 0] x2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] p(ok) = [1 0 0 1] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(proper) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(start) = [1 0 0 0] [1] [0 1 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(top) = [1 0 0 0] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] Following rules are strictly oriented: f(found(x)) = [1 1 0 1] [2] [0 1 0 0] x + [2] [0 0 0 0] [0] [0 0 0 0] [0] > [1 1 0 0] [1] [0 1 0 0] x + [2] [0 0 0 0] [0] [0 0 0 0] [0] = found(f(x)) Following rules are (at-least) weakly oriented: active(f(x)) = [1 1 0 0] [1] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = f(active(x)) active(f(x)) = [1 1 0 0] [1] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = mark(x) check(x) = [1 1 0 0] [1] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [1] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = start(match(f(X()),x)) check(f(x)) = [1 2 0 0] [2] [0 1 0 0] x + [2] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 2 0 0] [2] [0 1 0 0] x + [2] [0 0 0 0] [0] [0 0 0 0] [0] = f(check(x)) f(mark(x)) = [1 2 0 0] [2] [0 1 0 0] x + [2] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 2 0 0] [2] [0 1 0 0] x + [2] [0 0 0 0] [0] [0 0 0 0] [0] = mark(f(x)) f(ok(x)) = [1 1 0 1] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = ok(f(x)) match(X(),x) = [1 0 0 0] [0] [0 1 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 0 0] [0] [0 1 0 0] x + [0] [0 0 0 0] [0] [0 0 0 0] [0] = proper(x) match(f(x),f(y)) = [1 1 0 0] [1 1 0 0] [0] [0 0 0 0] x + [0 1 0 0] y + [1] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 0] [1 1 0 0] [0] [0 0 0 0] x + [0 1 0 0] y + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] = f(match(x,y)) proper(c()) = [0] [0] [0] [0] >= [0] [0] [0] [0] = ok(c()) proper(f(x)) = [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [0] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = f(proper(x)) start(ok(x)) = [1 0 0 1] [1] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 1] [1] [0 1 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = found(x) top(active(c())) = [1] [1] [0] [0] >= [1] [1] [0] [0] = top(mark(c())) top(found(x)) = [1 0 0 1] [1] [0 0 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [1] [0 0 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = top(active(x)) top(mark(x)) = [1 1 0 0] [1] [0 0 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = top(check(x)) * Step 11: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(f(x)) -> f(active(x)) check(x) -> start(match(f(X()),x)) proper(f(x)) -> f(proper(x)) - Weak TRS: active(f(x)) -> mark(x) check(f(x)) -> f(check(x)) f(found(x)) -> found(f(x)) f(mark(x)) -> mark(f(x)) f(ok(x)) -> ok(f(x)) match(X(),x) -> proper(x) match(f(x),f(y)) -> f(match(x,y)) proper(c()) -> ok(c()) start(ok(x)) -> found(x) top(active(c())) -> top(mark(c())) top(found(x)) -> top(active(x)) top(mark(x)) -> top(check(x)) - Signature: {active/1,check/1,f/1,match/2,proper/1,start/1,top/1} / {X/0,c/0,found/1,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {X,active,c,check,f,found,mark,match,ok,proper,start,top} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 2, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): Following symbols are considered usable: all TcT has computed the following interpretation: p(X) = [0] [0] [0] [0] p(active) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(c) = [0] [0] [0] [0] p(check) = [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(f) = [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [1] p(found) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(mark) = [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [1] p(match) = [1 0 0 0] [1 0 0 1] [0] [0 0 0 0] x1 + [0 0 0 1] x2 + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 1] [0] p(ok) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(proper) = [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(start) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(top) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [1] Following rules are strictly oriented: proper(f(x)) = [1 0 0 2] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] > [1 0 0 2] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] = f(proper(x)) Following rules are (at-least) weakly oriented: active(f(x)) = [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] >= [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] = f(active(x)) active(f(x)) = [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] >= [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] = mark(x) check(x) = [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [0] >= [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [0] = start(match(f(X()),x)) check(f(x)) = [1 0 0 2] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] >= [1 0 0 2] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] = f(check(x)) f(found(x)) = [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] >= [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] = found(f(x)) f(mark(x)) = [1 0 0 2] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [2] >= [1 0 0 2] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [2] = mark(f(x)) f(ok(x)) = [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] >= [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] = ok(f(x)) match(X(),x) = [1 0 0 1] [0] [0 0 0 1] x + [1] [0 0 0 0] [0] [0 0 0 1] [0] >= [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [0] = proper(x) match(f(x),f(y)) = [1 0 0 1] [1 0 0 2] [1] [0 0 0 0] x + [0 0 0 1] y + [2] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 1] [1] >= [1 0 0 0] [1 0 0 2] [0] [0 0 0 0] x + [0 0 0 0] y + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 1] [1] = f(match(x,y)) proper(c()) = [0] [0] [0] [0] >= [0] [0] [0] [0] = ok(c()) start(ok(x)) = [1 0 0 0] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [0] >= [1 0 0 0] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [0] = found(x) top(active(c())) = [0] [0] [1] [1] >= [0] [0] [1] [1] = top(mark(c())) top(found(x)) = [1 0 0 0] [0] [0 0 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [1] >= [1 0 0 0] [0] [0 0 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [1] = top(active(x)) top(mark(x)) = [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [1] >= [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [1] = top(check(x)) * Step 12: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(f(x)) -> f(active(x)) check(x) -> start(match(f(X()),x)) - Weak TRS: active(f(x)) -> mark(x) check(f(x)) -> f(check(x)) f(found(x)) -> found(f(x)) f(mark(x)) -> mark(f(x)) f(ok(x)) -> ok(f(x)) match(X(),x) -> proper(x) match(f(x),f(y)) -> f(match(x,y)) proper(c()) -> ok(c()) proper(f(x)) -> f(proper(x)) start(ok(x)) -> found(x) top(active(c())) -> top(mark(c())) top(found(x)) -> top(active(x)) top(mark(x)) -> top(check(x)) - Signature: {active/1,check/1,f/1,match/2,proper/1,start/1,top/1} / {X/0,c/0,found/1,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {X,active,c,check,f,found,mark,match,ok,proper,start,top} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 2, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): Following symbols are considered usable: all TcT has computed the following interpretation: p(X) = [0] [0] [0] [0] p(active) = [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(c) = [0] [0] [0] [0] p(check) = [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [1] p(f) = [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [1] p(found) = [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [1] p(mark) = [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [1] p(match) = [1 0 0 0] [1 0 0 1] [0] [0 0 0 0] x1 + [0 0 0 0] x2 + [0] [0 0 0 1] [0 0 0 0] [0] [0 0 0 0] [0 0 0 1] [1] p(ok) = [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [1] p(proper) = [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [1] p(start) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(top) = [1 0 0 0] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] Following rules are strictly oriented: active(f(x)) = [1 0 0 2] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] > [1 0 0 2] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] = f(active(x)) Following rules are (at-least) weakly oriented: active(f(x)) = [1 0 0 2] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] >= [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] = mark(x) check(x) = [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] >= [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] = start(match(f(X()),x)) check(f(x)) = [1 0 0 2] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [2] >= [1 0 0 2] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [2] = f(check(x)) f(found(x)) = [1 0 0 2] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [2] >= [1 0 0 2] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [2] = found(f(x)) f(mark(x)) = [1 0 0 2] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [2] >= [1 0 0 2] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [2] = mark(f(x)) f(ok(x)) = [1 0 0 2] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [2] >= [1 0 0 2] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [2] = ok(f(x)) match(X(),x) = [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] >= [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] = proper(x) match(f(x),f(y)) = [1 0 0 1] [1 0 0 2] [1] [0 0 0 0] x + [0 0 0 0] y + [0] [0 0 0 1] [0 0 0 0] [1] [0 0 0 0] [0 0 0 1] [2] >= [1 0 0 0] [1 0 0 2] [1] [0 0 0 0] x + [0 0 0 0] y + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 1] [2] = f(match(x,y)) proper(c()) = [0] [0] [0] [1] >= [0] [0] [0] [1] = ok(c()) proper(f(x)) = [1 0 0 2] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [2] >= [1 0 0 2] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [2] = f(proper(x)) start(ok(x)) = [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] >= [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 1] [1] = found(x) top(active(c())) = [0] [1] [0] [0] >= [0] [1] [0] [0] = top(mark(c())) top(found(x)) = [1 0 0 1] [0] [0 0 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 1] [0] [0 0 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = top(active(x)) top(mark(x)) = [1 0 0 1] [0] [0 0 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 1] [0] [0 0 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = top(check(x)) * Step 13: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: check(x) -> start(match(f(X()),x)) - Weak TRS: active(f(x)) -> f(active(x)) active(f(x)) -> mark(x) check(f(x)) -> f(check(x)) f(found(x)) -> found(f(x)) f(mark(x)) -> mark(f(x)) f(ok(x)) -> ok(f(x)) match(X(),x) -> proper(x) match(f(x),f(y)) -> f(match(x,y)) proper(c()) -> ok(c()) proper(f(x)) -> f(proper(x)) start(ok(x)) -> found(x) top(active(c())) -> top(mark(c())) top(found(x)) -> top(active(x)) top(mark(x)) -> top(check(x)) - Signature: {active/1,check/1,f/1,match/2,proper/1,start/1,top/1} / {X/0,c/0,found/1,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {X,active,c,check,f,found,mark,match,ok,proper,start,top} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 2, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): Following symbols are considered usable: all TcT has computed the following interpretation: p(X) = [0] [0] [0] [1] p(active) = [1 0 0 0] [0] [0 0 0 1] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(c) = [0] [0] [0] [1] p(check) = [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(f) = [1 0 1 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(found) = [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(mark) = [1 0 1 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(match) = [1 0 0 0] [1 0 0 0] [0] [0 0 0 1] x1 + [0 0 0 0] x2 + [0] [0 0 0 0] [0 0 1 0] [0] [0 0 0 0] [0 0 0 0] [0] p(ok) = [1 0 0 0] [0] [0 0 0 1] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(proper) = [1 0 0 0] [0] [0 0 0 0] x1 + [1] [0 0 1 0] [0] [0 0 0 0] [0] p(start) = [1 1 0 0] [0] [0 0 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(top) = [1 1 1 0] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [1] [0 0 0 0] [0] Following rules are strictly oriented: check(x) = [1 0 0 0] [1] [0 0 0 0] x + [0] [0 0 1 0] [0] [0 0 0 0] [0] > [1 0 0 0] [0] [0 0 0 0] x + [0] [0 0 1 0] [0] [0 0 0 0] [0] = start(match(f(X()),x)) Following rules are (at-least) weakly oriented: active(f(x)) = [1 0 1 0] [0] [0 0 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 1 0] [0] [0 0 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [0] = f(active(x)) active(f(x)) = [1 0 1 0] [0] [0 0 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 1 0] [0] [0 0 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [0] = mark(x) check(f(x)) = [1 0 1 0] [1] [0 0 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 1 0] [1] [0 0 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [0] = f(check(x)) f(found(x)) = [1 0 1 1] [0] [0 0 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 1 0] [0] [0 0 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [0] = found(f(x)) f(mark(x)) = [1 0 1 0] [1] [0 0 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 1 0] [1] [0 0 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [0] = mark(f(x)) f(ok(x)) = [1 0 1 0] [0] [0 0 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 1 0] [0] [0 0 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [0] = ok(f(x)) match(X(),x) = [1 0 0 0] [0] [0 0 0 0] x + [1] [0 0 1 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [0] [0 0 0 0] x + [1] [0 0 1 0] [0] [0 0 0 0] [0] = proper(x) match(f(x),f(y)) = [1 0 1 0] [1 0 1 0] [0] [0 0 0 0] x + [0 0 0 0] y + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 0] [1 0 1 0] [0] [0 0 0 0] x + [0 0 0 0] y + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] = f(match(x,y)) proper(c()) = [0] [1] [0] [0] >= [0] [1] [0] [0] = ok(c()) proper(f(x)) = [1 0 1 0] [0] [0 0 0 0] x + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 1 0] [0] [0 0 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [0] = f(proper(x)) start(ok(x)) = [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 1 0] [0] [0 0 0 0] [0] >= [1 0 0 1] [0] [0 0 0 0] x + [0] [0 0 1 0] [0] [0 0 0 0] [0] = found(x) top(active(c())) = [1] [1] [1] [0] >= [1] [1] [1] [0] = top(mark(c())) top(found(x)) = [1 0 1 1] [0] [0 0 0 0] x + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 1 1] [0] [0 0 0 0] x + [1] [0 0 0 0] [1] [0 0 0 0] [0] = top(active(x)) top(mark(x)) = [1 0 1 0] [1] [0 0 0 0] x + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 1 0] [1] [0 0 0 0] x + [1] [0 0 0 0] [1] [0 0 0 0] [0] = top(check(x)) * Step 14: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: active(f(x)) -> f(active(x)) active(f(x)) -> mark(x) check(x) -> start(match(f(X()),x)) check(f(x)) -> f(check(x)) f(found(x)) -> found(f(x)) f(mark(x)) -> mark(f(x)) f(ok(x)) -> ok(f(x)) match(X(),x) -> proper(x) match(f(x),f(y)) -> f(match(x,y)) proper(c()) -> ok(c()) proper(f(x)) -> f(proper(x)) start(ok(x)) -> found(x) top(active(c())) -> top(mark(c())) top(found(x)) -> top(active(x)) top(mark(x)) -> top(check(x)) - Signature: {active/1,check/1,f/1,match/2,proper/1,start/1,top/1} / {X/0,c/0,found/1,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {X,active,c,check,f,found,mark,match,ok,proper,start,top} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))