*** 1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
f(cons(x,k),l) -> g(k,l,cons(x,k))
f(empty(),l) -> l
g(a,b,c) -> f(a,cons(b,c))
Weak DP Rules:
Weak TRS Rules:
Signature:
{f/2,g/3} / {cons/2,empty/0}
Obligation:
Full
basic terms: {f,g}/{cons,empty}
Applied Processor:
DependencyPairs {dpKind_ = WIDP}
Proof:
We add the following weak dependency pairs:
Strict DPs
f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
f#(empty(),l) -> c_2(l)
g#(a,b,c) -> c_3(f#(a,cons(b,c)))
Weak DPs
and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
f#(empty(),l) -> c_2(l)
g#(a,b,c) -> c_3(f#(a,cons(b,c)))
Strict TRS Rules:
f(cons(x,k),l) -> g(k,l,cons(x,k))
f(empty(),l) -> l
g(a,b,c) -> f(a,cons(b,c))
Weak DP Rules:
Weak TRS Rules:
Signature:
{f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/1,c_3/1}
Obligation:
Full
basic terms: {f#,g#}/{cons,empty}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
f#(empty(),l) -> c_2(l)
g#(a,b,c) -> c_3(f#(a,cons(b,c)))
*** 1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
f#(empty(),l) -> c_2(l)
g#(a,b,c) -> c_3(f#(a,cons(b,c)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/1,c_3/1}
Obligation:
Full
basic terms: {f#,g#}/{cons,empty}
Applied Processor:
Succeeding
Proof:
()
*** 1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
f#(empty(),l) -> c_2(l)
g#(a,b,c) -> c_3(f#(a,cons(b,c)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/1,c_3/1}
Obligation:
Full
basic terms: {f#,g#}/{cons,empty}
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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: f#(cons(x,k),l) -> c_1(g#(k
,l
,cons(x,k)))
2: f#(empty(),l) -> c_2(l)
Consider the set of all dependency pairs
1: f#(cons(x,k),l) -> c_1(g#(k
,l
,cons(x,k)))
2: f#(empty(),l) -> c_2(l)
3: g#(a,b,c) -> c_3(f#(a
,cons(b,c)))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{1,2}
These cover all (indirect) predecessors of dependency pairs
{1,2,3}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
*** 1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
f#(empty(),l) -> c_2(l)
g#(a,b,c) -> c_3(f#(a,cons(b,c)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/1,c_3/1}
Obligation:
Full
basic terms: {f#,g#}/{cons,empty}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_1) = {1},
uargs(c_3) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(cons) = [1] x2 + [12]
p(empty) = [0]
p(f) = [1] x2 + [0]
p(g) = [1] x1 + [1] x2 + [1] x3 + [1]
p(f#) = [1] x1 + [6]
p(g#) = [1] x1 + [6]
p(c_1) = [1] x1 + [1]
p(c_2) = [5]
p(c_3) = [1] x1 + [0]
Following rules are strictly oriented:
f#(cons(x,k),l) = [1] k + [18]
> [1] k + [7]
= c_1(g#(k,l,cons(x,k)))
f#(empty(),l) = [6]
> [5]
= c_2(l)
Following rules are (at-least) weakly oriented:
g#(a,b,c) = [1] a + [6]
>= [1] a + [6]
= c_3(f#(a,cons(b,c)))
*** 1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
g#(a,b,c) -> c_3(f#(a,cons(b,c)))
Strict TRS Rules:
Weak DP Rules:
f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
f#(empty(),l) -> c_2(l)
Weak TRS Rules:
Signature:
{f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/1,c_3/1}
Obligation:
Full
basic terms: {f#,g#}/{cons,empty}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
f#(empty(),l) -> c_2(l)
g#(a,b,c) -> c_3(f#(a,cons(b,c)))
Weak TRS Rules:
Signature:
{f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/1,c_3/1}
Obligation:
Full
basic terms: {f#,g#}/{cons,empty}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
-->_1 g#(a,b,c) -> c_3(f#(a,cons(b,c))):3
2:W:f#(empty(),l) -> c_2(l)
-->_1 g#(a,b,c) -> c_3(f#(a,cons(b,c))):3
-->_1 f#(empty(),l) -> c_2(l):2
-->_1 f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k))):1
3:W:g#(a,b,c) -> c_3(f#(a,cons(b,c)))
-->_1 f#(empty(),l) -> c_2(l):2
-->_1 f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: f#(cons(x,k),l) -> c_1(g#(k
,l
,cons(x,k)))
3: g#(a,b,c) -> c_3(f#(a
,cons(b,c)))
2: f#(empty(),l) -> c_2(l)
*** 1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/1,c_3/1}
Obligation:
Full
basic terms: {f#,g#}/{cons,empty}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).