*** 1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
f(a()) -> b()
f(c()) -> d()
f(g(x,y)) -> g(f(x),f(y))
f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
g(x,x) -> h(e(),x)
Weak DP Rules:
Weak TRS Rules:
Signature:
{f/1,g/2} / {a/0,b/0,c/0,d/0,e/0,h/2}
Obligation:
Innermost
basic terms: {f,g}/{a,b,c,d,e,h}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
We add the following dependency tuples:
Strict DPs
f#(a()) -> c_1()
f#(c()) -> c_2()
f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y))
f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y))
g#(x,x) -> c_5()
Weak DPs
and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(a()) -> c_1()
f#(c()) -> c_2()
f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y))
f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y))
g#(x,x) -> c_5()
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
f(a()) -> b()
f(c()) -> d()
f(g(x,y)) -> g(f(x),f(y))
f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
g(x,x) -> h(e(),x)
Signature:
{f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/3,c_4/3,c_5/0}
Obligation:
Innermost
basic terms: {f#,g#}/{a,b,c,d,e,h}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{1,2,5}
by application of
Pre({1,2,5}) = {3,4}.
Here rules are labelled as follows:
1: f#(a()) -> c_1()
2: f#(c()) -> c_2()
3: f#(g(x,y)) -> c_3(g#(f(x),f(y))
,f#(x)
,f#(y))
4: f#(h(x,y)) -> c_4(g#(h(y,f(x))
,h(x,f(y)))
,f#(x)
,f#(y))
5: g#(x,x) -> c_5()
*** 1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y))
f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y))
Strict TRS Rules:
Weak DP Rules:
f#(a()) -> c_1()
f#(c()) -> c_2()
g#(x,x) -> c_5()
Weak TRS Rules:
f(a()) -> b()
f(c()) -> d()
f(g(x,y)) -> g(f(x),f(y))
f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
g(x,x) -> h(e(),x)
Signature:
{f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/3,c_4/3,c_5/0}
Obligation:
Innermost
basic terms: {f#,g#}/{a,b,c,d,e,h}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y))
-->_3 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2
-->_2 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2
-->_1 g#(x,x) -> c_5():5
-->_3 f#(c()) -> c_2():4
-->_2 f#(c()) -> c_2():4
-->_3 f#(a()) -> c_1():3
-->_2 f#(a()) -> c_1():3
-->_3 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1
-->_2 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1
2:S:f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y))
-->_1 g#(x,x) -> c_5():5
-->_3 f#(c()) -> c_2():4
-->_2 f#(c()) -> c_2():4
-->_3 f#(a()) -> c_1():3
-->_2 f#(a()) -> c_1():3
-->_3 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2
-->_2 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2
-->_3 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1
-->_2 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1
3:W:f#(a()) -> c_1()
4:W:f#(c()) -> c_2()
5:W:g#(x,x) -> c_5()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: f#(a()) -> c_1()
4: f#(c()) -> c_2()
5: g#(x,x) -> c_5()
*** 1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y))
f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
f(a()) -> b()
f(c()) -> d()
f(g(x,y)) -> g(f(x),f(y))
f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
g(x,x) -> h(e(),x)
Signature:
{f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/3,c_4/3,c_5/0}
Obligation:
Innermost
basic terms: {f#,g#}/{a,b,c,d,e,h}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y))
-->_3 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2
-->_2 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2
-->_3 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1
-->_2 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1
2:S:f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y))
-->_3 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2
-->_2 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2
-->_3 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1
-->_2 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
f#(g(x,y)) -> c_3(f#(x),f#(y))
f#(h(x,y)) -> c_4(f#(x),f#(y))
*** 1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(g(x,y)) -> c_3(f#(x),f#(y))
f#(h(x,y)) -> c_4(f#(x),f#(y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
f(a()) -> b()
f(c()) -> d()
f(g(x,y)) -> g(f(x),f(y))
f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
g(x,x) -> h(e(),x)
Signature:
{f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {f#,g#}/{a,b,c,d,e,h}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
f#(g(x,y)) -> c_3(f#(x),f#(y))
f#(h(x,y)) -> c_4(f#(x),f#(y))
*** 1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(g(x,y)) -> c_3(f#(x),f#(y))
f#(h(x,y)) -> c_4(f#(x),f#(y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {f#,g#}/{a,b,c,d,e,h}
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#(g(x,y)) -> c_3(f#(x),f#(y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(g(x,y)) -> c_3(f#(x),f#(y))
f#(h(x,y)) -> c_4(f#(x),f#(y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {f#,g#}/{a,b,c,d,e,h}
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_3) = {1,2},
uargs(c_4) = {1,2}
Following symbols are considered usable:
{f#,g#}
TcT has computed the following interpretation:
p(a) = [0]
p(b) = [4]
p(c) = [0]
p(d) = [0]
p(e) = [2]
p(f) = [1] x1 + [1]
p(g) = [5] x1 + [1] x2 + [8]
p(h) = [1] x1 + [1] x2 + [0]
p(f#) = [2] x1 + [0]
p(g#) = [0]
p(c_1) = [1]
p(c_2) = [0]
p(c_3) = [2] x1 + [1] x2 + [13]
p(c_4) = [1] x1 + [1] x2 + [0]
p(c_5) = [2]
Following rules are strictly oriented:
f#(g(x,y)) = [10] x + [2] y + [16]
> [4] x + [2] y + [13]
= c_3(f#(x),f#(y))
Following rules are (at-least) weakly oriented:
f#(h(x,y)) = [2] x + [2] y + [0]
>= [2] x + [2] y + [0]
= c_4(f#(x),f#(y))
*** 1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
f#(h(x,y)) -> c_4(f#(x),f#(y))
Strict TRS Rules:
Weak DP Rules:
f#(g(x,y)) -> c_3(f#(x),f#(y))
Weak TRS Rules:
Signature:
{f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {f#,g#}/{a,b,c,d,e,h}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(h(x,y)) -> c_4(f#(x),f#(y))
Strict TRS Rules:
Weak DP Rules:
f#(g(x,y)) -> c_3(f#(x),f#(y))
Weak TRS Rules:
Signature:
{f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {f#,g#}/{a,b,c,d,e,h}
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#(h(x,y)) -> c_4(f#(x),f#(y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(h(x,y)) -> c_4(f#(x),f#(y))
Strict TRS Rules:
Weak DP Rules:
f#(g(x,y)) -> c_3(f#(x),f#(y))
Weak TRS Rules:
Signature:
{f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {f#,g#}/{a,b,c,d,e,h}
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_3) = {1,2},
uargs(c_4) = {1,2}
Following symbols are considered usable:
{f#,g#}
TcT has computed the following interpretation:
p(a) = [1]
p(b) = [2]
p(c) = [1]
p(d) = [0]
p(e) = [0]
p(f) = [0]
p(g) = [2] x1 + [2] x2 + [0]
p(h) = [1] x1 + [1] x2 + [1]
p(f#) = [8] x1 + [0]
p(g#) = [8] x1 + [2]
p(c_1) = [0]
p(c_2) = [1]
p(c_3) = [1] x1 + [2] x2 + [0]
p(c_4) = [1] x1 + [1] x2 + [0]
p(c_5) = [1]
Following rules are strictly oriented:
f#(h(x,y)) = [8] x + [8] y + [8]
> [8] x + [8] y + [0]
= c_4(f#(x),f#(y))
Following rules are (at-least) weakly oriented:
f#(g(x,y)) = [16] x + [16] y + [0]
>= [8] x + [16] y + [0]
= c_3(f#(x),f#(y))
*** 1.1.1.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
f#(g(x,y)) -> c_3(f#(x),f#(y))
f#(h(x,y)) -> c_4(f#(x),f#(y))
Weak TRS Rules:
Signature:
{f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {f#,g#}/{a,b,c,d,e,h}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
f#(g(x,y)) -> c_3(f#(x),f#(y))
f#(h(x,y)) -> c_4(f#(x),f#(y))
Weak TRS Rules:
Signature:
{f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {f#,g#}/{a,b,c,d,e,h}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:f#(g(x,y)) -> c_3(f#(x),f#(y))
-->_2 f#(h(x,y)) -> c_4(f#(x),f#(y)):2
-->_1 f#(h(x,y)) -> c_4(f#(x),f#(y)):2
-->_2 f#(g(x,y)) -> c_3(f#(x),f#(y)):1
-->_1 f#(g(x,y)) -> c_3(f#(x),f#(y)):1
2:W:f#(h(x,y)) -> c_4(f#(x),f#(y))
-->_2 f#(h(x,y)) -> c_4(f#(x),f#(y)):2
-->_1 f#(h(x,y)) -> c_4(f#(x),f#(y)):2
-->_2 f#(g(x,y)) -> c_3(f#(x),f#(y)):1
-->_1 f#(g(x,y)) -> c_3(f#(x),f#(y)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: f#(g(x,y)) -> c_3(f#(x),f#(y))
2: f#(h(x,y)) -> c_4(f#(x),f#(y))
*** 1.1.1.1.1.1.2.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {f#,g#}/{a,b,c,d,e,h}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).