*** 1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
f(x,0()) -> x
f(0(),y) -> y
f(1(),g(x,y)) -> x
f(2(),g(x,y)) -> y
f(f(x,y),z) -> f(x,f(y,z))
f(g(x,y),z) -> g(f(x,z),f(y,z))
f(i(x),y) -> i(x)
Weak DP Rules:
Weak TRS Rules:
Signature:
{f/2} / {0/0,1/0,2/0,g/2,i/1}
Obligation:
Innermost
basic terms: {f}/{0,1,2,g,i}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
We add the following dependency tuples:
Strict DPs
f#(x,0()) -> c_1()
f#(0(),y) -> c_2()
f#(1(),g(x,y)) -> c_3()
f#(2(),g(x,y)) -> c_4()
f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
f#(i(x),y) -> c_7()
Weak DPs
and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(x,0()) -> c_1()
f#(0(),y) -> c_2()
f#(1(),g(x,y)) -> c_3()
f#(2(),g(x,y)) -> c_4()
f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
f#(i(x),y) -> c_7()
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
f(x,0()) -> x
f(0(),y) -> y
f(1(),g(x,y)) -> x
f(2(),g(x,y)) -> y
f(f(x,y),z) -> f(x,f(y,z))
f(g(x,y),z) -> g(f(x,z),f(y,z))
f(i(x),y) -> i(x)
Signature:
{f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/2,c_7/0}
Obligation:
Innermost
basic terms: {f#}/{0,1,2,g,i}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{1,2,3,4,7}
by application of
Pre({1,2,3,4,7}) = {5,6}.
Here rules are labelled as follows:
1: f#(x,0()) -> c_1()
2: f#(0(),y) -> c_2()
3: f#(1(),g(x,y)) -> c_3()
4: f#(2(),g(x,y)) -> c_4()
5: f#(f(x,y),z) -> c_5(f#(x,f(y,z))
,f#(y,z))
6: f#(g(x,y),z) -> c_6(f#(x,z)
,f#(y,z))
7: f#(i(x),y) -> c_7()
*** 1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
Strict TRS Rules:
Weak DP Rules:
f#(x,0()) -> c_1()
f#(0(),y) -> c_2()
f#(1(),g(x,y)) -> c_3()
f#(2(),g(x,y)) -> c_4()
f#(i(x),y) -> c_7()
Weak TRS Rules:
f(x,0()) -> x
f(0(),y) -> y
f(1(),g(x,y)) -> x
f(2(),g(x,y)) -> y
f(f(x,y),z) -> f(x,f(y,z))
f(g(x,y),z) -> g(f(x,z),f(y,z))
f(i(x),y) -> i(x)
Signature:
{f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/2,c_7/0}
Obligation:
Innermost
basic terms: {f#}/{0,1,2,g,i}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
-->_2 f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z)):2
-->_1 f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z)):2
-->_2 f#(i(x),y) -> c_7():7
-->_1 f#(i(x),y) -> c_7():7
-->_2 f#(2(),g(x,y)) -> c_4():6
-->_1 f#(2(),g(x,y)) -> c_4():6
-->_2 f#(1(),g(x,y)) -> c_3():5
-->_1 f#(1(),g(x,y)) -> c_3():5
-->_2 f#(0(),y) -> c_2():4
-->_1 f#(0(),y) -> c_2():4
-->_2 f#(x,0()) -> c_1():3
-->_1 f#(x,0()) -> c_1():3
-->_2 f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z)):1
-->_1 f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z)):1
2:S:f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
-->_2 f#(i(x),y) -> c_7():7
-->_1 f#(i(x),y) -> c_7():7
-->_2 f#(2(),g(x,y)) -> c_4():6
-->_1 f#(2(),g(x,y)) -> c_4():6
-->_2 f#(1(),g(x,y)) -> c_3():5
-->_1 f#(1(),g(x,y)) -> c_3():5
-->_2 f#(0(),y) -> c_2():4
-->_1 f#(0(),y) -> c_2():4
-->_2 f#(x,0()) -> c_1():3
-->_1 f#(x,0()) -> c_1():3
-->_2 f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z)):2
-->_1 f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z)):2
-->_2 f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z)):1
-->_1 f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z)):1
3:W:f#(x,0()) -> c_1()
4:W:f#(0(),y) -> c_2()
5:W:f#(1(),g(x,y)) -> c_3()
6:W:f#(2(),g(x,y)) -> c_4()
7:W:f#(i(x),y) -> c_7()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: f#(x,0()) -> c_1()
4: f#(0(),y) -> c_2()
5: f#(1(),g(x,y)) -> c_3()
6: f#(2(),g(x,y)) -> c_4()
7: f#(i(x),y) -> c_7()
*** 1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
f(x,0()) -> x
f(0(),y) -> y
f(1(),g(x,y)) -> x
f(2(),g(x,y)) -> y
f(f(x,y),z) -> f(x,f(y,z))
f(g(x,y),z) -> g(f(x,z),f(y,z))
f(i(x),y) -> i(x)
Signature:
{f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/2,c_7/0}
Obligation:
Innermost
basic terms: {f#}/{0,1,2,g,i}
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:
2: f#(g(x,y),z) -> c_6(f#(x,z)
,f#(y,z))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
f(x,0()) -> x
f(0(),y) -> y
f(1(),g(x,y)) -> x
f(2(),g(x,y)) -> y
f(f(x,y),z) -> f(x,f(y,z))
f(g(x,y),z) -> g(f(x,z),f(y,z))
f(i(x),y) -> i(x)
Signature:
{f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/2,c_7/0}
Obligation:
Innermost
basic terms: {f#}/{0,1,2,g,i}
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_5) = {1,2},
uargs(c_6) = {1,2}
Following symbols are considered usable:
{f#}
TcT has computed the following interpretation:
p(0) = [0]
p(1) = [0]
p(2) = [2]
p(f) = [4] x1 + [2] x2 + [2]
p(g) = [1] x1 + [1] x2 + [2]
p(i) = [0]
p(f#) = [2] x1 + [0]
p(c_1) = [4]
p(c_2) = [1]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [4] x1 + [2] x2 + [4]
p(c_6) = [1] x1 + [1] x2 + [0]
p(c_7) = [1]
Following rules are strictly oriented:
f#(g(x,y),z) = [2] x + [2] y + [4]
> [2] x + [2] y + [0]
= c_6(f#(x,z),f#(y,z))
Following rules are (at-least) weakly oriented:
f#(f(x,y),z) = [8] x + [4] y + [4]
>= [8] x + [4] y + [4]
= c_5(f#(x,f(y,z)),f#(y,z))
*** 1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
Strict TRS Rules:
Weak DP Rules:
f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
Weak TRS Rules:
f(x,0()) -> x
f(0(),y) -> y
f(1(),g(x,y)) -> x
f(2(),g(x,y)) -> y
f(f(x,y),z) -> f(x,f(y,z))
f(g(x,y),z) -> g(f(x,z),f(y,z))
f(i(x),y) -> i(x)
Signature:
{f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/2,c_7/0}
Obligation:
Innermost
basic terms: {f#}/{0,1,2,g,i}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
Strict TRS Rules:
Weak DP Rules:
f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
Weak TRS Rules:
f(x,0()) -> x
f(0(),y) -> y
f(1(),g(x,y)) -> x
f(2(),g(x,y)) -> y
f(f(x,y),z) -> f(x,f(y,z))
f(g(x,y),z) -> g(f(x,z),f(y,z))
f(i(x),y) -> i(x)
Signature:
{f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/2,c_7/0}
Obligation:
Innermost
basic terms: {f#}/{0,1,2,g,i}
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#(f(x,y),z) -> c_5(f#(x,f(y,z))
,f#(y,z))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
Strict TRS Rules:
Weak DP Rules:
f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
Weak TRS Rules:
f(x,0()) -> x
f(0(),y) -> y
f(1(),g(x,y)) -> x
f(2(),g(x,y)) -> y
f(f(x,y),z) -> f(x,f(y,z))
f(g(x,y),z) -> g(f(x,z),f(y,z))
f(i(x),y) -> i(x)
Signature:
{f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/2,c_7/0}
Obligation:
Innermost
basic terms: {f#}/{0,1,2,g,i}
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_5) = {1,2},
uargs(c_6) = {1,2}
Following symbols are considered usable:
{f#}
TcT has computed the following interpretation:
p(0) = [2]
p(1) = [2]
p(2) = [2]
p(f) = [4] x1 + [2] x2 + [4]
p(g) = [1] x1 + [1] x2 + [4]
p(i) = [3]
p(f#) = [4] x1 + [0]
p(c_1) = [1]
p(c_2) = [2]
p(c_3) = [1]
p(c_4) = [4]
p(c_5) = [4] x1 + [2] x2 + [12]
p(c_6) = [1] x1 + [1] x2 + [0]
p(c_7) = [0]
Following rules are strictly oriented:
f#(f(x,y),z) = [16] x + [8] y + [16]
> [16] x + [8] y + [12]
= c_5(f#(x,f(y,z)),f#(y,z))
Following rules are (at-least) weakly oriented:
f#(g(x,y),z) = [4] x + [4] y + [16]
>= [4] x + [4] y + [0]
= c_6(f#(x,z),f#(y,z))
*** 1.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
Weak TRS Rules:
f(x,0()) -> x
f(0(),y) -> y
f(1(),g(x,y)) -> x
f(2(),g(x,y)) -> y
f(f(x,y),z) -> f(x,f(y,z))
f(g(x,y),z) -> g(f(x,z),f(y,z))
f(i(x),y) -> i(x)
Signature:
{f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/2,c_7/0}
Obligation:
Innermost
basic terms: {f#}/{0,1,2,g,i}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
Weak TRS Rules:
f(x,0()) -> x
f(0(),y) -> y
f(1(),g(x,y)) -> x
f(2(),g(x,y)) -> y
f(f(x,y),z) -> f(x,f(y,z))
f(g(x,y),z) -> g(f(x,z),f(y,z))
f(i(x),y) -> i(x)
Signature:
{f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/2,c_7/0}
Obligation:
Innermost
basic terms: {f#}/{0,1,2,g,i}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
-->_2 f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z)):2
-->_1 f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z)):2
-->_2 f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z)):1
-->_1 f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z)):1
2:W:f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
-->_2 f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z)):2
-->_1 f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z)):2
-->_2 f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z)):1
-->_1 f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: f#(f(x,y),z) -> c_5(f#(x,f(y,z))
,f#(y,z))
2: f#(g(x,y),z) -> c_6(f#(x,z)
,f#(y,z))
*** 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:
f(x,0()) -> x
f(0(),y) -> y
f(1(),g(x,y)) -> x
f(2(),g(x,y)) -> y
f(f(x,y),z) -> f(x,f(y,z))
f(g(x,y),z) -> g(f(x,z),f(y,z))
f(i(x),y) -> i(x)
Signature:
{f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/2,c_7/0}
Obligation:
Innermost
basic terms: {f#}/{0,1,2,g,i}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).