*** 1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
f(g(f(x))) -> f(h(s(0()),x))
f(g(h(x,y))) -> f(h(s(x),y))
f(h(x,h(y,z))) -> f(h(+(x,y),z))
Weak DP Rules:
Weak TRS Rules:
Signature:
{+/2,f/1} / {0/0,g/1,h/2,s/1}
Obligation:
Innermost
basic terms: {+,f}/{0,g,h,s}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
We add the following dependency tuples:
Strict DPs
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,0()) -> c_2()
+#(x,s(y)) -> c_3(+#(x,y))
+#(0(),y) -> c_4()
+#(s(x),y) -> c_5(+#(x,y))
f#(g(f(x))) -> c_6(f#(h(s(0()),x)))
f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Weak DPs
and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,0()) -> c_2()
+#(x,s(y)) -> c_3(+#(x,y))
+#(0(),y) -> c_4()
+#(s(x),y) -> c_5(+#(x,y))
f#(g(f(x))) -> c_6(f#(h(s(0()),x)))
f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
f(g(f(x))) -> f(h(s(0()),x))
f(g(h(x,y))) -> f(h(s(x),y))
f(h(x,h(y,z))) -> f(h(+(x,y),z))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,0()) -> c_2()
+#(x,s(y)) -> c_3(+#(x,y))
+#(0(),y) -> c_4()
+#(s(x),y) -> c_5(+#(x,y))
f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
*** 1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,0()) -> c_2()
+#(x,s(y)) -> c_3(+#(x,y))
+#(0(),y) -> c_4()
+#(s(x),y) -> c_5(+#(x,y))
f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{2,4}
by application of
Pre({2,4}) = {1,3,5,7}.
Here rules are labelled as follows:
1: +#(x,+(y,z)) -> c_1(+#(+(x,y),z)
,+#(x,y))
2: +#(x,0()) -> c_2()
3: +#(x,s(y)) -> c_3(+#(x,y))
4: +#(0(),y) -> c_4()
5: +#(s(x),y) -> c_5(+#(x,y))
6: f#(g(h(x,y))) -> c_7(f#(h(s(x)
,y)))
7: f#(h(x,h(y,z))) -> c_8(f#(h(+(x
,y)
,z))
,+#(x,y))
*** 1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Strict TRS Rules:
Weak DP Rules:
+#(x,0()) -> c_2()
+#(0(),y) -> c_4()
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
-->_2 +#(s(x),y) -> c_5(+#(x,y)):3
-->_1 +#(s(x),y) -> c_5(+#(x,y)):3
-->_2 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_2 +#(0(),y) -> c_4():7
-->_1 +#(0(),y) -> c_4():7
-->_2 +#(x,0()) -> c_2():6
-->_1 +#(x,0()) -> c_2():6
-->_2 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
2:S:+#(x,s(y)) -> c_3(+#(x,y))
-->_1 +#(s(x),y) -> c_5(+#(x,y)):3
-->_1 +#(0(),y) -> c_4():7
-->_1 +#(x,0()) -> c_2():6
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
3:S:+#(s(x),y) -> c_5(+#(x,y))
-->_1 +#(0(),y) -> c_4():7
-->_1 +#(x,0()) -> c_2():6
-->_1 +#(s(x),y) -> c_5(+#(x,y)):3
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
4:S:f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
-->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):5
5:S:f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
-->_2 +#(0(),y) -> c_4():7
-->_2 +#(x,0()) -> c_2():6
-->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):5
-->_2 +#(s(x),y) -> c_5(+#(x,y)):3
-->_2 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_2 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
6:W:+#(x,0()) -> c_2()
7:W:+#(0(),y) -> c_4()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: +#(x,0()) -> c_2()
7: +#(0(),y) -> c_4()
*** 1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
RemoveHeads
Proof:
Consider the dependency graph
1:S:+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
-->_2 +#(s(x),y) -> c_5(+#(x,y)):3
-->_1 +#(s(x),y) -> c_5(+#(x,y)):3
-->_2 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_2 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
2:S:+#(x,s(y)) -> c_3(+#(x,y))
-->_1 +#(s(x),y) -> c_5(+#(x,y)):3
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
3:S:+#(s(x),y) -> c_5(+#(x,y))
-->_1 +#(s(x),y) -> c_5(+#(x,y)):3
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
4:S:f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
-->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):5
5:S:f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
-->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):5
-->_2 +#(s(x),y) -> c_5(+#(x,y)):3
-->_2 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_2 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
[(4,f#(g(h(x,y))) -> c_7(f#(h(s(x),y))))]
*** 1.1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
Strict DP Rules:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
Strict TRS Rules:
Weak DP Rules:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Problem (S)
Strict DP Rules:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Strict TRS Rules:
Weak DP Rules:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
*** 1.1.1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
Strict TRS Rules:
Weak DP Rules:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
2: +#(x,s(y)) -> c_3(+#(x,y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
Strict TRS Rules:
Weak DP Rules:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
NaturalMI {miDimension = 3, miDegree = 2, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_1) = {1,2},
uargs(c_3) = {1},
uargs(c_5) = {1},
uargs(c_8) = {1,2}
Following symbols are considered usable:
{+,+#,f#}
TcT has computed the following interpretation:
p(+) = [1 0 0] [1 0 0] [0]
[0 1 0] x1 + [1 1 1] x2 + [1]
[0 0 1] [0 0 1] [0]
p(0) = [0]
[0]
[0]
p(f) = [0]
[0]
[0]
p(g) = [0 0 0] [0]
[0 0 0] x1 + [0]
[0 0 1] [0]
p(h) = [0 0 0] [0 0 1] [1]
[0 0 1] x1 + [0 1 1] x2 + [1]
[0 0 1] [0 0 0] [0]
p(s) = [0 0 0] [0]
[0 0 0] x1 + [0]
[0 0 1] [1]
p(+#) = [0 0 1] [0]
[0 0 1] x2 + [0]
[0 0 0] [0]
p(f#) = [0 1 0] [1]
[0 1 1] x1 + [1]
[0 1 1] [0]
p(c_1) = [1 0 0] [1 0 0] [0]
[0 0 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 0] [0]
p(c_2) = [0]
[0]
[0]
p(c_3) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 0] [0]
p(c_4) = [0]
[0]
[0]
p(c_5) = [1 0 0] [0]
[1 0 0] x1 + [0]
[0 0 0] [0]
p(c_6) = [0]
[0]
[0]
p(c_7) = [0]
[0]
[0]
p(c_8) = [1 0 0] [1 0 0] [1]
[0 0 0] x1 + [1 0 0] x2 + [0]
[0 1 0] [0 0 0] [0]
Following rules are strictly oriented:
+#(x,s(y)) = [0 0 1] [1]
[0 0 1] y + [1]
[0 0 0] [0]
> [0 0 1] [0]
[0 0 0] y + [0]
[0 0 0] [0]
= c_3(+#(x,y))
Following rules are (at-least) weakly oriented:
+#(x,+(y,z)) = [0 0 1] [0 0 1] [0]
[0 0 1] y + [0 0 1] z + [0]
[0 0 0] [0 0 0] [0]
>= [0 0 1] [0 0 1] [0]
[0 0 1] y + [0 0 0] z + [0]
[0 0 0] [0 0 0] [0]
= c_1(+#(+(x,y),z),+#(x,y))
+#(s(x),y) = [0 0 1] [0]
[0 0 1] y + [0]
[0 0 0] [0]
>= [0 0 1] [0]
[0 0 1] y + [0]
[0 0 0] [0]
= c_5(+#(x,y))
f#(h(x,h(y,z))) = [0 0 1] [0 0 2] [0 1
1] [3]
[0 0 2] x + [0 0 2] y + [0 1
1] z + [3]
[0 0 2] [0 0 2] [0 1
1] [2]
>= [0 0 1] [0 0 2] [0 1
1] [3]
[0 0 0] x + [0 0 1] y + [0 0
0] z + [0]
[0 0 2] [0 0 2] [0 1
1] [2]
= c_8(f#(h(+(x,y),z)),+#(x,y))
+(x,+(y,z)) = [1 0 0] [1 0 0] [1 0
0] [0]
[0 1 0] x + [1 1 1] y + [2 1
2] z + [2]
[0 0 1] [0 0 1] [0 0
1] [0]
>= [1 0 0] [1 0 0] [1 0
0] [0]
[0 1 0] x + [1 1 1] y + [1 1
1] z + [2]
[0 0 1] [0 0 1] [0 0
1] [0]
= +(+(x,y),z)
+(x,0()) = [1 0 0] [0]
[0 1 0] x + [1]
[0 0 1] [0]
>= [1 0 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= x
+(x,s(y)) = [1 0 0] [0 0 0] [0]
[0 1 0] x + [0 0 1] y + [2]
[0 0 1] [0 0 1] [1]
>= [0 0 0] [0 0 0] [0]
[0 0 0] x + [0 0 0] y + [0]
[0 0 1] [0 0 1] [1]
= s(+(x,y))
+(0(),y) = [1 0 0] [0]
[1 1 1] y + [1]
[0 0 1] [0]
>= [1 0 0] [0]
[0 1 0] y + [0]
[0 0 1] [0]
= y
+(s(x),y) = [0 0 0] [1 0 0] [0]
[0 0 0] x + [1 1 1] y + [1]
[0 0 1] [0 0 1] [1]
>= [0 0 0] [0 0 0] [0]
[0 0 0] x + [0 0 0] y + [0]
[0 0 1] [0 0 1] [1]
= s(+(x,y))
*** 1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
Strict TRS Rules:
Weak DP Rules:
+#(x,s(y)) -> c_3(+#(x,y))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.2 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
Strict TRS Rules:
Weak DP Rules:
+#(x,s(y)) -> c_3(+#(x,y))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: +#(x,+(y,z)) -> c_1(+#(+(x,y),z)
,+#(x,y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
Strict TRS Rules:
Weak DP Rules:
+#(x,s(y)) -> c_3(+#(x,y))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
NaturalMI {miDimension = 3, miDegree = 2, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_1) = {1,2},
uargs(c_3) = {1},
uargs(c_5) = {1},
uargs(c_8) = {1,2}
Following symbols are considered usable:
{+,+#,f#}
TcT has computed the following interpretation:
p(+) = [1 0 0] [1 0 1] [0]
[0 1 0] x1 + [1 1 1] x2 + [0]
[0 0 1] [0 0 1] [1]
p(0) = [1]
[1]
[0]
p(f) = [0]
[0]
[0]
p(g) = [0]
[0]
[0]
p(h) = [0 0 0] [0 1 1] [1]
[0 0 0] x1 + [0 0 1] x2 + [1]
[0 0 1] [0 0 1] [0]
p(s) = [0 0 0] [0]
[0 0 0] x1 + [1]
[0 0 1] [1]
p(+#) = [0 0 1] [0]
[0 0 1] x2 + [0]
[0 0 1] [1]
p(f#) = [0 1 0] [1]
[0 1 0] x1 + [1]
[1 0 0] [0]
p(c_1) = [1 0 0] [1 0 0] [0]
[0 0 0] x1 + [0 0 1] x2 + [0]
[0 0 1] [0 0 0] [1]
p(c_2) = [0]
[0]
[0]
p(c_3) = [1 0 0] [0]
[0 0 1] x1 + [0]
[0 0 0] [0]
p(c_4) = [0]
[0]
[0]
p(c_5) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 0] [1]
p(c_6) = [0]
[0]
[0]
p(c_7) = [0]
[0]
[0]
p(c_8) = [1 0 0] [1 0 0] [0]
[1 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
Following rules are strictly oriented:
+#(x,+(y,z)) = [0 0 1] [0 0 1] [1]
[0 0 1] y + [0 0 1] z + [1]
[0 0 1] [0 0 1] [2]
> [0 0 1] [0 0 1] [0]
[0 0 1] y + [0 0 0] z + [1]
[0 0 0] [0 0 1] [2]
= c_1(+#(+(x,y),z),+#(x,y))
Following rules are (at-least) weakly oriented:
+#(x,s(y)) = [0 0 1] [1]
[0 0 1] y + [1]
[0 0 1] [2]
>= [0 0 1] [0]
[0 0 1] y + [1]
[0 0 0] [0]
= c_3(+#(x,y))
+#(s(x),y) = [0 0 1] [0]
[0 0 1] y + [0]
[0 0 1] [1]
>= [0 0 1] [0]
[0 0 0] y + [0]
[0 0 0] [1]
= c_5(+#(x,y))
f#(h(x,h(y,z))) = [0 0 1] [0 0 1] [2]
[0 0 1] y + [0 0 1] z + [2]
[0 0 1] [0 0 2] [2]
>= [0 0 1] [0 0 1] [2]
[0 0 0] y + [0 0 1] z + [2]
[0 0 0] [0 0 0] [0]
= c_8(f#(h(+(x,y),z)),+#(x,y))
+(x,+(y,z)) = [1 0 0] [1 0 1] [1 0
2] [1]
[0 1 0] x + [1 1 1] y + [2 1
3] z + [1]
[0 0 1] [0 0 1] [0 0
1] [2]
>= [1 0 0] [1 0 1] [1 0
1] [0]
[0 1 0] x + [1 1 1] y + [1 1
1] z + [0]
[0 0 1] [0 0 1] [0 0
1] [2]
= +(+(x,y),z)
+(x,0()) = [1 0 0] [1]
[0 1 0] x + [2]
[0 0 1] [1]
>= [1 0 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= x
+(x,s(y)) = [1 0 0] [0 0 1] [1]
[0 1 0] x + [0 0 1] y + [2]
[0 0 1] [0 0 1] [2]
>= [0 0 0] [0 0 0] [0]
[0 0 0] x + [0 0 0] y + [1]
[0 0 1] [0 0 1] [2]
= s(+(x,y))
+(0(),y) = [1 0 1] [1]
[1 1 1] y + [1]
[0 0 1] [1]
>= [1 0 0] [0]
[0 1 0] y + [0]
[0 0 1] [0]
= y
+(s(x),y) = [0 0 0] [1 0 1] [0]
[0 0 0] x + [1 1 1] y + [1]
[0 0 1] [0 0 1] [2]
>= [0 0 0] [0 0 0] [0]
[0 0 0] x + [0 0 0] y + [1]
[0 0 1] [0 0 1] [2]
= s(+(x,y))
*** 1.1.1.1.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
+#(s(x),y) -> c_5(+#(x,y))
Strict TRS Rules:
Weak DP Rules:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.2.2 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
+#(s(x),y) -> c_5(+#(x,y))
Strict TRS Rules:
Weak DP Rules:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
Proof:
We decompose the input problem according to the dependency graph into the upper component
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
and a lower component
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
Further, following extension rules are added to the lower component.
f#(h(x,h(y,z))) -> +#(x,y)
f#(h(x,h(y,z))) -> f#(h(+(x,y),z))
*** 1.1.1.1.1.1.1.2.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,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, 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,h(y,z))) -> c_8(f#(h(+(x
,y)
,z))
,+#(x,y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1.2.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
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_8) = {1}
Following symbols are considered usable:
{+#,f#}
TcT has computed the following interpretation:
p(+) = [1] x2 + [8]
p(0) = [0]
p(f) = [1] x1 + [4]
p(g) = [1] x1 + [0]
p(h) = [1] x2 + [2]
p(s) = [1] x1 + [2]
p(+#) = [1] x2 + [0]
p(f#) = [4] x1 + [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [1] x1 + [1]
p(c_4) = [1]
p(c_5) = [1] x1 + [0]
p(c_6) = [2]
p(c_7) = [1]
p(c_8) = [1] x1 + [6]
Following rules are strictly oriented:
f#(h(x,h(y,z))) = [4] z + [16]
> [4] z + [14]
= c_8(f#(h(+(x,y),z)),+#(x,y))
Following rules are (at-least) weakly oriented:
*** 1.1.1.1.1.1.1.2.2.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.2.2.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
-->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: f#(h(x,h(y,z))) -> c_8(f#(h(+(x
,y)
,z))
,+#(x,y))
*** 1.1.1.1.1.1.1.2.2.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.1.2.2.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
+#(s(x),y) -> c_5(+#(x,y))
Strict TRS Rules:
Weak DP Rules:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
f#(h(x,h(y,z))) -> +#(x,y)
f#(h(x,h(y,z))) -> f#(h(+(x,y),z))
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: +#(s(x),y) -> c_5(+#(x,y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1.2.2.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
+#(s(x),y) -> c_5(+#(x,y))
Strict TRS Rules:
Weak DP Rules:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
f#(h(x,h(y,z))) -> +#(x,y)
f#(h(x,h(y,z))) -> f#(h(+(x,y),z))
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_1) = {1,2},
uargs(c_3) = {1},
uargs(c_5) = {1}
Following symbols are considered usable:
{+,+#,f#}
TcT has computed the following interpretation:
p(+) = 1 + x1 + x2
p(0) = 0
p(f) = x1^2
p(g) = x1
p(h) = 1 + x1 + x2
p(s) = 1 + x1
p(+#) = x1 + 2*x1*x2 + 3*x2 + x2^2
p(f#) = 2 + x1^2
p(c_1) = 1 + x1 + x2
p(c_2) = 0
p(c_3) = 1 + x1
p(c_4) = 0
p(c_5) = x1
p(c_6) = 1 + x1
p(c_7) = x1
p(c_8) = 1
Following rules are strictly oriented:
+#(s(x),y) = 1 + x + 2*x*y + 5*y + y^2
> x + 2*x*y + 3*y + y^2
= c_5(+#(x,y))
Following rules are (at-least) weakly oriented:
+#(x,+(y,z)) = 4 + 3*x + 2*x*y + 2*x*z + 5*y + 2*y*z + y^2 + 5*z + z^2
>= 2 + 2*x + 2*x*y + 2*x*z + 4*y + 2*y*z + y^2 + 5*z + z^2
= c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) = 4 + 3*x + 2*x*y + 5*y + y^2
>= 1 + x + 2*x*y + 3*y + y^2
= c_3(+#(x,y))
f#(h(x,h(y,z))) = 6 + 4*x + 2*x*y + 2*x*z + x^2 + 4*y + 2*y*z + y^2 + 4*z + z^2
>= x + 2*x*y + 3*y + y^2
= +#(x,y)
f#(h(x,h(y,z))) = 6 + 4*x + 2*x*y + 2*x*z + x^2 + 4*y + 2*y*z + y^2 + 4*z + z^2
>= 6 + 4*x + 2*x*y + 2*x*z + x^2 + 4*y + 2*y*z + y^2 + 4*z + z^2
= f#(h(+(x,y),z))
+(x,+(y,z)) = 2 + x + y + z
>= 2 + x + y + z
= +(+(x,y),z)
+(x,0()) = 1 + x
>= x
= x
+(x,s(y)) = 2 + x + y
>= 2 + x + y
= s(+(x,y))
+(0(),y) = 1 + y
>= y
= y
+(s(x),y) = 2 + x + y
>= 2 + x + y
= s(+(x,y))
*** 1.1.1.1.1.1.1.2.2.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
f#(h(x,h(y,z))) -> +#(x,y)
f#(h(x,h(y,z))) -> f#(h(+(x,y),z))
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.2.2.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
f#(h(x,h(y,z))) -> +#(x,y)
f#(h(x,h(y,z))) -> f#(h(+(x,y),z))
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
-->_2 +#(s(x),y) -> c_5(+#(x,y)):3
-->_1 +#(s(x),y) -> c_5(+#(x,y)):3
-->_2 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_2 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
2:W:+#(x,s(y)) -> c_3(+#(x,y))
-->_1 +#(s(x),y) -> c_5(+#(x,y)):3
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
3:W:+#(s(x),y) -> c_5(+#(x,y))
-->_1 +#(s(x),y) -> c_5(+#(x,y)):3
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
4:W:f#(h(x,h(y,z))) -> +#(x,y)
-->_1 +#(s(x),y) -> c_5(+#(x,y)):3
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
5:W:f#(h(x,h(y,z))) -> f#(h(+(x,y),z))
-->_1 f#(h(x,h(y,z))) -> f#(h(+(x,y),z)):5
-->_1 f#(h(x,h(y,z))) -> +#(x,y):4
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: f#(h(x,h(y,z))) -> f#(h(+(x,y)
,z))
4: f#(h(x,h(y,z))) -> +#(x,y)
1: +#(x,+(y,z)) -> c_1(+#(+(x,y),z)
,+#(x,y))
3: +#(s(x),y) -> c_5(+#(x,y))
2: +#(x,s(y)) -> c_3(+#(x,y))
*** 1.1.1.1.1.1.1.2.2.2.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Strict TRS Rules:
Weak DP Rules:
+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
+#(x,s(y)) -> c_3(+#(x,y))
+#(s(x),y) -> c_5(+#(x,y))
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
-->_2 +#(s(x),y) -> c_5(+#(x,y)):4
-->_2 +#(x,s(y)) -> c_3(+#(x,y)):3
-->_2 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):2
-->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):1
2:W:+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
-->_2 +#(s(x),y) -> c_5(+#(x,y)):4
-->_1 +#(s(x),y) -> c_5(+#(x,y)):4
-->_2 +#(x,s(y)) -> c_3(+#(x,y)):3
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):3
-->_2 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):2
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):2
3:W:+#(x,s(y)) -> c_3(+#(x,y))
-->_1 +#(s(x),y) -> c_5(+#(x,y)):4
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):3
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):2
4:W:+#(s(x),y) -> c_5(+#(x,y))
-->_1 +#(s(x),y) -> c_5(+#(x,y)):4
-->_1 +#(x,s(y)) -> c_3(+#(x,y)):3
-->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: +#(s(x),y) -> c_5(+#(x,y))
3: +#(x,s(y)) -> c_3(+#(x,y))
2: +#(x,+(y,z)) -> c_1(+#(+(x,y),z)
,+#(x,y))
*** 1.1.1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
-->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)))
*** 1.1.1.1.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/1}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,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, 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,h(y,z))) -> c_8(f#(h(+(x
,y)
,z)))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/1}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
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_8) = {1}
Following symbols are considered usable:
{+#,f#}
TcT has computed the following interpretation:
p(+) = [0]
p(0) = [1]
p(f) = [2]
p(g) = [1] x1 + [1]
p(h) = [1] x2 + [1]
p(s) = [0]
p(+#) = [1] x1 + [2] x2 + [1]
p(f#) = [4] x1 + [8]
p(c_1) = [1] x2 + [1]
p(c_2) = [1]
p(c_3) = [2] x1 + [2]
p(c_4) = [8]
p(c_5) = [4] x1 + [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [2]
p(c_8) = [1] x1 + [0]
Following rules are strictly oriented:
f#(h(x,h(y,z))) = [4] z + [16]
> [4] z + [12]
= c_8(f#(h(+(x,y),z)))
Following rules are (at-least) weakly oriented:
*** 1.1.1.1.1.1.2.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)))
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/1}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)))
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/1}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)))
-->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,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#(h(x,h(y,z))) -> c_8(f#(h(+(x
,y)
,z)))
*** 1.1.1.1.1.1.2.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
+(x,+(y,z)) -> +(+(x,y),z)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
Signature:
{+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/1}
Obligation:
Innermost
basic terms: {+#,f#}/{0,g,h,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).