*** 1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
first(0(),X) -> nil()
first(s(X),cons(Y)) -> cons(Y)
half(0()) -> 0()
half(dbl(X)) -> X
half(s(0())) -> 0()
half(s(s(X))) -> s(half(X))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
terms(N) -> cons(recip(sqr(N)))
Weak DP Rules:
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1} / {0/0,cons/1,nil/0,recip/1,s/1}
Obligation:
Innermost
basic terms: {add,dbl,first,half,sqr,terms}/{0,cons,nil,recip,s}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
We add the following dependency tuples:
Strict DPs
add#(0(),X) -> c_1()
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(0()) -> c_3()
dbl#(s(X)) -> c_4(dbl#(X))
first#(0(),X) -> c_5()
first#(s(X),cons(Y)) -> c_6()
half#(0()) -> c_7()
half#(dbl(X)) -> c_8()
half#(s(0())) -> c_9()
half#(s(s(X))) -> c_10(half#(X))
sqr#(0()) -> c_11()
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N))
Weak DPs
and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
add#(0(),X) -> c_1()
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(0()) -> c_3()
dbl#(s(X)) -> c_4(dbl#(X))
first#(0(),X) -> c_5()
first#(s(X),cons(Y)) -> c_6()
half#(0()) -> c_7()
half#(dbl(X)) -> c_8()
half#(s(0())) -> c_9()
half#(s(s(X))) -> c_10(half#(X))
sqr#(0()) -> c_11()
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
first(0(),X) -> nil()
first(s(X),cons(Y)) -> cons(Y)
half(0()) -> 0()
half(dbl(X)) -> X
half(s(0())) -> 0()
half(s(s(X))) -> s(half(X))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
terms(N) -> cons(recip(sqr(N)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
add#(0(),X) -> c_1()
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(0()) -> c_3()
dbl#(s(X)) -> c_4(dbl#(X))
first#(0(),X) -> c_5()
first#(s(X),cons(Y)) -> c_6()
half#(0()) -> c_7()
half#(dbl(X)) -> c_8()
half#(s(0())) -> c_9()
half#(s(s(X))) -> c_10(half#(X))
sqr#(0()) -> c_11()
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N))
*** 1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
add#(0(),X) -> c_1()
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(0()) -> c_3()
dbl#(s(X)) -> c_4(dbl#(X))
first#(0(),X) -> c_5()
first#(s(X),cons(Y)) -> c_6()
half#(0()) -> c_7()
half#(dbl(X)) -> c_8()
half#(s(0())) -> c_9()
half#(s(s(X))) -> c_10(half#(X))
sqr#(0()) -> c_11()
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{1,3,5,6,7,8,9,11}
by application of
Pre({1,3,5,6,7,8,9,11}) = {2,4,10,12,13}.
Here rules are labelled as follows:
1: add#(0(),X) -> c_1()
2: add#(s(X),Y) -> c_2(add#(X,Y))
3: dbl#(0()) -> c_3()
4: dbl#(s(X)) -> c_4(dbl#(X))
5: first#(0(),X) -> c_5()
6: first#(s(X),cons(Y)) -> c_6()
7: half#(0()) -> c_7()
8: half#(dbl(X)) -> c_8()
9: half#(s(0())) -> c_9()
10: half#(s(s(X))) -> c_10(half#(X))
11: sqr#(0()) -> c_11()
12: sqr#(s(X)) -> c_12(add#(sqr(X)
,dbl(X))
,sqr#(X)
,dbl#(X))
13: terms#(N) -> c_13(sqr#(N))
*** 1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N))
Strict TRS Rules:
Weak DP Rules:
add#(0(),X) -> c_1()
dbl#(0()) -> c_3()
first#(0(),X) -> c_5()
first#(s(X),cons(Y)) -> c_6()
half#(0()) -> c_7()
half#(dbl(X)) -> c_8()
half#(s(0())) -> c_9()
sqr#(0()) -> c_11()
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:add#(s(X),Y) -> c_2(add#(X,Y))
-->_1 add#(0(),X) -> c_1():6
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
2:S:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(0()) -> c_3():7
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):2
3:S:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(0())) -> c_9():12
-->_1 half#(dbl(X)) -> c_8():11
-->_1 half#(0()) -> c_7():10
-->_1 half#(s(s(X))) -> c_10(half#(X)):3
4:S:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_2 sqr#(0()) -> c_11():13
-->_3 dbl#(0()) -> c_3():7
-->_1 add#(0(),X) -> c_1():6
-->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4
-->_3 dbl#(s(X)) -> c_4(dbl#(X)):2
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
5:S:terms#(N) -> c_13(sqr#(N))
-->_1 sqr#(0()) -> c_11():13
-->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4
6:W:add#(0(),X) -> c_1()
7:W:dbl#(0()) -> c_3()
8:W:first#(0(),X) -> c_5()
9:W:first#(s(X),cons(Y)) -> c_6()
10:W:half#(0()) -> c_7()
11:W:half#(dbl(X)) -> c_8()
12:W:half#(s(0())) -> c_9()
13:W:sqr#(0()) -> c_11()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
9: first#(s(X),cons(Y)) -> c_6()
8: first#(0(),X) -> c_5()
13: sqr#(0()) -> c_11()
10: half#(0()) -> c_7()
11: half#(dbl(X)) -> c_8()
12: half#(s(0())) -> c_9()
7: dbl#(0()) -> c_3()
6: add#(0(),X) -> c_1()
*** 1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
RemoveHeads
Proof:
Consider the dependency graph
1:S:add#(s(X),Y) -> c_2(add#(X,Y))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
2:S:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):2
3:S:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):3
4:S:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4
-->_3 dbl#(s(X)) -> c_4(dbl#(X)):2
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
5:S:terms#(N) -> c_13(sqr#(N))
-->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4
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).
[(5,terms#(N) -> c_13(sqr#(N)))]
*** 1.1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,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:
add#(s(X),Y) -> c_2(add#(X,Y))
Strict TRS Rules:
Weak DP Rules:
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Problem (S)
Strict DP Rules:
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
Strict TRS Rules:
Weak DP Rules:
add#(s(X),Y) -> c_2(add#(X,Y))
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
*** 1.1.1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
add#(s(X),Y) -> c_2(add#(X,Y))
Strict TRS Rules:
Weak DP Rules:
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:add#(s(X),Y) -> c_2(add#(X,Y))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
2:W:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):2
3:W:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):3
4:W:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_3 dbl#(s(X)) -> c_4(dbl#(X)):2
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
-->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: half#(s(s(X))) -> c_10(half#(X))
2: dbl#(s(X)) -> c_4(dbl#(X))
*** 1.1.1.1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
add#(s(X),Y) -> c_2(add#(X,Y))
Strict TRS Rules:
Weak DP Rules:
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:add#(s(X),Y) -> c_2(add#(X,Y))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
4:W:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
-->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X))
*** 1.1.1.1.1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
add#(s(X),Y) -> c_2(add#(X,Y))
Strict TRS Rules:
Weak DP Rules:
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X))
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,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
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X))
and a lower component
add#(s(X),Y) -> c_2(add#(X,Y))
Further, following extension rules are added to the lower component.
sqr#(s(X)) -> add#(sqr(X),dbl(X))
sqr#(s(X)) -> sqr#(X)
*** 1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,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: sqr#(s(X)) -> c_12(add#(sqr(X)
,dbl(X))
,sqr#(X))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,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_12) = {1,2}
Following symbols are considered usable:
{add#,dbl#,first#,half#,sqr#,terms#}
TcT has computed the following interpretation:
p(0) = [3]
p(add) = [2]
p(cons) = [1] x1 + [0]
p(dbl) = [1] x1 + [0]
p(first) = [0]
p(half) = [4] x1 + [0]
p(nil) = [0]
p(recip) = [0]
p(s) = [1] x1 + [6]
p(sqr) = [2]
p(terms) = [0]
p(add#) = [5]
p(dbl#) = [0]
p(first#) = [1] x1 + [8] x2 + [0]
p(half#) = [1] x1 + [0]
p(sqr#) = [4] x1 + [2]
p(terms#) = [8] x1 + [0]
p(c_1) = [0]
p(c_2) = [4] x1 + [0]
p(c_3) = [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [0]
p(c_10) = [1] x1 + [0]
p(c_11) = [2]
p(c_12) = [2] x1 + [1] x2 + [6]
p(c_13) = [0]
Following rules are strictly oriented:
sqr#(s(X)) = [4] X + [26]
> [4] X + [18]
= c_12(add#(sqr(X),dbl(X))
,sqr#(X))
Following rules are (at-least) weakly oriented:
*** 1.1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X))
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X))
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X))
-->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: sqr#(s(X)) -> c_12(add#(sqr(X)
,dbl(X))
,sqr#(X))
*** 1.1.1.1.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.1.1.1.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
add#(s(X),Y) -> c_2(add#(X,Y))
Strict TRS Rules:
Weak DP Rules:
sqr#(s(X)) -> add#(sqr(X),dbl(X))
sqr#(s(X)) -> sqr#(X)
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,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: add#(s(X),Y) -> c_2(add#(X,Y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1.1.1.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
add#(s(X),Y) -> c_2(add#(X,Y))
Strict TRS Rules:
Weak DP Rules:
sqr#(s(X)) -> add#(sqr(X),dbl(X))
sqr#(s(X)) -> sqr#(X)
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,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_2) = {1}
Following symbols are considered usable:
{add,dbl,sqr,add#,dbl#,first#,half#,sqr#,terms#}
TcT has computed the following interpretation:
p(0) = 0
p(add) = x1 + x2
p(cons) = 0
p(dbl) = 2*x1
p(first) = 0
p(half) = 1 + x1
p(nil) = 1
p(recip) = 0
p(s) = 1 + x1
p(sqr) = 2*x1^2
p(terms) = 2 + x1^2
p(add#) = 4 + 2*x1
p(dbl#) = 4 + x1^2
p(first#) = 2*x1 + 4*x2
p(half#) = 1 + x1 + 4*x1^2
p(sqr#) = 4*x1^2
p(terms#) = 4*x1^2
p(c_1) = 1
p(c_2) = x1
p(c_3) = 0
p(c_4) = 1 + x1
p(c_5) = 1
p(c_6) = 1
p(c_7) = 1
p(c_8) = 1
p(c_9) = 1
p(c_10) = 1
p(c_11) = 0
p(c_12) = 0
p(c_13) = 0
Following rules are strictly oriented:
add#(s(X),Y) = 6 + 2*X
> 4 + 2*X
= c_2(add#(X,Y))
Following rules are (at-least) weakly oriented:
sqr#(s(X)) = 4 + 8*X + 4*X^2
>= 4 + 4*X^2
= add#(sqr(X),dbl(X))
sqr#(s(X)) = 4 + 8*X + 4*X^2
>= 4*X^2
= sqr#(X)
add(0(),X) = X
>= X
= X
add(s(X),Y) = 1 + X + Y
>= 1 + X + Y
= s(add(X,Y))
dbl(0()) = 0
>= 0
= 0()
dbl(s(X)) = 2 + 2*X
>= 2 + 2*X
= s(s(dbl(X)))
sqr(0()) = 0
>= 0
= 0()
sqr(s(X)) = 2 + 4*X + 2*X^2
>= 1 + 2*X + 2*X^2
= s(add(sqr(X),dbl(X)))
*** 1.1.1.1.1.1.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
add#(s(X),Y) -> c_2(add#(X,Y))
sqr#(s(X)) -> add#(sqr(X),dbl(X))
sqr#(s(X)) -> sqr#(X)
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.1.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
add#(s(X),Y) -> c_2(add#(X,Y))
sqr#(s(X)) -> add#(sqr(X),dbl(X))
sqr#(s(X)) -> sqr#(X)
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:add#(s(X),Y) -> c_2(add#(X,Y))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
2:W:sqr#(s(X)) -> add#(sqr(X),dbl(X))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
3:W:sqr#(s(X)) -> sqr#(X)
-->_1 sqr#(s(X)) -> sqr#(X):3
-->_1 sqr#(s(X)) -> add#(sqr(X),dbl(X)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: sqr#(s(X)) -> sqr#(X)
2: sqr#(s(X)) -> add#(sqr(X)
,dbl(X))
1: add#(s(X),Y) -> c_2(add#(X,Y))
*** 1.1.1.1.1.1.1.1.1.2.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,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^2))] ***
Considered Problem:
Strict DP Rules:
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
Strict TRS Rules:
Weak DP Rules:
add#(s(X),Y) -> c_2(add#(X,Y))
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):1
2:S:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):2
3:S:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):4
-->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
-->_3 dbl#(s(X)) -> c_4(dbl#(X)):1
4:W:add#(s(X),Y) -> c_2(add#(X,Y))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):4
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: add#(s(X),Y) -> c_2(add#(X,Y))
*** 1.1.1.1.1.1.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):1
2:S:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):2
3:S:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
-->_3 dbl#(s(X)) -> c_4(dbl#(X)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
*** 1.1.1.1.1.1.2.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
*** 1.1.1.1.1.1.2.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
dbl#(s(X)) -> c_4(dbl#(X))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,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:
dbl#(s(X)) -> c_4(dbl#(X))
Strict TRS Rules:
Weak DP Rules:
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Problem (S)
Strict DP Rules:
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
Strict TRS Rules:
Weak DP Rules:
dbl#(s(X)) -> c_4(dbl#(X))
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
*** 1.1.1.1.1.1.2.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
dbl#(s(X)) -> c_4(dbl#(X))
Strict TRS Rules:
Weak DP Rules:
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):1
2:W:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):2
3:W:sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
-->_2 dbl#(s(X)) -> c_4(dbl#(X)):1
-->_1 sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: half#(s(s(X))) -> c_10(half#(X))
*** 1.1.1.1.1.1.2.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
dbl#(s(X)) -> c_4(dbl#(X))
Strict TRS Rules:
Weak DP Rules:
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,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: dbl#(s(X)) -> c_4(dbl#(X))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
dbl#(s(X)) -> c_4(dbl#(X))
Strict TRS Rules:
Weak DP Rules:
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,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_4) = {1},
uargs(c_12) = {1,2}
Following symbols are considered usable:
{add#,dbl#,first#,half#,sqr#,terms#}
TcT has computed the following interpretation:
p(0) = 0
p(add) = 2 + x1 + x1*x2
p(cons) = x1
p(dbl) = 1 + x1 + x1^2
p(first) = 2 + 2*x1 + x1^2 + x2 + x2^2
p(half) = 1 + 8*x1 + x1^2
p(nil) = 0
p(recip) = 1
p(s) = 1 + x1
p(sqr) = x1^2
p(terms) = 1 + 4*x1 + x1^2
p(add#) = 4*x1 + x1*x2 + 2*x1^2 + x2^2
p(dbl#) = 1 + 2*x1
p(first#) = 2 + x1 + 8*x1^2
p(half#) = 0
p(sqr#) = 12 + 7*x1 + 2*x1^2
p(terms#) = 1 + x1 + x1^2
p(c_1) = 0
p(c_2) = 1 + x1
p(c_3) = 0
p(c_4) = 1 + x1
p(c_5) = 0
p(c_6) = 1
p(c_7) = 1
p(c_8) = 0
p(c_9) = 1
p(c_10) = 1
p(c_11) = 0
p(c_12) = x1 + x2
p(c_13) = x1
Following rules are strictly oriented:
dbl#(s(X)) = 3 + 2*X
> 2 + 2*X
= c_4(dbl#(X))
Following rules are (at-least) weakly oriented:
sqr#(s(X)) = 21 + 11*X + 2*X^2
>= 13 + 9*X + 2*X^2
= c_12(sqr#(X),dbl#(X))
*** 1.1.1.1.1.1.2.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):1
2:W:sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
-->_1 sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)):2
-->_2 dbl#(s(X)) -> c_4(dbl#(X)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: sqr#(s(X)) -> c_12(sqr#(X)
,dbl#(X))
1: dbl#(s(X)) -> c_4(dbl#(X))
*** 1.1.1.1.1.1.2.1.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.2.1.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
Strict TRS Rules:
Weak DP Rules:
dbl#(s(X)) -> c_4(dbl#(X))
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):1
2:S:sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
-->_2 dbl#(s(X)) -> c_4(dbl#(X)):3
-->_1 sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)):2
3:W:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: dbl#(s(X)) -> c_4(dbl#(X))
*** 1.1.1.1.1.1.2.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):1
2:S:sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
-->_1 sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)):2
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
sqr#(s(X)) -> c_12(sqr#(X))
*** 1.1.1.1.1.1.2.1.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,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:
half#(s(s(X))) -> c_10(half#(X))
Strict TRS Rules:
Weak DP Rules:
sqr#(s(X)) -> c_12(sqr#(X))
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Problem (S)
Strict DP Rules:
sqr#(s(X)) -> c_12(sqr#(X))
Strict TRS Rules:
Weak DP Rules:
half#(s(s(X))) -> c_10(half#(X))
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
*** 1.1.1.1.1.1.2.1.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
half#(s(s(X))) -> c_10(half#(X))
Strict TRS Rules:
Weak DP Rules:
sqr#(s(X)) -> c_12(sqr#(X))
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):1
2:W:sqr#(s(X)) -> c_12(sqr#(X))
-->_1 sqr#(s(X)) -> c_12(sqr#(X)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: sqr#(s(X)) -> c_12(sqr#(X))
*** 1.1.1.1.1.1.2.1.1.1.2.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
half#(s(s(X))) -> c_10(half#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,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: half#(s(s(X))) -> c_10(half#(X))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.1.1.1.2.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
half#(s(s(X))) -> c_10(half#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,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_10) = {1}
Following symbols are considered usable:
{add#,dbl#,first#,half#,sqr#,terms#}
TcT has computed the following interpretation:
p(0) = [2]
p(add) = [2]
p(cons) = [1]
p(dbl) = [2] x1 + [1]
p(first) = [8] x1 + [1] x2 + [1]
p(half) = [0]
p(nil) = [0]
p(recip) = [2]
p(s) = [1] x1 + [8]
p(sqr) = [2] x1 + [2]
p(terms) = [2] x1 + [1]
p(add#) = [2] x1 + [1] x2 + [0]
p(dbl#) = [0]
p(first#) = [1] x1 + [1] x2 + [0]
p(half#) = [1] x1 + [4]
p(sqr#) = [8]
p(terms#) = [1]
p(c_1) = [1]
p(c_2) = [2] x1 + [1]
p(c_3) = [1]
p(c_4) = [2] x1 + [4]
p(c_5) = [4]
p(c_6) = [0]
p(c_7) = [8]
p(c_8) = [4]
p(c_9) = [0]
p(c_10) = [1] x1 + [10]
p(c_11) = [2]
p(c_12) = [1] x1 + [1]
p(c_13) = [4]
Following rules are strictly oriented:
half#(s(s(X))) = [1] X + [20]
> [1] X + [14]
= c_10(half#(X))
Following rules are (at-least) weakly oriented:
*** 1.1.1.1.1.1.2.1.1.1.2.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
half#(s(s(X))) -> c_10(half#(X))
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.1.1.1.2.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
half#(s(s(X))) -> c_10(half#(X))
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: half#(s(s(X))) -> c_10(half#(X))
*** 1.1.1.1.1.1.2.1.1.1.2.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.2.1.1.1.2.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sqr#(s(X)) -> c_12(sqr#(X))
Strict TRS Rules:
Weak DP Rules:
half#(s(s(X))) -> c_10(half#(X))
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:sqr#(s(X)) -> c_12(sqr#(X))
-->_1 sqr#(s(X)) -> c_12(sqr#(X)):1
2:W:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: half#(s(s(X))) -> c_10(half#(X))
*** 1.1.1.1.1.1.2.1.1.1.2.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sqr#(s(X)) -> c_12(sqr#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,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: sqr#(s(X)) -> c_12(sqr#(X))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.1.1.1.2.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sqr#(s(X)) -> c_12(sqr#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,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_12) = {1}
Following symbols are considered usable:
{add#,dbl#,first#,half#,sqr#,terms#}
TcT has computed the following interpretation:
p(0) = [0]
p(add) = [0]
p(cons) = [1] x1 + [0]
p(dbl) = [0]
p(first) = [0]
p(half) = [0]
p(nil) = [0]
p(recip) = [2]
p(s) = [1] x1 + [5]
p(sqr) = [0]
p(terms) = [0]
p(add#) = [0]
p(dbl#) = [0]
p(first#) = [0]
p(half#) = [0]
p(sqr#) = [5] x1 + [0]
p(terms#) = [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [0]
p(c_10) = [8] x1 + [0]
p(c_11) = [0]
p(c_12) = [1] x1 + [0]
p(c_13) = [1]
Following rules are strictly oriented:
sqr#(s(X)) = [5] X + [25]
> [5] X + [0]
= c_12(sqr#(X))
Following rules are (at-least) weakly oriented:
*** 1.1.1.1.1.1.2.1.1.1.2.1.1.2.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
sqr#(s(X)) -> c_12(sqr#(X))
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.1.1.1.2.1.1.2.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
sqr#(s(X)) -> c_12(sqr#(X))
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:sqr#(s(X)) -> c_12(sqr#(X))
-->_1 sqr#(s(X)) -> c_12(sqr#(X)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: sqr#(s(X)) -> c_12(sqr#(X))
*** 1.1.1.1.1.1.2.1.1.1.2.1.1.2.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1}
Obligation:
Innermost
basic terms: {add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,nil,recip,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).