*** 1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
Weak DP Rules:
Weak TRS Rules:
Signature:
{minus/2,plus/2,quot/2} / {0/0,s/1}
Obligation:
Innermost
basic terms: {minus,plus,quot}/{0,s}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
We add the following dependency tuples:
Strict DPs
minus#(x,0()) -> c_1()
minus#(s(x),s(y)) -> c_2(minus#(x,y))
plus#(0(),y) -> c_3()
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(y,s(s(z))),minus#(x,s(0())))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))),plus#(y,s(s(z))),plus#(x,s(0())))
plus#(s(x),y) -> c_6(plus#(x,y))
quot#(0(),s(y)) -> c_7()
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y))
Weak DPs
and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
minus#(x,0()) -> c_1()
minus#(s(x),s(y)) -> c_2(minus#(x,y))
plus#(0(),y) -> c_3()
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(y,s(s(z))),minus#(x,s(0())))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))),plus#(y,s(s(z))),plus#(x,s(0())))
plus#(s(x),y) -> c_6(plus#(x,y))
quot#(0(),s(y)) -> c_7()
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/3,c_5/3,c_6/1,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
minus#(x,0()) -> c_1()
minus#(s(x),s(y)) -> c_2(minus#(x,y))
plus#(0(),y) -> c_3()
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(y,s(s(z))),minus#(x,s(0())))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))),plus#(y,s(s(z))),plus#(x,s(0())))
plus#(s(x),y) -> c_6(plus#(x,y))
quot#(0(),s(y)) -> c_7()
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y))
*** 1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
minus#(x,0()) -> c_1()
minus#(s(x),s(y)) -> c_2(minus#(x,y))
plus#(0(),y) -> c_3()
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(y,s(s(z))),minus#(x,s(0())))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))),plus#(y,s(s(z))),plus#(x,s(0())))
plus#(s(x),y) -> c_6(plus#(x,y))
quot#(0(),s(y)) -> c_7()
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/3,c_5/3,c_6/1,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{1,3,7}
by application of
Pre({1,3,7}) = {2,4,5,6,8}.
Here rules are labelled as follows:
1: minus#(x,0()) -> c_1()
2: minus#(s(x),s(y)) ->
c_2(minus#(x,y))
3: plus#(0(),y) -> c_3()
4: plus#(minus(x,s(0()))
,minus(y,s(s(z)))) ->
c_4(plus#(minus(y,s(s(z)))
,minus(x,s(0())))
,minus#(y,s(s(z)))
,minus#(x,s(0())))
5: plus#(plus(x,s(0()))
,plus(y,s(s(z)))) ->
c_5(plus#(plus(y,s(s(z)))
,plus(x,s(0())))
,plus#(y,s(s(z)))
,plus#(x,s(0())))
6: plus#(s(x),y) -> c_6(plus#(x,y))
7: quot#(0(),s(y)) -> c_7()
8: quot#(s(x),s(y)) ->
c_8(quot#(minus(x,y),s(y))
,minus#(x,y))
*** 1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(y,s(s(z))),minus#(x,s(0())))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))),plus#(y,s(s(z))),plus#(x,s(0())))
plus#(s(x),y) -> c_6(plus#(x,y))
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
minus#(x,0()) -> c_1()
plus#(0(),y) -> c_3()
quot#(0(),s(y)) -> c_7()
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/3,c_5/3,c_6/1,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:minus#(s(x),s(y)) -> c_2(minus#(x,y))
-->_1 minus#(x,0()) -> c_1():6
-->_1 minus#(s(x),s(y)) -> c_2(minus#(x,y)):1
2:S:plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(y,s(s(z))),minus#(x,s(0())))
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):4
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))),plus#(y,s(s(z))),plus#(x,s(0()))):3
-->_1 plus#(0(),y) -> c_3():7
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(y,s(s(z))),minus#(x,s(0()))):2
-->_3 minus#(s(x),s(y)) -> c_2(minus#(x,y)):1
3:S:plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))),plus#(y,s(s(z))),plus#(x,s(0())))
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):4
-->_1 plus#(0(),y) -> c_3():7
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))),plus#(y,s(s(z))),plus#(x,s(0()))):3
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(y,s(s(z))),minus#(x,s(0()))):2
4:S:plus#(s(x),y) -> c_6(plus#(x,y))
-->_1 plus#(0(),y) -> c_3():7
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):4
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))),plus#(y,s(s(z))),plus#(x,s(0()))):3
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(y,s(s(z))),minus#(x,s(0()))):2
5:S:quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(0(),s(y)) -> c_7():8
-->_2 minus#(x,0()) -> c_1():6
-->_1 quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y)):5
-->_2 minus#(s(x),s(y)) -> c_2(minus#(x,y)):1
6:W:minus#(x,0()) -> c_1()
7:W:plus#(0(),y) -> c_3()
8:W:quot#(0(),s(y)) -> c_7()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
8: quot#(0(),s(y)) -> c_7()
7: plus#(0(),y) -> c_3()
6: minus#(x,0()) -> c_1()
*** 1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(y,s(s(z))),minus#(x,s(0())))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))),plus#(y,s(s(z))),plus#(x,s(0())))
plus#(s(x),y) -> c_6(plus#(x,y))
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/3,c_5/3,c_6/1,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:minus#(s(x),s(y)) -> c_2(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_2(minus#(x,y)):1
2:S:plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(y,s(s(z))),minus#(x,s(0())))
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):4
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))),plus#(y,s(s(z))),plus#(x,s(0()))):3
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(y,s(s(z))),minus#(x,s(0()))):2
-->_3 minus#(s(x),s(y)) -> c_2(minus#(x,y)):1
3:S:plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))),plus#(y,s(s(z))),plus#(x,s(0())))
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):4
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))),plus#(y,s(s(z))),plus#(x,s(0()))):3
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(y,s(s(z))),minus#(x,s(0()))):2
4:S:plus#(s(x),y) -> c_6(plus#(x,y))
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):4
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))),plus#(y,s(s(z))),plus#(x,s(0()))):3
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(y,s(s(z))),minus#(x,s(0()))):2
5:S:quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y)):5
-->_2 minus#(s(x),s(y)) -> c_2(minus#(x,y)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0())))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
*** 1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0())))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/1,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,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:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0())))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/1,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Problem (S)
Strict DP Rules:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0())))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/1,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
*** 1.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0())))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/1,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: minus#(s(x),s(y)) ->
c_2(minus#(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:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0())))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/1,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
NaturalMI {miDimension = 2, 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:
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_4) = {2},
uargs(c_6) = {1},
uargs(c_8) = {1,2}
Following symbols are considered usable:
{minus,minus#,plus#,quot#}
TcT has computed the following interpretation:
p(0) = [3]
[1]
p(minus) = [1 0] x1 + [0]
[0 1] [0]
p(plus) = [0 0] x1 + [0]
[2 0] [0]
p(quot) = [2]
[2]
p(s) = [1 2] x1 + [0]
[0 1] [2]
p(minus#) = [0 2] x1 + [0]
[0 0] [1]
p(plus#) = [0 3] x1 + [0 0] x2 + [1]
[0 0] [0 1] [0]
p(quot#) = [2 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [0]
p(c_1) = [0]
[2]
p(c_2) = [1 2] x1 + [0]
[0 0] [0]
p(c_3) = [1]
[2]
p(c_4) = [0 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
p(c_5) = [0]
[0]
p(c_6) = [1 0] x1 + [0]
[0 0] [0]
p(c_7) = [0]
[2]
p(c_8) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
Following rules are strictly oriented:
minus#(s(x),s(y)) = [0 2] x + [4]
[0 0] [1]
> [0 2] x + [2]
[0 0] [0]
= c_2(minus#(x,y))
Following rules are (at-least) weakly oriented:
plus#(minus(x,s(0())) = [0 3] x + [0 0] y + [1]
,minus(y,s(s(z)))) [0 0] [0 1] [0]
>= [0 3] x + [0]
[0 0] [0]
= c_4(plus#(minus(y,s(s(z)))
,minus(x,s(0())))
,minus#(x,s(0())))
plus#(plus(x,s(0())) = [6 0] x + [0 0] y + [1]
,plus(y,s(s(z)))) [0 0] [2 0] [0]
>= [0]
[0]
= c_5(plus#(plus(y,s(s(z)))
,plus(x,s(0()))))
plus#(s(x),y) = [0 3] x + [0 0] y + [7]
[0 0] [0 1] [0]
>= [0 3] x + [1]
[0 0] [0]
= c_6(plus#(x,y))
quot#(s(x),s(y)) = [2 4] x + [1 2] y + [0]
[0 1] [0 0] [2]
>= [2 4] x + [1 2] y + [0]
[0 0] [0 0] [0]
= c_8(quot#(minus(x,y),s(y))
,minus#(x,y))
minus(x,0()) = [1 0] x + [0]
[0 1] [0]
>= [1 0] x + [0]
[0 1] [0]
= x
minus(s(x),s(y)) = [1 2] x + [0]
[0 1] [2]
>= [1 0] x + [0]
[0 1] [0]
= minus(x,y)
*** 1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0())))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/1,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0())))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/1,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:minus#(s(x),s(y)) -> c_2(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_2(minus#(x,y)):1
2:W:plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0())))
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):4
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0())))):3
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0()))):2
-->_2 minus#(s(x),s(y)) -> c_2(minus#(x,y)):1
3:W:plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):4
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0())))):3
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0()))):2
4:W:plus#(s(x),y) -> c_6(plus#(x,y))
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):4
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0())))):3
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0()))):2
5:W:quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y)):5
-->_2 minus#(s(x),s(y)) -> c_2(minus#(x,y)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: quot#(s(x),s(y)) ->
c_8(quot#(minus(x,y),s(y))
,minus#(x,y))
2: plus#(minus(x,s(0()))
,minus(y,s(s(z)))) ->
c_4(plus#(minus(y,s(s(z)))
,minus(x,s(0())))
,minus#(x,s(0())))
4: plus#(s(x),y) -> c_6(plus#(x,y))
3: plus#(plus(x,s(0()))
,plus(y,s(s(z)))) ->
c_5(plus#(plus(y,s(s(z)))
,plus(x,s(0()))))
1: minus#(s(x),s(y)) ->
c_2(minus#(x,y))
*** 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:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/1,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,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:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0())))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/1,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0())))
-->_2 minus#(s(x),s(y)) -> c_2(minus#(x,y)):5
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):3
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0())))):2
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0()))):1
2:S:plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):3
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0())))):2
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0()))):1
3:S:plus#(s(x),y) -> c_6(plus#(x,y))
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):3
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0())))):2
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0()))):1
4:S:quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y))
-->_2 minus#(s(x),s(y)) -> c_2(minus#(x,y)):5
-->_1 quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y)):4
5:W:minus#(s(x),s(y)) -> c_2(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_2(minus#(x,y)):5
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: minus#(s(x),s(y)) ->
c_2(minus#(x,y))
*** 1.1.1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0())))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/1,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0())))
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):3
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0())))):2
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0()))):1
2:S:plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):3
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0())))):2
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0()))):1
3:S:plus#(s(x),y) -> c_6(plus#(x,y))
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):3
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0())))):2
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))),minus#(x,s(0()))):1
4:S:quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)),minus#(x,y)):4
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)))
*** 1.1.1.1.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,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:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
Strict TRS Rules:
Weak DP Rules:
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Problem (S)
Strict DP Rules:
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)))
Strict TRS Rules:
Weak DP Rules:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
*** 1.1.1.1.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
Strict TRS Rules:
Weak DP Rules:
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0())))):2
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):3
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0())))):1
2:S:plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):3
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0())))):2
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0())))):1
3:S:plus#(s(x),y) -> c_6(plus#(x,y))
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0())))):2
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):3
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0())))):1
4:W:quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)))
-->_1 quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y))):4
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: quot#(s(x),s(y)) ->
c_8(quot#(minus(x,y),s(y)))
*** 1.1.1.1.1.1.2.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,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: plus#(minus(x,s(0()))
,minus(y,s(s(z)))) ->
c_4(plus#(minus(y,s(s(z)))
,minus(x,s(0()))))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,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_6) = {1}
Following symbols are considered usable:
{minus#,plus#,quot#}
TcT has computed the following interpretation:
p(0) = [2]
p(minus) = [1] x2 + [0]
p(plus) = [1] x1 + [0]
p(quot) = [8]
p(s) = [1]
p(minus#) = [1] x1 + [1] x2 + [1]
p(plus#) = [8] x2 + [0]
p(quot#) = [1] x2 + [2]
p(c_1) = [1]
p(c_2) = [1] x1 + [2]
p(c_3) = [1]
p(c_4) = [4]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1]
p(c_8) = [1] x1 + [1]
Following rules are strictly oriented:
plus#(minus(x,s(0())) = [8]
,minus(y,s(s(z))))
> [4]
= c_4(plus#(minus(y,s(s(z)))
,minus(x,s(0()))))
Following rules are (at-least) weakly oriented:
plus#(plus(x,s(0())) = [8] y + [0]
,plus(y,s(s(z))))
>= [0]
= c_5(plus#(plus(y,s(s(z)))
,plus(x,s(0()))))
plus#(s(x),y) = [8] y + [0]
>= [8] y + [0]
= c_6(plus#(x,y))
*** 1.1.1.1.1.1.2.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
Strict TRS Rules:
Weak DP Rules:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.1.1.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
Strict TRS Rules:
Weak DP Rules:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,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: plus#(plus(x,s(0()))
,plus(y,s(s(z)))) ->
c_5(plus#(plus(y,s(s(z)))
,plus(x,s(0()))))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.1.1.1.1.2.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
Strict TRS Rules:
Weak DP Rules:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,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_6) = {1}
Following symbols are considered usable:
{minus#,plus#,quot#}
TcT has computed the following interpretation:
p(0) = [0]
p(minus) = [5]
p(plus) = [1] x1 + [11]
p(quot) = [1] x2 + [0]
p(s) = [4]
p(minus#) = [1] x1 + [0]
p(plus#) = [2] x2 + [1]
p(quot#) = [1] x2 + [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [0]
p(c_3) = [1]
p(c_4) = [11]
p(c_5) = [1]
p(c_6) = [1] x1 + [0]
p(c_7) = [0]
p(c_8) = [2]
Following rules are strictly oriented:
plus#(plus(x,s(0())) = [2] y + [23]
,plus(y,s(s(z))))
> [1]
= c_5(plus#(plus(y,s(s(z)))
,plus(x,s(0()))))
Following rules are (at-least) weakly oriented:
plus#(minus(x,s(0())) = [11]
,minus(y,s(s(z))))
>= [11]
= c_4(plus#(minus(y,s(s(z)))
,minus(x,s(0()))))
plus#(s(x),y) = [2] y + [1]
>= [2] y + [1]
= c_6(plus#(x,y))
*** 1.1.1.1.1.1.2.1.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
plus#(s(x),y) -> c_6(plus#(x,y))
Strict TRS Rules:
Weak DP Rules:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.1.1.1.1.2.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
plus#(s(x),y) -> c_6(plus#(x,y))
Strict TRS Rules:
Weak DP Rules:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,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: plus#(s(x),y) -> c_6(plus#(x,y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.1.1.1.1.2.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
plus#(s(x),y) -> c_6(plus#(x,y))
Strict TRS Rules:
Weak DP Rules:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,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_6) = {1}
Following symbols are considered usable:
{minus#,plus#,quot#}
TcT has computed the following interpretation:
p(0) = [1]
p(minus) = [1]
p(plus) = [1] x2 + [2]
p(quot) = [0]
p(s) = [1] x1 + [4]
p(minus#) = [0]
p(plus#) = [2] x1 + [8]
p(quot#) = [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [1]
p(c_6) = [1] x1 + [3]
p(c_7) = [0]
p(c_8) = [0]
Following rules are strictly oriented:
plus#(s(x),y) = [2] x + [16]
> [2] x + [11]
= c_6(plus#(x,y))
Following rules are (at-least) weakly oriented:
plus#(minus(x,s(0())) = [10]
,minus(y,s(s(z))))
>= [0]
= c_4(plus#(minus(y,s(s(z)))
,minus(x,s(0()))))
plus#(plus(x,s(0())) = [22]
,plus(y,s(s(z))))
>= [1]
= c_5(plus#(plus(y,s(s(z)))
,plus(x,s(0()))))
*** 1.1.1.1.1.1.2.1.1.1.1.2.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.1.1.1.1.2.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):3
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0())))):2
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0())))):1
2:W:plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):3
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0())))):2
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0())))):1
3:W:plus#(s(x),y) -> c_6(plus#(x,y))
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):3
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0())))):2
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0())))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: plus#(minus(x,s(0()))
,minus(y,s(s(z)))) ->
c_4(plus#(minus(y,s(s(z)))
,minus(x,s(0()))))
3: plus#(s(x),y) -> c_6(plus#(x,y))
2: plus#(plus(x,s(0()))
,plus(y,s(s(z)))) ->
c_5(plus#(plus(y,s(s(z)))
,plus(x,s(0()))))
*** 1.1.1.1.1.1.2.1.1.1.1.2.2.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.2.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)))
Strict TRS Rules:
Weak DP Rules:
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
plus#(s(x),y) -> c_6(plus#(x,y))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)))
-->_1 quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y))):1
2:W:plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):4
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0())))):3
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0())))):2
3:W:plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0()))))
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):4
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0())))):3
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0())))):2
4:W:plus#(s(x),y) -> c_6(plus#(x,y))
-->_1 plus#(s(x),y) -> c_6(plus#(x,y)):4
-->_1 plus#(plus(x,s(0())),plus(y,s(s(z)))) -> c_5(plus#(plus(y,s(s(z))),plus(x,s(0())))):3
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0())))):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: plus#(minus(x,s(0()))
,minus(y,s(s(z)))) ->
c_4(plus#(minus(y,s(s(z)))
,minus(x,s(0()))))
4: plus#(s(x),y) -> c_6(plus#(x,y))
3: plus#(plus(x,s(0()))
,plus(y,s(s(z)))) ->
c_5(plus#(plus(y,s(s(z)))
,plus(x,s(0()))))
*** 1.1.1.1.1.1.2.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)))
*** 1.1.1.1.1.1.2.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,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: quot#(s(x),s(y)) ->
c_8(quot#(minus(x,y),s(y)))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,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:
{minus,minus#,plus#,quot#}
TcT has computed the following interpretation:
p(0) = [0]
p(minus) = [1] x1 + [0]
p(plus) = [1] x1 + [2]
p(quot) = [2] x1 + [1] x2 + [0]
p(s) = [1] x1 + [4]
p(minus#) = [1] x2 + [0]
p(plus#) = [2] x1 + [1] x2 + [0]
p(quot#) = [4] x1 + [2] x2 + [0]
p(c_1) = [1]
p(c_2) = [2]
p(c_3) = [8]
p(c_4) = [1]
p(c_5) = [0]
p(c_6) = [1]
p(c_7) = [1]
p(c_8) = [1] x1 + [3]
Following rules are strictly oriented:
quot#(s(x),s(y)) = [4] x + [2] y + [24]
> [4] x + [2] y + [11]
= c_8(quot#(minus(x,y),s(y)))
Following rules are (at-least) weakly oriented:
minus(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
minus(s(x),s(y)) = [1] x + [4]
>= [1] x + [0]
= minus(x,y)
*** 1.1.1.1.1.1.2.1.1.2.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.1.1.2.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)))
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y)))
-->_1 quot#(s(x),s(y)) -> c_8(quot#(minus(x,y),s(y))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: quot#(s(x),s(y)) ->
c_8(quot#(minus(x,y),s(y)))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {minus#,plus#,quot#}/{0,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).