*** 1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
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,pred/1,quot/2} / {0/0,s/1}
Obligation:
Innermost
basic terms: {minus,pred,quot}/{0,s}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
We add the following dependency tuples:
Strict DPs
minus#(x,0()) -> c_1()
minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y))
pred#(s(x)) -> c_3()
quot#(0(),s(y)) -> c_4()
quot#(s(x),s(y)) -> c_5(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#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y))
pred#(s(x)) -> c_3()
quot#(0(),s(y)) -> c_4()
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2}
Obligation:
Innermost
basic terms: {minus#,pred#,quot#}/{0,s}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
minus#(x,0()) -> c_1()
minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y))
pred#(s(x)) -> c_3()
quot#(0(),s(y)) -> c_4()
quot#(s(x),s(y)) -> c_5(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#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y))
pred#(s(x)) -> c_3()
quot#(0(),s(y)) -> c_4()
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2}
Obligation:
Innermost
basic terms: {minus#,pred#,quot#}/{0,s}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{1,3,4}
by application of
Pre({1,3,4}) = {2,5}.
Here rules are labelled as follows:
1: minus#(x,0()) -> c_1()
2: minus#(x,s(y)) ->
c_2(pred#(minus(x,y))
,minus#(x,y))
3: pred#(s(x)) -> c_3()
4: quot#(0(),s(y)) -> c_4()
5: quot#(s(x),s(y)) ->
c_5(quot#(minus(x,y),s(y))
,minus#(x,y))
*** 1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y))
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
minus#(x,0()) -> c_1()
pred#(s(x)) -> c_3()
quot#(0(),s(y)) -> c_4()
Weak TRS Rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2}
Obligation:
Innermost
basic terms: {minus#,pred#,quot#}/{0,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y))
-->_1 pred#(s(x)) -> c_3():4
-->_2 minus#(x,0()) -> c_1():3
-->_2 minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y)):1
2:S:quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(0(),s(y)) -> c_4():5
-->_2 minus#(x,0()) -> c_1():3
-->_1 quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)):2
-->_2 minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y)):1
3:W:minus#(x,0()) -> c_1()
4:W:pred#(s(x)) -> c_3()
5:W:quot#(0(),s(y)) -> c_4()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: quot#(0(),s(y)) -> c_4()
3: minus#(x,0()) -> c_1()
4: pred#(s(x)) -> c_3()
*** 1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y))
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2}
Obligation:
Innermost
basic terms: {minus#,pred#,quot#}/{0,s}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y))
-->_2 minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y)):1
2:S:quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)):2
-->_2 minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
minus#(x,s(y)) -> c_2(minus#(x,y))
*** 1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
minus#(x,s(y)) -> c_2(minus#(x,y))
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
Obligation:
Innermost
basic terms: {minus#,pred#,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#(x,s(y)) -> c_2(minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
Weak TRS Rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
Obligation:
Innermost
basic terms: {minus#,pred#,quot#}/{0,s}
Problem (S)
Strict DP Rules:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
minus#(x,s(y)) -> c_2(minus#(x,y))
Weak TRS Rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
Obligation:
Innermost
basic terms: {minus#,pred#,quot#}/{0,s}
*** 1.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
minus#(x,s(y)) -> c_2(minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
Weak TRS Rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
Obligation:
Innermost
basic terms: {minus#,pred#,quot#}/{0,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: minus#(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#(x,s(y)) -> c_2(minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
Weak TRS Rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
Obligation:
Innermost
basic terms: {minus#,pred#,quot#}/{0,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},
uargs(c_5) = {1,2}
Following symbols are considered usable:
{minus,pred,minus#,pred#,quot#}
TcT has computed the following interpretation:
p(0) = 1
p(minus) = x1
p(pred) = x1
p(quot) = 1 + x2
p(s) = 1 + x1
p(minus#) = 5*x1 + 2*x2
p(pred#) = 2 + 2*x1^2
p(quot#) = 2*x1*x2 + 4*x1^2 + 4*x2 + 2*x2^2
p(c_1) = 0
p(c_2) = x1
p(c_3) = 1
p(c_4) = 0
p(c_5) = 1 + x1 + x2
Following rules are strictly oriented:
minus#(x,s(y)) = 2 + 5*x + 2*y
> 5*x + 2*y
= c_2(minus#(x,y))
Following rules are (at-least) weakly oriented:
quot#(s(x),s(y)) = 12 + 10*x + 2*x*y + 4*x^2 + 10*y + 2*y^2
>= 7 + 7*x + 2*x*y + 4*x^2 + 10*y + 2*y^2
= c_5(quot#(minus(x,y),s(y))
,minus#(x,y))
minus(x,0()) = x
>= x
= x
minus(x,s(y)) = x
>= x
= pred(minus(x,y))
pred(s(x)) = 1 + x
>= x
= x
*** 1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
minus#(x,s(y)) -> c_2(minus#(x,y))
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
Weak TRS Rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
Obligation:
Innermost
basic terms: {minus#,pred#,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#(x,s(y)) -> c_2(minus#(x,y))
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
Weak TRS Rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
Obligation:
Innermost
basic terms: {minus#,pred#,quot#}/{0,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:minus#(x,s(y)) -> c_2(minus#(x,y))
-->_1 minus#(x,s(y)) -> c_2(minus#(x,y)):1
2:W:quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)):2
-->_2 minus#(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.
2: quot#(s(x),s(y)) ->
c_5(quot#(minus(x,y),s(y))
,minus#(x,y))
1: minus#(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(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
Obligation:
Innermost
basic terms: {minus#,pred#,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:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
minus#(x,s(y)) -> c_2(minus#(x,y))
Weak TRS Rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
Obligation:
Innermost
basic terms: {minus#,pred#,quot#}/{0,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
-->_2 minus#(x,s(y)) -> c_2(minus#(x,y)):2
-->_1 quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)):1
2:W:minus#(x,s(y)) -> c_2(minus#(x,y))
-->_1 minus#(x,s(y)) -> c_2(minus#(x,y)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: minus#(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:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
Obligation:
Innermost
basic terms: {minus#,pred#,quot#}/{0,s}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)))
*** 1.1.1.1.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {minus#,pred#,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_5(quot#(minus(x,y),s(y)))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {minus#,pred#,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_5) = {1}
Following symbols are considered usable:
{minus,pred,minus#,pred#,quot#}
TcT has computed the following interpretation:
p(0) = [0]
p(minus) = [1] x1 + [0]
p(pred) = [1] x1 + [0]
p(quot) = [2] x2 + [1]
p(s) = [1] x1 + [2]
p(minus#) = [8] x1 + [2]
p(pred#) = [2] x1 + [1]
p(quot#) = [2] x1 + [4] x2 + [3]
p(c_1) = [2]
p(c_2) = [2]
p(c_3) = [4]
p(c_4) = [0]
p(c_5) = [1] x1 + [0]
Following rules are strictly oriented:
quot#(s(x),s(y)) = [2] x + [4] y + [15]
> [2] x + [4] y + [11]
= c_5(quot#(minus(x,y),s(y)))
Following rules are (at-least) weakly oriented:
minus(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
minus(x,s(y)) = [1] x + [0]
>= [1] x + [0]
= pred(minus(x,y))
pred(s(x)) = [1] x + [2]
>= [1] x + [0]
= x
*** 1.1.1.1.1.1.2.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)))
Weak TRS Rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {minus#,pred#,quot#}/{0,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)))
Weak TRS Rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {minus#,pred#,quot#}/{0,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)))
-->_1 quot#(s(x),s(y)) -> c_5(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_5(quot#(minus(x,y),s(y)))
*** 1.1.1.1.1.1.2.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {minus#,pred#,quot#}/{0,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).