*** 1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
if_mod(false(),s(x),s(y)) -> s(x)
if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
mod(0(),y) -> 0()
mod(s(x),0()) -> 0()
mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
Weak DP Rules:
Weak TRS Rules:
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {if_minus,if_mod,le,minus,mod}/{0,false,s,true}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
We add the following dependency tuples:
Strict DPs
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
if_minus#(true(),s(x),y) -> c_2()
if_mod#(false(),s(x),s(y)) -> c_3()
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
le#(0(),y) -> c_5()
le#(s(x),0()) -> c_6()
le#(s(x),s(y)) -> c_7(le#(x,y))
minus#(0(),y) -> c_8()
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
mod#(0(),y) -> c_10()
mod#(s(x),0()) -> c_11()
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
Weak DPs
and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
if_minus#(true(),s(x),y) -> c_2()
if_mod#(false(),s(x),s(y)) -> c_3()
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
le#(0(),y) -> c_5()
le#(s(x),0()) -> c_6()
le#(s(x),s(y)) -> c_7(le#(x,y))
minus#(0(),y) -> c_8()
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
mod#(0(),y) -> c_10()
mod#(s(x),0()) -> c_11()
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
if_mod(false(),s(x),s(y)) -> s(x)
if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
mod(0(),y) -> 0()
mod(s(x),0()) -> 0()
mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
if_minus#(true(),s(x),y) -> c_2()
if_mod#(false(),s(x),s(y)) -> c_3()
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
le#(0(),y) -> c_5()
le#(s(x),0()) -> c_6()
le#(s(x),s(y)) -> c_7(le#(x,y))
minus#(0(),y) -> c_8()
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
mod#(0(),y) -> c_10()
mod#(s(x),0()) -> c_11()
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
*** 1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
if_minus#(true(),s(x),y) -> c_2()
if_mod#(false(),s(x),s(y)) -> c_3()
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
le#(0(),y) -> c_5()
le#(s(x),0()) -> c_6()
le#(s(x),s(y)) -> c_7(le#(x,y))
minus#(0(),y) -> c_8()
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
mod#(0(),y) -> c_10()
mod#(s(x),0()) -> c_11()
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{2,3,5,6,8,10,11}
by application of
Pre({2,3,5,6,8,10,11}) = {1,4,7,9,12}.
Here rules are labelled as follows:
1: if_minus#(false(),s(x),y) ->
c_1(minus#(x,y))
2: if_minus#(true(),s(x),y) ->
c_2()
3: if_mod#(false(),s(x),s(y)) ->
c_3()
4: if_mod#(true(),s(x),s(y)) ->
c_4(mod#(minus(x,y),s(y))
,minus#(x,y))
5: le#(0(),y) -> c_5()
6: le#(s(x),0()) -> c_6()
7: le#(s(x),s(y)) -> c_7(le#(x,y))
8: minus#(0(),y) -> c_8()
9: minus#(s(x),y) ->
c_9(if_minus#(le(s(x),y),s(x),y)
,le#(s(x),y))
10: mod#(0(),y) -> c_10()
11: mod#(s(x),0()) -> c_11()
12: mod#(s(x),s(y)) ->
c_12(if_mod#(le(y,x),s(x),s(y))
,le#(y,x))
*** 1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
le#(s(x),s(y)) -> c_7(le#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
Strict TRS Rules:
Weak DP Rules:
if_minus#(true(),s(x),y) -> c_2()
if_mod#(false(),s(x),s(y)) -> c_3()
le#(0(),y) -> c_5()
le#(s(x),0()) -> c_6()
minus#(0(),y) -> c_8()
mod#(0(),y) -> c_10()
mod#(s(x),0()) -> c_11()
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
-->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):4
-->_1 minus#(0(),y) -> c_8():10
2:S:if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
-->_1 mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)):5
-->_2 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):4
-->_1 mod#(0(),y) -> c_10():11
-->_2 minus#(0(),y) -> c_8():10
3:S:le#(s(x),s(y)) -> c_7(le#(x,y))
-->_1 le#(s(x),0()) -> c_6():9
-->_1 le#(0(),y) -> c_5():8
-->_1 le#(s(x),s(y)) -> c_7(le#(x,y)):3
4:S:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
-->_2 le#(s(x),0()) -> c_6():9
-->_1 if_minus#(true(),s(x),y) -> c_2():6
-->_2 le#(s(x),s(y)) -> c_7(le#(x,y)):3
-->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):1
5:S:mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
-->_2 le#(s(x),0()) -> c_6():9
-->_2 le#(0(),y) -> c_5():8
-->_1 if_mod#(false(),s(x),s(y)) -> c_3():7
-->_2 le#(s(x),s(y)) -> c_7(le#(x,y)):3
-->_1 if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y)):2
6:W:if_minus#(true(),s(x),y) -> c_2()
7:W:if_mod#(false(),s(x),s(y)) -> c_3()
8:W:le#(0(),y) -> c_5()
9:W:le#(s(x),0()) -> c_6()
10:W:minus#(0(),y) -> c_8()
11:W:mod#(0(),y) -> c_10()
12:W:mod#(s(x),0()) -> c_11()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
12: mod#(s(x),0()) -> c_11()
11: mod#(0(),y) -> c_10()
7: if_mod#(false(),s(x),s(y)) ->
c_3()
10: minus#(0(),y) -> c_8()
8: le#(0(),y) -> c_5()
6: if_minus#(true(),s(x),y) ->
c_2()
9: le#(s(x),0()) -> c_6()
*** 1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
le#(s(x),s(y)) -> c_7(le#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
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:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
le#(s(x),s(y)) -> c_7(le#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
Strict TRS Rules:
Weak DP Rules:
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Problem (S)
Strict DP Rules:
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
Strict TRS Rules:
Weak DP Rules:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
le#(s(x),s(y)) -> c_7(le#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
*** 1.1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
le#(s(x),s(y)) -> c_7(le#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
Strict TRS Rules:
Weak DP Rules:
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
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
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
and a lower component
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
le#(s(x),s(y)) -> c_7(le#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
Further, following extension rules are added to the lower component.
if_mod#(true(),s(x),s(y)) -> minus#(x,y)
if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
mod#(s(x),s(y)) -> le#(y,x)
*** 1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: if_mod#(true(),s(x),s(y)) ->
c_4(mod#(minus(x,y),s(y))
,minus#(x,y))
Consider the set of all dependency pairs
1: if_mod#(true(),s(x),s(y)) ->
c_4(mod#(minus(x,y),s(y))
,minus#(x,y))
2: mod#(s(x),s(y)) ->
c_12(if_mod#(le(y,x),s(x),s(y))
,le#(y,x))
Processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{1}
These cover all (indirect) predecessors of dependency pairs
{1,2}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
*** 1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Applied Processor:
NaturalMI {miDimension = 3, 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 (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_4) = {1},
uargs(c_12) = {1}
Following symbols are considered usable:
{if_minus,minus,if_minus#,if_mod#,le#,minus#,mod#}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[1]
p(false) = [0]
[0]
[0]
p(if_minus) = [1 0 0] [0]
[1 0 0] x2 + [0]
[0 0 0] [1]
p(if_mod) = [0]
[0]
[0]
p(le) = [0 0 0] [0]
[0 1 0] x2 + [0]
[0 0 0] [0]
p(minus) = [0 1 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(mod) = [0]
[0]
[0]
p(s) = [0 1 1] [0]
[0 1 1] x1 + [0]
[0 0 0] [1]
p(true) = [0]
[0]
[0]
p(if_minus#) = [0]
[0]
[0]
p(if_mod#) = [1 0 0] [0 0 1] [0]
[1 1 1] x2 + [0 0 1] x3 + [1]
[1 0 0] [0 0 0] [0]
p(le#) = [0 0 0] [0]
[0 0 0] x2 + [0]
[0 1 0] [1]
p(minus#) = [0 0 1] [0]
[0 0 0] x1 + [0]
[0 1 0] [0]
p(mod#) = [1 0 1] [0 0 0] [0]
[1 1 1] x1 + [0 1 0] x2 + [1]
[0 1 0] [0 0 0] [1]
p(c_1) = [0]
[0]
[0]
p(c_2) = [0]
[0]
[0]
p(c_3) = [0]
[0]
[0]
p(c_4) = [1 0 0] [0 0 0] [0]
[0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [1 0 1] [0]
p(c_5) = [0]
[0]
[0]
p(c_6) = [0]
[0]
[0]
p(c_7) = [0]
[0]
[0]
p(c_8) = [0]
[0]
[0]
p(c_9) = [0]
[0]
[0]
p(c_10) = [0]
[0]
[0]
p(c_11) = [0]
[0]
[0]
p(c_12) = [1 0 0] [0 0 0] [0]
[1 0 0] x1 + [0 0 1] x2 + [0]
[0 0 1] [0 0 0] [0]
Following rules are strictly oriented:
if_mod#(true(),s(x),s(y)) = [0 1 1] [1]
[0 2 2] x + [3]
[0 1 1] [0]
> [0 1 1] [0]
[0 0 0] x + [0]
[0 1 1] [0]
= c_4(mod#(minus(x,y),s(y))
,minus#(x,y))
Following rules are (at-least) weakly oriented:
mod#(s(x),s(y)) = [0 1 1] [0 0 0] [1]
[0 2 2] x + [0 1 1] y + [2]
[0 1 1] [0 0 0] [1]
>= [0 1 1] [1]
[0 2 1] x + [2]
[0 1 1] [0]
= c_12(if_mod#(le(y,x),s(x),s(y))
,le#(y,x))
if_minus(false(),s(x),y) = [0 1 1] [0]
[0 1 1] x + [0]
[0 0 0] [1]
>= [0 1 1] [0]
[0 1 1] x + [0]
[0 0 0] [1]
= s(minus(x,y))
if_minus(true(),s(x),y) = [0 1 1] [0]
[0 1 1] x + [0]
[0 0 0] [1]
>= [0]
[0]
[1]
= 0()
minus(0(),y) = [0]
[0]
[1]
>= [0]
[0]
[1]
= 0()
minus(s(x),y) = [0 1 1] [0]
[0 1 1] x + [0]
[0 0 0] [1]
>= [0 1 1] [0]
[0 1 1] x + [0]
[0 0 0] [1]
= if_minus(le(s(x),y),s(x),y)
*** 1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
Strict TRS Rules:
Weak DP Rules:
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
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:
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
-->_1 mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)):2
2:W:mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
-->_1 if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: if_mod#(true(),s(x),s(y)) ->
c_4(mod#(minus(x,y),s(y))
,minus#(x,y))
2: mod#(s(x),s(y)) ->
c_12(if_mod#(le(y,x),s(x),s(y))
,le#(y,x))
*** 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:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
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:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
le#(s(x),s(y)) -> c_7(le#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
Strict TRS Rules:
Weak DP Rules:
if_mod#(true(),s(x),s(y)) -> minus#(x,y)
if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
mod#(s(x),s(y)) -> le#(y,x)
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
2: le#(s(x),s(y)) -> c_7(le#(x,y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
le#(s(x),s(y)) -> c_7(le#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
Strict TRS Rules:
Weak DP Rules:
if_mod#(true(),s(x),s(y)) -> minus#(x,y)
if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
mod#(s(x),s(y)) -> le#(y,x)
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Applied Processor:
NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_1) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1,2}
Following symbols are considered usable:
{if_minus,minus,if_minus#,if_mod#,le#,minus#,mod#}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(false) = [0]
[0]
[0]
p(if_minus) = [0 1 0] [0]
[1 0 0] x2 + [0]
[0 0 1] [0]
p(if_mod) = [0]
[0]
[0]
p(le) = [1 0 0] [0]
[0 1 1] x1 + [1]
[0 0 0] [0]
p(minus) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(mod) = [0]
[0]
[0]
p(s) = [0 1 1] [0]
[0 1 1] x1 + [0]
[0 0 1] [1]
p(true) = [0]
[0]
[0]
p(if_minus#) = [1 0 0] [0 0 0] [0]
[0 0 0] x2 + [0 0 1] x3 + [0]
[1 0 0] [0 1 0] [1]
p(if_mod#) = [0 1 1] [0 0 1] [0]
[0 0 0] x2 + [1 0 0] x3 + [0]
[1 0 0] [0 0 1] [0]
p(le#) = [0 0 1] [0 0 0] [0]
[0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 1 1] [0]
p(minus#) = [0 1 1] [0 0 0] [0]
[0 0 0] x1 + [0 0 1] x2 + [0]
[0 0 1] [0 0 1] [1]
p(mod#) = [0 1 1] [0 0 1] [1]
[0 0 0] x1 + [0 1 0] x2 + [0]
[0 1 0] [0 0 1] [0]
p(c_1) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 0] [0]
p(c_2) = [0]
[0]
[0]
p(c_3) = [0]
[0]
[0]
p(c_4) = [0]
[0]
[0]
p(c_5) = [0]
[0]
[0]
p(c_6) = [0]
[0]
[0]
p(c_7) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 1] [1]
p(c_8) = [0]
[0]
[0]
p(c_9) = [1 0 0] [1 0 0] [0]
[0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [1 0 0] [0]
p(c_10) = [0]
[0]
[0]
p(c_11) = [0]
[0]
[0]
p(c_12) = [0]
[0]
[0]
Following rules are strictly oriented:
le#(s(x),s(y)) = [0 0 1] [0 0 0] [1]
[0 0 0] x + [0 0 0] y + [0]
[0 0 0] [0 1 2] [1]
> [0 0 1] [0 0 0] [0]
[0 0 0] x + [0 0 0] y + [0]
[0 0 0] [0 1 1] [1]
= c_7(le#(x,y))
Following rules are (at-least) weakly oriented:
if_minus#(false(),s(x),y) = [0 1 1] [0 0 0] [0]
[0 0 0] x + [0 0 1] y + [0]
[0 1 1] [0 1 0] [1]
>= [0 1 1] [0]
[0 0 0] x + [0]
[0 0 0] [0]
= c_1(minus#(x,y))
if_mod#(true(),s(x),s(y)) = [0 1 2] [0 0 1] [2]
[0 0 0] x + [0 1 1] y + [0]
[0 1 1] [0 0 1] [1]
>= [0 1 1] [0 0 0] [0]
[0 0 0] x + [0 0 1] y + [0]
[0 0 1] [0 0 1] [1]
= minus#(x,y)
if_mod#(true(),s(x),s(y)) = [0 1 2] [0 0 1] [2]
[0 0 0] x + [0 1 1] y + [0]
[0 1 1] [0 0 1] [1]
>= [0 1 1] [0 0 1] [2]
[0 0 0] x + [0 1 1] y + [0]
[0 1 0] [0 0 1] [1]
= mod#(minus(x,y),s(y))
minus#(s(x),y) = [0 1 2] [0 0 0] [1]
[0 0 0] x + [0 0 1] y + [0]
[0 0 1] [0 0 1] [2]
>= [0 1 2] [1]
[0 0 0] x + [0]
[0 0 1] [1]
= c_9(if_minus#(le(s(x),y),s(x),y)
,le#(s(x),y))
mod#(s(x),s(y)) = [0 1 2] [0 0 1] [3]
[0 0 0] x + [0 1 1] y + [0]
[0 1 1] [0 0 1] [1]
>= [0 1 2] [0 0 1] [2]
[0 0 0] x + [0 1 1] y + [0]
[0 1 1] [0 0 1] [1]
= if_mod#(le(y,x),s(x),s(y))
mod#(s(x),s(y)) = [0 1 2] [0 0 1] [3]
[0 0 0] x + [0 1 1] y + [0]
[0 1 1] [0 0 1] [1]
>= [0 0 0] [0 0 1] [0]
[0 0 0] x + [0 0 0] y + [0]
[0 1 1] [0 0 0] [0]
= le#(y,x)
if_minus(false(),s(x),y) = [0 1 1] [0]
[0 1 1] x + [0]
[0 0 1] [1]
>= [0 1 1] [0]
[0 1 1] x + [0]
[0 0 1] [1]
= s(minus(x,y))
if_minus(true(),s(x),y) = [0 1 1] [0]
[0 1 1] x + [0]
[0 0 1] [1]
>= [0]
[0]
[0]
= 0()
minus(0(),y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
minus(s(x),y) = [0 1 1] [0]
[0 1 1] x + [0]
[0 0 1] [1]
>= [0 1 1] [0]
[0 1 1] x + [0]
[0 0 1] [1]
= if_minus(le(s(x),y),s(x),y)
*** 1.1.1.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
Strict TRS Rules:
Weak DP Rules:
if_mod#(true(),s(x),s(y)) -> minus#(x,y)
if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
le#(s(x),s(y)) -> c_7(le#(x,y))
mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
mod#(s(x),s(y)) -> le#(y,x)
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
Strict TRS Rules:
Weak DP Rules:
if_mod#(true(),s(x),s(y)) -> minus#(x,y)
if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
le#(s(x),s(y)) -> c_7(le#(x,y))
mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
mod#(s(x),s(y)) -> le#(y,x)
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
-->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
2:S:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
-->_2 le#(s(x),s(y)) -> c_7(le#(x,y)):5
-->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):1
3:W:if_mod#(true(),s(x),s(y)) -> minus#(x,y)
-->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
4:W:if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
-->_1 mod#(s(x),s(y)) -> le#(y,x):7
-->_1 mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)):6
5:W:le#(s(x),s(y)) -> c_7(le#(x,y))
-->_1 le#(s(x),s(y)) -> c_7(le#(x,y)):5
6:W:mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
-->_1 if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)):4
-->_1 if_mod#(true(),s(x),s(y)) -> minus#(x,y):3
7:W:mod#(s(x),s(y)) -> le#(y,x)
-->_1 le#(s(x),s(y)) -> c_7(le#(x,y)):5
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
7: mod#(s(x),s(y)) -> le#(y,x)
5: le#(s(x),s(y)) -> c_7(le#(x,y))
*** 1.1.1.1.1.1.2.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
Strict TRS Rules:
Weak DP Rules:
if_mod#(true(),s(x),s(y)) -> minus#(x,y)
if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
-->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
2:S:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
-->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):1
3:W:if_mod#(true(),s(x),s(y)) -> minus#(x,y)
-->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
4:W:if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
-->_1 mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)):6
6:W:mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
-->_1 if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)):4
-->_1 if_mod#(true(),s(x),s(y)) -> minus#(x,y):3
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
*** 1.1.1.1.1.1.2.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
Strict TRS Rules:
Weak DP Rules:
if_mod#(true(),s(x),s(y)) -> minus#(x,y)
if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
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: if_minus#(false(),s(x),y) ->
c_1(minus#(x,y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
Strict TRS Rules:
Weak DP Rules:
if_mod#(true(),s(x),s(y)) -> minus#(x,y)
if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
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_1) = {1},
uargs(c_9) = {1}
Following symbols are considered usable:
{if_minus,minus,if_minus#,if_mod#,le#,minus#,mod#}
TcT has computed the following interpretation:
p(0) = [2]
p(false) = [0]
p(if_minus) = [1] x2 + [1]
p(if_mod) = [2] x1 + [2]
p(le) = [0]
p(minus) = [1] x1 + [1]
p(mod) = [2]
p(s) = [1] x1 + [2]
p(true) = [0]
p(if_minus#) = [8] x2 + [1]
p(if_mod#) = [9] x2 + [1] x3 + [4]
p(le#) = [1] x1 + [1] x2 + [8]
p(minus#) = [8] x1 + [4]
p(mod#) = [9] x1 + [1] x2 + [4]
p(c_1) = [1] x1 + [12]
p(c_2) = [1]
p(c_3) = [1]
p(c_4) = [2] x2 + [1]
p(c_5) = [0]
p(c_6) = [1]
p(c_7) = [2] x1 + [1]
p(c_8) = [1]
p(c_9) = [1] x1 + [3]
p(c_10) = [4]
p(c_11) = [0]
p(c_12) = [1] x2 + [2]
Following rules are strictly oriented:
if_minus#(false(),s(x),y) = [8] x + [17]
> [8] x + [16]
= c_1(minus#(x,y))
Following rules are (at-least) weakly oriented:
if_mod#(true(),s(x),s(y)) = [9] x + [1] y + [24]
>= [8] x + [4]
= minus#(x,y)
if_mod#(true(),s(x),s(y)) = [9] x + [1] y + [24]
>= [9] x + [1] y + [15]
= mod#(minus(x,y),s(y))
minus#(s(x),y) = [8] x + [20]
>= [8] x + [20]
= c_9(if_minus#(le(s(x),y)
,s(x)
,y))
mod#(s(x),s(y)) = [9] x + [1] y + [24]
>= [9] x + [1] y + [24]
= if_mod#(le(y,x),s(x),s(y))
if_minus(false(),s(x),y) = [1] x + [3]
>= [1] x + [3]
= s(minus(x,y))
if_minus(true(),s(x),y) = [1] x + [3]
>= [2]
= 0()
minus(0(),y) = [3]
>= [2]
= 0()
minus(s(x),y) = [1] x + [3]
>= [1] x + [3]
= if_minus(le(s(x),y),s(x),y)
*** 1.1.1.1.1.1.2.2.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
Strict TRS Rules:
Weak DP Rules:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
if_mod#(true(),s(x),s(y)) -> minus#(x,y)
if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.2.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
Strict TRS Rules:
Weak DP Rules:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
if_mod#(true(),s(x),s(y)) -> minus#(x,y)
if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
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: minus#(s(x),y) ->
c_9(if_minus#(le(s(x),y)
,s(x)
,y))
Consider the set of all dependency pairs
1: minus#(s(x),y) ->
c_9(if_minus#(le(s(x),y)
,s(x)
,y))
2: if_minus#(false(),s(x),y) ->
c_1(minus#(x,y))
3: if_mod#(true(),s(x),s(y)) ->
minus#(x,y)
4: if_mod#(true(),s(x),s(y)) ->
mod#(minus(x,y),s(y))
5: mod#(s(x),s(y)) -> if_mod#(le(y
,x)
,s(x)
,s(y))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{1}
These cover all (indirect) predecessors of dependency pairs
{1,2}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
*** 1.1.1.1.1.1.2.2.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
Strict TRS Rules:
Weak DP Rules:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
if_mod#(true(),s(x),s(y)) -> minus#(x,y)
if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
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_1) = {1},
uargs(c_9) = {1}
Following symbols are considered usable:
{if_minus,minus,if_minus#,if_mod#,le#,minus#,mod#}
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [0]
p(if_minus) = [1] x2 + [2]
p(if_mod) = [1] x1 + [8] x2 + [8] x3 + [2]
p(le) = [0]
p(minus) = [1] x1 + [2]
p(mod) = [0]
p(s) = [1] x1 + [2]
p(true) = [0]
p(if_minus#) = [1] x2 + [0]
p(if_mod#) = [8] x2 + [4] x3 + [2]
p(le#) = [2] x2 + [2]
p(minus#) = [1] x1 + [1]
p(mod#) = [8] x1 + [4] x2 + [2]
p(c_1) = [1] x1 + [0]
p(c_2) = [1]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [1]
p(c_6) = [1]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [2]
p(c_12) = [2]
Following rules are strictly oriented:
minus#(s(x),y) = [1] x + [3]
> [1] x + [2]
= c_9(if_minus#(le(s(x),y)
,s(x)
,y))
Following rules are (at-least) weakly oriented:
if_minus#(false(),s(x),y) = [1] x + [2]
>= [1] x + [1]
= c_1(minus#(x,y))
if_mod#(true(),s(x),s(y)) = [8] x + [4] y + [26]
>= [1] x + [1]
= minus#(x,y)
if_mod#(true(),s(x),s(y)) = [8] x + [4] y + [26]
>= [8] x + [4] y + [26]
= mod#(minus(x,y),s(y))
mod#(s(x),s(y)) = [8] x + [4] y + [26]
>= [8] x + [4] y + [26]
= if_mod#(le(y,x),s(x),s(y))
if_minus(false(),s(x),y) = [1] x + [4]
>= [1] x + [4]
= s(minus(x,y))
if_minus(true(),s(x),y) = [1] x + [4]
>= [0]
= 0()
minus(0(),y) = [2]
>= [0]
= 0()
minus(s(x),y) = [1] x + [4]
>= [1] x + [4]
= if_minus(le(s(x),y),s(x),y)
*** 1.1.1.1.1.1.2.2.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
if_mod#(true(),s(x),s(y)) -> minus#(x,y)
if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.2.1.1.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
if_mod#(true(),s(x),s(y)) -> minus#(x,y)
if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
-->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y)):4
2:W:if_mod#(true(),s(x),s(y)) -> minus#(x,y)
-->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y)):4
3:W:if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
-->_1 mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)):5
4:W:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
-->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):1
5:W:mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
-->_1 if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)):3
-->_1 if_mod#(true(),s(x),s(y)) -> minus#(x,y):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: if_mod#(true(),s(x),s(y)) ->
mod#(minus(x,y),s(y))
5: mod#(s(x),s(y)) -> if_mod#(le(y
,x)
,s(x)
,s(y))
2: if_mod#(true(),s(x),s(y)) ->
minus#(x,y)
1: if_minus#(false(),s(x),y) ->
c_1(minus#(x,y))
4: minus#(s(x),y) ->
c_9(if_minus#(le(s(x),y)
,s(x)
,y))
*** 1.1.1.1.1.1.2.2.1.1.2.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
Strict TRS Rules:
Weak DP Rules:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
le#(s(x),s(y)) -> c_7(le#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
-->_2 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):5
-->_1 mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)):2
2:S:mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
-->_2 le#(s(x),s(y)) -> c_7(le#(x,y)):4
-->_1 if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y)):1
3:W:if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
-->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):5
4:W:le#(s(x),s(y)) -> c_7(le#(x,y))
-->_1 le#(s(x),s(y)) -> c_7(le#(x,y)):4
5:W:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
-->_2 le#(s(x),s(y)) -> c_7(le#(x,y)):4
-->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: minus#(s(x),y) ->
c_9(if_minus#(le(s(x),y),s(x),y)
,le#(s(x),y))
3: if_minus#(false(),s(x),y) ->
c_1(minus#(x,y))
4: le#(s(x),s(y)) -> c_7(le#(x,y))
*** 1.1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
-->_1 mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)):2
2:S:mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
-->_1 if_mod#(true(),s(x),s(y)) -> c_4(mod#(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:
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)))
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)))
*** 1.1.1.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)))
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/1}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
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: if_mod#(true(),s(x),s(y)) ->
c_4(mod#(minus(x,y),s(y)))
Consider the set of all dependency pairs
1: if_mod#(true(),s(x),s(y)) ->
c_4(mod#(minus(x,y),s(y)))
2: mod#(s(x),s(y)) ->
c_12(if_mod#(le(y,x),s(x),s(y)))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{1}
These cover all (indirect) predecessors of dependency pairs
{1,2}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
*** 1.1.1.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)))
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/1}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
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_4) = {1},
uargs(c_12) = {1}
Following symbols are considered usable:
{if_minus,minus,if_minus#,if_mod#,le#,minus#,mod#}
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [0]
p(if_minus) = [1] x2 + [0]
p(if_mod) = [1] x1 + [1] x3 + [1]
p(le) = [0]
p(minus) = [1] x1 + [0]
p(mod) = [2] x1 + [1] x2 + [1]
p(s) = [1] x1 + [4]
p(true) = [0]
p(if_minus#) = [2] x1 + [0]
p(if_mod#) = [4] x2 + [3] x3 + [0]
p(le#) = [1] x2 + [0]
p(minus#) = [0]
p(mod#) = [4] x1 + [3] x2 + [0]
p(c_1) = [1] x1 + [2]
p(c_2) = [0]
p(c_3) = [1]
p(c_4) = [1] x1 + [15]
p(c_5) = [1]
p(c_6) = [1]
p(c_7) = [8] x1 + [2]
p(c_8) = [0]
p(c_9) = [1] x2 + [0]
p(c_10) = [2]
p(c_11) = [1]
p(c_12) = [1] x1 + [0]
Following rules are strictly oriented:
if_mod#(true(),s(x),s(y)) = [4] x + [3] y + [28]
> [4] x + [3] y + [27]
= c_4(mod#(minus(x,y),s(y)))
Following rules are (at-least) weakly oriented:
mod#(s(x),s(y)) = [4] x + [3] y + [28]
>= [4] x + [3] y + [28]
= c_12(if_mod#(le(y,x),s(x),s(y)))
if_minus(false(),s(x),y) = [1] x + [4]
>= [1] x + [4]
= s(minus(x,y))
if_minus(true(),s(x),y) = [1] x + [4]
>= [0]
= 0()
minus(0(),y) = [0]
>= [0]
= 0()
minus(s(x),y) = [1] x + [4]
>= [1] x + [4]
= if_minus(le(s(x),y),s(x),y)
*** 1.1.1.1.1.2.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)))
Strict TRS Rules:
Weak DP Rules:
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)))
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/1}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Applied Processor:
Assumption
Proof:
()
*** 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:
if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)))
mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)))
Weak TRS Rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/1}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)))
-->_1 mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y))):2
2:W:mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)))
-->_1 if_mod#(true(),s(x),s(y)) -> c_4(mod#(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: if_mod#(true(),s(x),s(y)) ->
c_4(mod#(minus(x,y),s(y)))
2: mod#(s(x),s(y)) ->
c_12(if_mod#(le(y,x),s(x),s(y)))
*** 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:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
Signature:
{if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/1}
Obligation:
Innermost
basic terms: {if_minus#,if_mod#,le#,minus#,mod#}/{0,false,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).