*** 1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
gcd(0(),y) -> y
gcd(s(x),0()) -> s(x)
gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
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)
Weak DP Rules:
Weak TRS Rules:
Signature:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {gcd,if_gcd,if_minus,le,minus}/{0,false,s,true}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
We add the following dependency tuples:
Strict DPs
gcd#(0(),y) -> c_1()
gcd#(s(x),0()) -> c_2()
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
if_minus#(false(),s(x),y) -> c_6(minus#(x,y))
if_minus#(true(),s(x),y) -> c_7()
le#(0(),y) -> c_8()
le#(s(x),0()) -> c_9()
le#(s(x),s(y)) -> c_10(le#(x,y))
minus#(0(),y) -> c_11()
minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
Weak DPs
and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
gcd#(0(),y) -> c_1()
gcd#(s(x),0()) -> c_2()
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
if_minus#(false(),s(x),y) -> c_6(minus#(x,y))
if_minus#(true(),s(x),y) -> c_7()
le#(0(),y) -> c_8()
le#(s(x),0()) -> c_9()
le#(s(x),s(y)) -> c_10(le#(x,y))
minus#(0(),y) -> c_11()
minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
gcd(0(),y) -> y
gcd(s(x),0()) -> s(x)
gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{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)
gcd#(0(),y) -> c_1()
gcd#(s(x),0()) -> c_2()
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
if_minus#(false(),s(x),y) -> c_6(minus#(x,y))
if_minus#(true(),s(x),y) -> c_7()
le#(0(),y) -> c_8()
le#(s(x),0()) -> c_9()
le#(s(x),s(y)) -> c_10(le#(x,y))
minus#(0(),y) -> c_11()
minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
*** 1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
gcd#(0(),y) -> c_1()
gcd#(s(x),0()) -> c_2()
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
if_minus#(false(),s(x),y) -> c_6(minus#(x,y))
if_minus#(true(),s(x),y) -> c_7()
le#(0(),y) -> c_8()
le#(s(x),0()) -> c_9()
le#(s(x),s(y)) -> c_10(le#(x,y))
minus#(0(),y) -> c_11()
minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{0,false,s,true}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{1,2,7,8,9,11}
by application of
Pre({1,2,7,8,9,11}) = {3,4,5,6,10,12}.
Here rules are labelled as follows:
1: gcd#(0(),y) -> c_1()
2: gcd#(s(x),0()) -> c_2()
3: gcd#(s(x),s(y)) ->
c_3(if_gcd#(le(y,x),s(x),s(y))
,le#(y,x))
4: if_gcd#(false(),s(x),s(y)) ->
c_4(gcd#(minus(y,x),s(x))
,minus#(y,x))
5: if_gcd#(true(),s(x),s(y)) ->
c_5(gcd#(minus(x,y),s(y))
,minus#(x,y))
6: if_minus#(false(),s(x),y) ->
c_6(minus#(x,y))
7: if_minus#(true(),s(x),y) ->
c_7()
8: le#(0(),y) -> c_8()
9: le#(s(x),0()) -> c_9()
10: le#(s(x),s(y)) -> c_10(le#(x,y))
11: minus#(0(),y) -> c_11()
12: minus#(s(x),y) ->
c_12(if_minus#(le(s(x),y)
,s(x)
,y)
,le#(s(x),y))
*** 1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
if_minus#(false(),s(x),y) -> c_6(minus#(x,y))
le#(s(x),s(y)) -> c_10(le#(x,y))
minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
Strict TRS Rules:
Weak DP Rules:
gcd#(0(),y) -> c_1()
gcd#(s(x),0()) -> c_2()
if_minus#(true(),s(x),y) -> c_7()
le#(0(),y) -> c_8()
le#(s(x),0()) -> c_9()
minus#(0(),y) -> 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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{0,false,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
-->_2 le#(s(x),s(y)) -> c_10(le#(x,y)):5
-->_1 if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)):3
-->_1 if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)):2
-->_2 le#(s(x),0()) -> c_9():11
-->_2 le#(0(),y) -> c_8():10
2:S:if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
-->_2 minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):6
-->_2 minus#(0(),y) -> c_11():12
-->_1 gcd#(0(),y) -> c_1():7
-->_1 gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)):1
3:S:if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
-->_2 minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):6
-->_2 minus#(0(),y) -> c_11():12
-->_1 gcd#(0(),y) -> c_1():7
-->_1 gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)):1
4:S:if_minus#(false(),s(x),y) -> c_6(minus#(x,y))
-->_1 minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):6
-->_1 minus#(0(),y) -> c_11():12
5:S:le#(s(x),s(y)) -> c_10(le#(x,y))
-->_1 le#(s(x),0()) -> c_9():11
-->_1 le#(0(),y) -> c_8():10
-->_1 le#(s(x),s(y)) -> c_10(le#(x,y)):5
6:S:minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
-->_2 le#(s(x),0()) -> c_9():11
-->_1 if_minus#(true(),s(x),y) -> c_7():9
-->_2 le#(s(x),s(y)) -> c_10(le#(x,y)):5
-->_1 if_minus#(false(),s(x),y) -> c_6(minus#(x,y)):4
7:W:gcd#(0(),y) -> c_1()
8:W:gcd#(s(x),0()) -> c_2()
9:W:if_minus#(true(),s(x),y) -> c_7()
10:W:le#(0(),y) -> c_8()
11:W:le#(s(x),0()) -> c_9()
12:W:minus#(0(),y) -> c_11()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
8: gcd#(s(x),0()) -> c_2()
7: gcd#(0(),y) -> c_1()
12: minus#(0(),y) -> c_11()
9: if_minus#(true(),s(x),y) ->
c_7()
10: le#(0(),y) -> c_8()
11: le#(s(x),0()) -> c_9()
*** 1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
if_minus#(false(),s(x),y) -> c_6(minus#(x,y))
le#(s(x),s(y)) -> c_10(le#(x,y))
minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{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
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
and a lower component
if_minus#(false(),s(x),y) -> c_6(minus#(x,y))
le#(s(x),s(y)) -> c_10(le#(x,y))
minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
Further, following extension rules are added to the lower component.
gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
gcd#(s(x),s(y)) -> le#(y,x)
if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
*** 1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{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:
2: if_gcd#(false(),s(x),s(y)) ->
c_4(gcd#(minus(y,x),s(x))
,minus#(y,x))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{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_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1}
Following symbols are considered usable:
{if_minus,minus,gcd#,if_gcd#,if_minus#,le#,minus#}
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [0]
p(gcd) = [2] x2 + [0]
p(if_gcd) = [2] x1 + [1] x2 + [1] x3 + [8]
p(if_minus) = [1] x2 + [0]
p(le) = [0]
p(minus) = [1] x1 + [0]
p(s) = [1] x1 + [2]
p(true) = [0]
p(gcd#) = [4] x1 + [4] x2 + [12]
p(if_gcd#) = [4] x2 + [4] x3 + [8]
p(if_minus#) = [1] x2 + [1]
p(le#) = [1] x1 + [0]
p(minus#) = [8] x2 + [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [1] x1 + [4]
p(c_4) = [1] x1 + [0]
p(c_5) = [1] x1 + [4]
p(c_6) = [1] x1 + [2]
p(c_7) = [4]
p(c_8) = [2]
p(c_9) = [8]
p(c_10) = [1] x1 + [0]
p(c_11) = [0]
p(c_12) = [2] x1 + [2]
Following rules are strictly oriented:
if_gcd#(false(),s(x),s(y)) = [4] x + [4] y + [24]
> [4] x + [4] y + [20]
= c_4(gcd#(minus(y,x),s(x))
,minus#(y,x))
Following rules are (at-least) weakly oriented:
gcd#(s(x),s(y)) = [4] x + [4] y + [28]
>= [4] x + [4] y + [28]
= c_3(if_gcd#(le(y,x),s(x),s(y))
,le#(y,x))
if_gcd#(true(),s(x),s(y)) = [4] x + [4] y + [24]
>= [4] x + [4] y + [24]
= c_5(gcd#(minus(x,y),s(y))
,minus#(x,y))
if_minus(false(),s(x),y) = [1] x + [2]
>= [1] x + [2]
= s(minus(x,y))
if_minus(true(),s(x),y) = [1] x + [2]
>= [0]
= 0()
minus(0(),y) = [0]
>= [0]
= 0()
minus(s(x),y) = [1] x + [2]
>= [1] x + [2]
= if_minus(le(s(x),y),s(x),y)
*** 1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{0,false,s,true}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{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:
2: if_gcd#(true(),s(x),s(y)) ->
c_5(gcd#(minus(x,y),s(y))
,minus#(x,y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{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_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1}
Following symbols are considered usable:
{if_minus,minus,gcd#,if_gcd#,if_minus#,le#,minus#}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[1]
p(false) = [0]
[0]
[0]
p(gcd) = [0]
[0]
[0]
p(if_gcd) = [0]
[0]
[0]
p(if_minus) = [0 1 0] [0]
[0 1 0] x2 + [0]
[0 0 0] [1]
p(le) = [0 0 1] [0]
[0 0 0] x2 + [0]
[0 0 0] [1]
p(minus) = [0 1 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(s) = [0 1 0] [0]
[0 1 1] x1 + [0]
[0 0 0] [1]
p(true) = [0]
[0]
[0]
p(gcd#) = [1 1 1] [1 1 0] [1]
[0 0 0] x1 + [0 0 1] x2 + [1]
[1 0 1] [0 0 0] [0]
p(if_gcd#) = [1 1 0] [1 1 1] [1]
[0 1 0] x2 + [1 1 1] x3 + [1]
[1 1 1] [1 1 1] [0]
p(if_minus#) = [0]
[0]
[0]
p(le#) = [0]
[0]
[0]
p(minus#) = [0 1 0] [0 0 0] [1]
[0 0 0] x1 + [0 0 1] x2 + [0]
[0 1 0] [0 1 0] [0]
p(c_1) = [0]
[0]
[0]
p(c_2) = [0]
[0]
[0]
p(c_3) = [1 0 0] [0]
[0 0 0] x1 + [1]
[0 0 0] [0]
p(c_4) = [1 0 0] [0 0 0] [0]
[0 0 1] x1 + [1 1 0] x2 + [0]
[0 1 1] [0 1 0] [0]
p(c_5) = [1 0 0] [0 0 0] [0]
[0 1 0] x1 + [0 1 1] x2 + [0]
[0 0 0] [1 1 1] [1]
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) = [0]
[0]
[0]
Following rules are strictly oriented:
if_gcd#(true(),s(x),s(y)) = [0 2 1] [0 2 1] [2]
[0 1 1] x + [0 2 1] y + [2]
[0 2 1] [0 2 1] [2]
> [0 2 1] [0 2 1] [1]
[0 1 0] x + [0 1 1] y + [2]
[0 2 0] [0 1 1] [2]
= c_5(gcd#(minus(x,y),s(y))
,minus#(x,y))
Following rules are (at-least) weakly oriented:
gcd#(s(x),s(y)) = [0 2 1] [0 2 1] [2]
[0 0 0] x + [0 0 0] y + [2]
[0 1 0] [0 0 0] [1]
>= [0 2 1] [0 2 1] [2]
[0 0 0] x + [0 0 0] y + [1]
[0 0 0] [0 0 0] [0]
= c_3(if_gcd#(le(y,x),s(x),s(y))
,le#(y,x))
if_gcd#(false(),s(x),s(y)) = [0 2 1] [0 2 1] [2]
[0 1 1] x + [0 2 1] y + [2]
[0 2 1] [0 2 1] [2]
>= [0 2 1] [0 2 1] [1]
[0 0 1] x + [0 2 1] y + [1]
[0 0 1] [0 1 1] [2]
= c_4(gcd#(minus(y,x),s(x))
,minus#(y,x))
if_minus(false(),s(x),y) = [0 1 1] [0]
[0 1 1] x + [0]
[0 0 0] [1]
>= [0 1 0] [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)
*** 1.1.1.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
Strict TRS Rules:
Weak DP Rules:
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{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:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
Strict TRS Rules:
Weak DP Rules:
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{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: gcd#(s(x),s(y)) ->
c_3(if_gcd#(le(y,x),s(x),s(y))
,le#(y,x))
Consider the set of all dependency pairs
1: gcd#(s(x),s(y)) ->
c_3(if_gcd#(le(y,x),s(x),s(y))
,le#(y,x))
2: if_gcd#(false(),s(x),s(y)) ->
c_4(gcd#(minus(y,x),s(x))
,minus#(y,x))
3: if_gcd#(true(),s(x),s(y)) ->
c_5(gcd#(minus(x,y),s(y))
,minus#(x,y))
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
{1,2,3}
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 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
Strict TRS Rules:
Weak DP Rules:
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{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_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1}
Following symbols are considered usable:
{if_minus,minus,gcd#,if_gcd#,if_minus#,le#,minus#}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(false) = [0]
[0]
[0]
p(gcd) = [0]
[0]
[0]
p(if_gcd) = [0]
[0]
[0]
p(if_minus) = [1 0 1] [0]
[0 1 0] x2 + [0]
[0 0 0] [1]
p(le) = [0]
[0]
[0]
p(minus) = [0 1 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [1]
p(s) = [0 1 0] [0]
[0 1 0] x1 + [1]
[0 0 0] [1]
p(true) = [0]
[0]
[0]
p(gcd#) = [1 1 0] [1 1 1] [0]
[0 0 1] x1 + [0 0 1] x2 + [0]
[0 0 0] [1 0 0] [1]
p(if_gcd#) = [1 1 0] [1 1 0] [0]
[0 1 0] x2 + [1 0 0] x3 + [0]
[1 1 0] [1 1 1] [0]
p(if_minus#) = [0]
[0]
[0]
p(le#) = [0]
[0]
[0]
p(minus#) = [0]
[1]
[0]
p(c_1) = [0]
[0]
[0]
p(c_2) = [0]
[0]
[0]
p(c_3) = [1 0 0] [0]
[0 0 0] x1 + [1]
[0 0 0] [1]
p(c_4) = [1 0 0] [0]
[0 0 1] x1 + [0]
[1 0 0] [0]
p(c_5) = [1 0 0] [0 0 0] [0]
[0 0 0] x1 + [0 1 0] x2 + [0]
[1 0 0] [0 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) = [0]
[0]
[0]
Following rules are strictly oriented:
gcd#(s(x),s(y)) = [0 2 0] [0 2 0] [3]
[0 0 0] x + [0 0 0] y + [2]
[0 0 0] [0 1 0] [1]
> [0 2 0] [0 2 0] [2]
[0 0 0] x + [0 0 0] y + [1]
[0 0 0] [0 0 0] [1]
= c_3(if_gcd#(le(y,x),s(x),s(y))
,le#(y,x))
Following rules are (at-least) weakly oriented:
if_gcd#(false(),s(x),s(y)) = [0 2 0] [0 2 0] [2]
[0 1 0] x + [0 1 0] y + [1]
[0 2 0] [0 2 0] [3]
>= [0 2 0] [0 2 0] [2]
[0 1 0] x + [0 0 0] y + [1]
[0 2 0] [0 2 0] [2]
= c_4(gcd#(minus(y,x),s(x))
,minus#(y,x))
if_gcd#(true(),s(x),s(y)) = [0 2 0] [0 2 0] [2]
[0 1 0] x + [0 1 0] y + [1]
[0 2 0] [0 2 0] [3]
>= [0 2 0] [0 2 0] [2]
[0 0 0] x + [0 0 0] y + [1]
[0 2 0] [0 2 0] [2]
= c_5(gcd#(minus(x,y),s(y))
,minus#(x,y))
if_minus(false(),s(x),y) = [0 1 0] [1]
[0 1 0] x + [1]
[0 0 0] [1]
>= [0 1 0] [0]
[0 1 0] x + [1]
[0 0 0] [1]
= s(minus(x,y))
if_minus(true(),s(x),y) = [0 1 0] [1]
[0 1 0] x + [1]
[0 0 0] [1]
>= [0]
[0]
[0]
= 0()
minus(0(),y) = [0]
[0]
[1]
>= [0]
[0]
[0]
= 0()
minus(s(x),y) = [0 1 0] [1]
[0 1 0] x + [1]
[0 0 0] [2]
>= [0 1 0] [1]
[0 1 0] x + [1]
[0 0 0] [1]
= if_minus(le(s(x),y),s(x),y)
*** 1.1.1.1.1.1.2.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{0,false,s,true}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{0,false,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x))
-->_1 if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)):3
-->_1 if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)):2
2:W:if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x))
-->_1 gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)):1
3:W:if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y))
-->_1 gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: gcd#(s(x),s(y)) ->
c_3(if_gcd#(le(y,x),s(x),s(y))
,le#(y,x))
3: if_gcd#(true(),s(x),s(y)) ->
c_5(gcd#(minus(x,y),s(y))
,minus#(x,y))
2: if_gcd#(false(),s(x),s(y)) ->
c_4(gcd#(minus(y,x),s(x))
,minus#(y,x))
*** 1.1.1.1.1.1.2.2.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{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^2))] ***
Considered Problem:
Strict DP Rules:
if_minus#(false(),s(x),y) -> c_6(minus#(x,y))
le#(s(x),s(y)) -> c_10(le#(x,y))
minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
Strict TRS Rules:
Weak DP Rules:
gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
gcd#(s(x),s(y)) -> le#(y,x)
if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
if_gcd#(true(),s(x),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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{0,false,s,true}
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:
2: le#(s(x),s(y)) -> c_10(le#(x,y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
if_minus#(false(),s(x),y) -> c_6(minus#(x,y))
le#(s(x),s(y)) -> c_10(le#(x,y))
minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
Strict TRS Rules:
Weak DP Rules:
gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
gcd#(s(x),s(y)) -> le#(y,x)
if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
if_gcd#(true(),s(x),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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{0,false,s,true}
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_6) = {1},
uargs(c_10) = {1},
uargs(c_12) = {1,2}
Following symbols are considered usable:
{if_minus,minus,gcd#,if_gcd#,if_minus#,le#,minus#}
TcT has computed the following interpretation:
p(0) = [0]
[0]
p(false) = [0]
[0]
p(gcd) = [1 0] x1 + [2]
[0 0] [2]
p(if_gcd) = [0 1] x2 + [0 0] x3 + [0]
[2 1] [0 2] [0]
p(if_minus) = [1 0] x2 + [0]
[0 1] [0]
p(le) = [0]
[0]
p(minus) = [1 0] x1 + [0]
[0 1] [0]
p(s) = [1 2] x1 + [0]
[0 1] [1]
p(true) = [0]
[0]
p(gcd#) = [2 2] x1 + [2 0] x2 + [2]
[0 1] [0 1] [3]
p(if_gcd#) = [2 1] x2 + [2 0] x3 + [3]
[0 1] [0 1] [3]
p(if_minus#) = [2 0] x2 + [2 2] x3 + [0]
[1 1] [2 0] [0]
p(le#) = [0 1] x1 + [0]
[0 0] [0]
p(minus#) = [2 2] x1 + [2 2] x2 + [0]
[0 1] [0 0] [0]
p(c_1) = [1]
[0]
p(c_2) = [1]
[2]
p(c_3) = [0 1] x2 + [0]
[2 2] [0]
p(c_4) = [0]
[0]
p(c_5) = [0 0] x2 + [1]
[0 2] [0]
p(c_6) = [1 2] x1 + [0]
[0 0] [0]
p(c_7) = [0]
[0]
p(c_8) = [1]
[0]
p(c_9) = [2]
[0]
p(c_10) = [1 0] x1 + [0]
[0 0] [0]
p(c_11) = [0]
[2]
p(c_12) = [1 0] x1 + [2 0] x2 + [0]
[0 0] [0 0] [1]
Following rules are strictly oriented:
le#(s(x),s(y)) = [0 1] x + [1]
[0 0] [0]
> [0 1] x + [0]
[0 0] [0]
= c_10(le#(x,y))
Following rules are (at-least) weakly oriented:
gcd#(s(x),s(y)) = [2 6] x + [2 4] y + [4]
[0 1] [0 1] [5]
>= [2 5] x + [2 4] y + [4]
[0 1] [0 1] [5]
= if_gcd#(le(y,x),s(x),s(y))
gcd#(s(x),s(y)) = [2 6] x + [2 4] y + [4]
[0 1] [0 1] [5]
>= [0 1] y + [0]
[0 0] [0]
= le#(y,x)
if_gcd#(false(),s(x),s(y)) = [2 5] x + [2 4] y + [4]
[0 1] [0 1] [5]
>= [2 4] x + [2 2] y + [2]
[0 1] [0 1] [4]
= gcd#(minus(y,x),s(x))
if_gcd#(false(),s(x),s(y)) = [2 5] x + [2 4] y + [4]
[0 1] [0 1] [5]
>= [2 2] x + [2 2] y + [0]
[0 0] [0 1] [0]
= minus#(y,x)
if_gcd#(true(),s(x),s(y)) = [2 5] x + [2 4] y + [4]
[0 1] [0 1] [5]
>= [2 2] x + [2 4] y + [2]
[0 1] [0 1] [4]
= gcd#(minus(x,y),s(y))
if_gcd#(true(),s(x),s(y)) = [2 5] x + [2 4] y + [4]
[0 1] [0 1] [5]
>= [2 2] x + [2 2] y + [0]
[0 1] [0 0] [0]
= minus#(x,y)
if_minus#(false(),s(x),y) = [2 4] x + [2 2] y + [0]
[1 3] [2 0] [1]
>= [2 4] x + [2 2] y + [0]
[0 0] [0 0] [0]
= c_6(minus#(x,y))
minus#(s(x),y) = [2 6] x + [2 2] y + [2]
[0 1] [0 0] [1]
>= [2 6] x + [2 2] y + [2]
[0 0] [0 0] [1]
= c_12(if_minus#(le(s(x),y)
,s(x)
,y)
,le#(s(x),y))
if_minus(false(),s(x),y) = [1 2] x + [0]
[0 1] [1]
>= [1 2] x + [0]
[0 1] [1]
= s(minus(x,y))
if_minus(true(),s(x),y) = [1 2] x + [0]
[0 1] [1]
>= [0]
[0]
= 0()
minus(0(),y) = [0]
[0]
>= [0]
[0]
= 0()
minus(s(x),y) = [1 2] x + [0]
[0 1] [1]
>= [1 2] x + [0]
[0 1] [1]
= if_minus(le(s(x),y),s(x),y)
*** 1.1.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
if_minus#(false(),s(x),y) -> c_6(minus#(x,y))
minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
Strict TRS Rules:
Weak DP Rules:
gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
gcd#(s(x),s(y)) -> le#(y,x)
if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
le#(s(x),s(y)) -> c_10(le#(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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{0,false,s,true}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.2.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
if_minus#(false(),s(x),y) -> c_6(minus#(x,y))
minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
Strict TRS Rules:
Weak DP Rules:
gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
gcd#(s(x),s(y)) -> le#(y,x)
if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
le#(s(x),s(y)) -> c_10(le#(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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{0,false,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:if_minus#(false(),s(x),y) -> c_6(minus#(x,y))
-->_1 minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
2:S:minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
-->_2 le#(s(x),s(y)) -> c_10(le#(x,y)):9
-->_1 if_minus#(false(),s(x),y) -> c_6(minus#(x,y)):1
3:W:gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
-->_1 if_gcd#(true(),s(x),s(y)) -> minus#(x,y):8
-->_1 if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y)):7
-->_1 if_gcd#(false(),s(x),s(y)) -> minus#(y,x):6
-->_1 if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x)):5
4:W:gcd#(s(x),s(y)) -> le#(y,x)
-->_1 le#(s(x),s(y)) -> c_10(le#(x,y)):9
5:W:if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
-->_1 gcd#(s(x),s(y)) -> le#(y,x):4
-->_1 gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)):3
6:W:if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
-->_1 minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
7:W:if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
-->_1 gcd#(s(x),s(y)) -> le#(y,x):4
-->_1 gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)):3
8:W:if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
-->_1 minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
9:W:le#(s(x),s(y)) -> c_10(le#(x,y))
-->_1 le#(s(x),s(y)) -> c_10(le#(x,y)):9
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: gcd#(s(x),s(y)) -> le#(y,x)
9: le#(s(x),s(y)) -> c_10(le#(x,y))
*** 1.1.1.1.1.2.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
if_minus#(false(),s(x),y) -> c_6(minus#(x,y))
minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
Strict TRS Rules:
Weak DP Rules:
gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
if_gcd#(true(),s(x),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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{0,false,s,true}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:if_minus#(false(),s(x),y) -> c_6(minus#(x,y))
-->_1 minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
2:S:minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
-->_1 if_minus#(false(),s(x),y) -> c_6(minus#(x,y)):1
3:W:gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
-->_1 if_gcd#(true(),s(x),s(y)) -> minus#(x,y):8
-->_1 if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y)):7
-->_1 if_gcd#(false(),s(x),s(y)) -> minus#(y,x):6
-->_1 if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x)):5
5:W:if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
-->_1 gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)):3
6:W:if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
-->_1 minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
7:W:if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
-->_1 gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)):3
8:W:if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
-->_1 minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y))
*** 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_6(minus#(x,y))
minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y))
Strict TRS Rules:
Weak DP Rules:
gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
if_gcd#(true(),s(x),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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{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_6(minus#(x,y))
The strictly oriented rules are moved into the weak component.
*** 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_6(minus#(x,y))
minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y))
Strict TRS Rules:
Weak DP Rules:
gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
if_gcd#(true(),s(x),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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{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_6) = {1},
uargs(c_12) = {1}
Following symbols are considered usable:
{if_minus,minus,gcd#,if_gcd#,if_minus#,le#,minus#}
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [0]
p(gcd) = [1] x1 + [1] x2 + [1]
p(if_gcd) = [1] x2 + [4]
p(if_minus) = [1] x2 + [0]
p(le) = [0]
p(minus) = [1] x1 + [0]
p(s) = [1] x1 + [2]
p(true) = [2]
p(gcd#) = [1] x1 + [1] x2 + [14]
p(if_gcd#) = [1] x2 + [1] x3 + [12]
p(if_minus#) = [1] x2 + [9]
p(le#) = [1] x2 + [0]
p(minus#) = [1] x1 + [9]
p(c_1) = [4]
p(c_2) = [1]
p(c_3) = [2] x1 + [1]
p(c_4) = [1]
p(c_5) = [1] x1 + [1] x2 + [2]
p(c_6) = [1] x1 + [0]
p(c_7) = [1]
p(c_8) = [8]
p(c_9) = [0]
p(c_10) = [2]
p(c_11) = [0]
p(c_12) = [1] x1 + [0]
Following rules are strictly oriented:
if_minus#(false(),s(x),y) = [1] x + [11]
> [1] x + [9]
= c_6(minus#(x,y))
Following rules are (at-least) weakly oriented:
gcd#(s(x),s(y)) = [1] x + [1] y + [18]
>= [1] x + [1] y + [16]
= if_gcd#(le(y,x),s(x),s(y))
if_gcd#(false(),s(x),s(y)) = [1] x + [1] y + [16]
>= [1] x + [1] y + [16]
= gcd#(minus(y,x),s(x))
if_gcd#(false(),s(x),s(y)) = [1] x + [1] y + [16]
>= [1] y + [9]
= minus#(y,x)
if_gcd#(true(),s(x),s(y)) = [1] x + [1] y + [16]
>= [1] x + [1] y + [16]
= gcd#(minus(x,y),s(y))
if_gcd#(true(),s(x),s(y)) = [1] x + [1] y + [16]
>= [1] x + [9]
= minus#(x,y)
minus#(s(x),y) = [1] x + [11]
>= [1] x + [11]
= c_12(if_minus#(le(s(x),y)
,s(x)
,y))
if_minus(false(),s(x),y) = [1] x + [2]
>= [1] x + [2]
= s(minus(x,y))
if_minus(true(),s(x),y) = [1] x + [2]
>= [0]
= 0()
minus(0(),y) = [0]
>= [0]
= 0()
minus(s(x),y) = [1] x + [2]
>= [1] x + [2]
= if_minus(le(s(x),y),s(x),y)
*** 1.1.1.1.1.2.2.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y))
Strict TRS Rules:
Weak DP Rules:
gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
if_minus#(false(),s(x),y) -> c_6(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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{0,false,s,true}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.2.2.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y))
Strict TRS Rules:
Weak DP Rules:
gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
if_minus#(false(),s(x),y) -> c_6(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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{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_12(if_minus#(le(s(x),y)
,s(x)
,y))
Consider the set of all dependency pairs
1: minus#(s(x),y) ->
c_12(if_minus#(le(s(x),y)
,s(x)
,y))
2: gcd#(s(x),s(y)) -> if_gcd#(le(y
,x)
,s(x)
,s(y))
3: if_gcd#(false(),s(x),s(y)) ->
gcd#(minus(y,x),s(x))
4: if_gcd#(false(),s(x),s(y)) ->
minus#(y,x)
5: if_gcd#(true(),s(x),s(y)) ->
gcd#(minus(x,y),s(y))
6: if_gcd#(true(),s(x),s(y)) ->
minus#(x,y)
7: if_minus#(false(),s(x),y) ->
c_6(minus#(x,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
{1,7}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
*** 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_12(if_minus#(le(s(x),y),s(x),y))
Strict TRS Rules:
Weak DP Rules:
gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
if_minus#(false(),s(x),y) -> c_6(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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{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_6) = {1},
uargs(c_12) = {1}
Following symbols are considered usable:
{if_minus,minus,gcd#,if_gcd#,if_minus#,le#,minus#}
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [0]
p(gcd) = [1] x1 + [1] x2 + [2]
p(if_gcd) = [2] x3 + [1]
p(if_minus) = [1] x2 + [0]
p(le) = [0]
p(minus) = [1] x1 + [0]
p(s) = [1] x1 + [1]
p(true) = [0]
p(gcd#) = [1] x1 + [1] x2 + [5]
p(if_gcd#) = [1] x2 + [1] x3 + [5]
p(if_minus#) = [1] x2 + [1] x3 + [0]
p(le#) = [2] x2 + [0]
p(minus#) = [1] x1 + [1] x2 + [1]
p(c_1) = [2]
p(c_2) = [0]
p(c_3) = [2] x1 + [8] x2 + [1]
p(c_4) = [2] x1 + [2] x2 + [8]
p(c_5) = [1] x2 + [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1]
p(c_8) = [2]
p(c_9) = [1]
p(c_10) = [0]
p(c_11) = [4]
p(c_12) = [1] x1 + [0]
Following rules are strictly oriented:
minus#(s(x),y) = [1] x + [1] y + [2]
> [1] x + [1] y + [1]
= c_12(if_minus#(le(s(x),y)
,s(x)
,y))
Following rules are (at-least) weakly oriented:
gcd#(s(x),s(y)) = [1] x + [1] y + [7]
>= [1] x + [1] y + [7]
= if_gcd#(le(y,x),s(x),s(y))
if_gcd#(false(),s(x),s(y)) = [1] x + [1] y + [7]
>= [1] x + [1] y + [6]
= gcd#(minus(y,x),s(x))
if_gcd#(false(),s(x),s(y)) = [1] x + [1] y + [7]
>= [1] x + [1] y + [1]
= minus#(y,x)
if_gcd#(true(),s(x),s(y)) = [1] x + [1] y + [7]
>= [1] x + [1] y + [6]
= gcd#(minus(x,y),s(y))
if_gcd#(true(),s(x),s(y)) = [1] x + [1] y + [7]
>= [1] x + [1] y + [1]
= minus#(x,y)
if_minus#(false(),s(x),y) = [1] x + [1] y + [1]
>= [1] x + [1] y + [1]
= c_6(minus#(x,y))
if_minus(false(),s(x),y) = [1] x + [1]
>= [1] x + [1]
= s(minus(x,y))
if_minus(true(),s(x),y) = [1] x + [1]
>= [0]
= 0()
minus(0(),y) = [0]
>= [0]
= 0()
minus(s(x),y) = [1] x + [1]
>= [1] x + [1]
= if_minus(le(s(x),y),s(x),y)
*** 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:
gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
if_minus#(false(),s(x),y) -> c_6(minus#(x,y))
minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{0,false,s,true}
Applied Processor:
Assumption
Proof:
()
*** 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:
gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
if_minus#(false(),s(x),y) -> c_6(minus#(x,y))
minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{0,false,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
-->_1 if_gcd#(true(),s(x),s(y)) -> minus#(x,y):5
-->_1 if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y)):4
-->_1 if_gcd#(false(),s(x),s(y)) -> minus#(y,x):3
-->_1 if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x)):2
2:W:if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
-->_1 gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)):1
3:W:if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
-->_1 minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y)):7
4:W:if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
-->_1 gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)):1
5:W:if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
-->_1 minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y)):7
6:W:if_minus#(false(),s(x),y) -> c_6(minus#(x,y))
-->_1 minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y)):7
7:W:minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y))
-->_1 if_minus#(false(),s(x),y) -> c_6(minus#(x,y)):6
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: gcd#(s(x),s(y)) -> if_gcd#(le(y
,x)
,s(x)
,s(y))
4: if_gcd#(true(),s(x),s(y)) ->
gcd#(minus(x,y),s(y))
2: if_gcd#(false(),s(x),s(y)) ->
gcd#(minus(y,x),s(x))
3: if_gcd#(false(),s(x),s(y)) ->
minus#(y,x)
5: if_gcd#(true(),s(x),s(y)) ->
minus#(x,y)
7: minus#(s(x),y) ->
c_12(if_minus#(le(s(x),y)
,s(x)
,y))
6: if_minus#(false(),s(x),y) ->
c_6(minus#(x,y))
*** 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:
{gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1}
Obligation:
Innermost
basic terms: {gcd#,if_gcd#,if_minus#,le#,minus#}/{0,false,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).