*** 1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
log(s(0())) -> 0()
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
min(X,0()) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
Weak DP Rules:
Weak TRS Rules:
Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
Obligation:
Innermost
basic terms: {log,min,quot}/{0,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(log) = {1},
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [4]
p(log) = [1] x1 + [0]
p(min) = [1] x1 + [4]
p(quot) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
min(X,0()) = [1] X + [4]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
log(s(0())) = [4]
>= [4]
= 0()
log(s(s(X))) = [1] X + [0]
>= [1] X + [4]
= s(log(s(quot(X,s(s(0()))))))
min(s(X),s(Y)) = [1] X + [4]
>= [1] X + [4]
= min(X,Y)
quot(0(),s(Y)) = [1] Y + [4]
>= [4]
= 0()
quot(s(X),s(Y)) = [1] X + [1] Y + [0]
>= [1] X + [1] Y + [4]
= s(quot(min(X,Y),s(Y)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
log(s(0())) -> 0()
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
Weak DP Rules:
Weak TRS Rules:
min(X,0()) -> X
Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
Obligation:
Innermost
basic terms: {log,min,quot}/{0,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(log) = {1},
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [14]
p(log) = [1] x1 + [8]
p(min) = [1] x1 + [7]
p(quot) = [1] x1 + [0]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
log(s(0())) = [22]
> [14]
= 0()
Following rules are (at-least) weakly oriented:
log(s(s(X))) = [1] X + [8]
>= [1] X + [8]
= s(log(s(quot(X,s(s(0()))))))
min(X,0()) = [1] X + [7]
>= [1] X + [0]
= X
min(s(X),s(Y)) = [1] X + [7]
>= [1] X + [7]
= min(X,Y)
quot(0(),s(Y)) = [14]
>= [14]
= 0()
quot(s(X),s(Y)) = [1] X + [0]
>= [1] X + [7]
= s(quot(min(X,Y),s(Y)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
Weak DP Rules:
Weak TRS Rules:
log(s(0())) -> 0()
min(X,0()) -> X
Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
Obligation:
Innermost
basic terms: {log,min,quot}/{0,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(log) = {1},
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [3]
p(log) = [1] x1 + [0]
p(min) = [1] x1 + [0]
p(quot) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [1]
Following rules are strictly oriented:
min(s(X),s(Y)) = [1] X + [1]
> [1] X + [0]
= min(X,Y)
quot(0(),s(Y)) = [1] Y + [4]
> [3]
= 0()
Following rules are (at-least) weakly oriented:
log(s(0())) = [4]
>= [3]
= 0()
log(s(s(X))) = [1] X + [2]
>= [1] X + [7]
= s(log(s(quot(X,s(s(0()))))))
min(X,0()) = [1] X + [0]
>= [1] X + [0]
= X
quot(s(X),s(Y)) = [1] X + [1] Y + [2]
>= [1] X + [1] Y + [2]
= s(quot(min(X,Y),s(Y)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
Weak DP Rules:
Weak TRS Rules:
log(s(0())) -> 0()
min(X,0()) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
Obligation:
Innermost
basic terms: {log,min,quot}/{0,s}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(log) = {1},
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{log,min,quot}
TcT has computed the following interpretation:
p(0) = [0]
p(log) = [4] x1 + [0]
p(min) = [1] x1 + [0]
p(quot) = [1] x1 + [0]
p(s) = [1] x1 + [2]
Following rules are strictly oriented:
log(s(s(X))) = [4] X + [16]
> [4] X + [10]
= s(log(s(quot(X,s(s(0()))))))
Following rules are (at-least) weakly oriented:
log(s(0())) = [8]
>= [0]
= 0()
min(X,0()) = [1] X + [0]
>= [1] X + [0]
= X
min(s(X),s(Y)) = [1] X + [2]
>= [1] X + [0]
= min(X,Y)
quot(0(),s(Y)) = [0]
>= [0]
= 0()
quot(s(X),s(Y)) = [1] X + [2]
>= [1] X + [2]
= s(quot(min(X,Y),s(Y)))
*** 1.1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
Weak DP Rules:
Weak TRS Rules:
log(s(0())) -> 0()
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
min(X,0()) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
Obligation:
Innermost
basic terms: {log,min,quot}/{0,s}
Applied Processor:
NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(log) = {1},
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{log,min,quot}
TcT has computed the following interpretation:
p(0) = [6]
[0]
p(log) = [2 0] x1 + [0]
[0 1] [0]
p(min) = [1 0] x1 + [0]
[0 1] [0]
p(quot) = [1 1] x1 + [0]
[0 1] [0]
p(s) = [1 2] x1 + [0]
[0 1] [1]
Following rules are strictly oriented:
quot(s(X),s(Y)) = [1 3] X + [1]
[0 1] [1]
> [1 3] X + [0]
[0 1] [1]
= s(quot(min(X,Y),s(Y)))
Following rules are (at-least) weakly oriented:
log(s(0())) = [12]
[1]
>= [6]
[0]
= 0()
log(s(s(X))) = [2 8] X + [4]
[0 1] [2]
>= [2 8] X + [2]
[0 1] [2]
= s(log(s(quot(X,s(s(0()))))))
min(X,0()) = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= X
min(s(X),s(Y)) = [1 2] X + [0]
[0 1] [1]
>= [1 0] X + [0]
[0 1] [0]
= min(X,Y)
quot(0(),s(Y)) = [6]
[0]
>= [6]
[0]
= 0()
*** 1.1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
log(s(0())) -> 0()
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
min(X,0()) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
Obligation:
Innermost
basic terms: {log,min,quot}/{0,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).