* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
cond(true(),y) -> y
- Signature:
{admit/2,cond/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {admit,cond} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
cond(true(),y) -> y
- Signature:
{admit/2,cond/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {admit,cond} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
admit(x,u){u -> .(y,.(z,.(w(),u)))} =
admit(x,.(y,.(z,.(w(),u)))) ->^+ cond(=(sum(x,y,z),w()),.(y,.(z,.(w(),admit(carry(x,y,z),u)))))
= C[admit(carry(x,y,z),u) = admit(x,u){x -> carry(x,y,z)}]
** Step 1.b:1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
cond(true(),y) -> y
- Signature:
{admit/2,cond/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {admit,cond} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
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(.) = {2},
uargs(cond) = {2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(.) = [1] x2 + [0]
p(=) = [1] x2 + [1]
p(admit) = [0]
p(carry) = [1] x1 + [1] x2 + [1] x3 + [0]
p(cond) = [8] x1 + [1] x2 + [7]
p(nil) = [0]
p(sum) = [1] x1 + [1] x2 + [1] x3 + [0]
p(true) = [0]
p(w) = [0]
Following rules are strictly oriented:
cond(true(),y) = [1] y + [7]
> [1] y + [0]
= y
Following rules are (at-least) weakly oriented:
admit(x,.(u,.(v,.(w(),z)))) = [0]
>= [15]
= cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) = [0]
>= [0]
= nil()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
- Weak TRS:
cond(true(),y) -> y
- Signature:
{admit/2,cond/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {admit,cond} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
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(.) = {2},
uargs(cond) = {2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(.) = [1] x2 + [0]
p(=) = [1] x2 + [7]
p(admit) = [1]
p(carry) = [1] x1 + [1] x2 + [1] x3 + [0]
p(cond) = [3] x1 + [1] x2 + [0]
p(nil) = [0]
p(sum) = [1] x1 + [1] x2 + [1] x3 + [0]
p(true) = [0]
p(w) = [0]
Following rules are strictly oriented:
admit(x,nil()) = [1]
> [0]
= nil()
Following rules are (at-least) weakly oriented:
admit(x,.(u,.(v,.(w(),z)))) = [1]
>= [22]
= cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
cond(true(),y) = [1] y + [0]
>= [1] y + [0]
= y
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
- Weak TRS:
admit(x,nil()) -> nil()
cond(true(),y) -> y
- Signature:
{admit/2,cond/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {admit,cond} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
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(.) = {2},
uargs(cond) = {2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(.) = [1] x1 + [1] x2 + [1]
p(=) = [1] x2 + [0]
p(admit) = [8] x2 + [0]
p(carry) = [1] x2 + [1] x3 + [1]
p(cond) = [8] x1 + [1] x2 + [1]
p(nil) = [2]
p(sum) = [1] x1 + [1] x2 + [0]
p(true) = [0]
p(w) = [0]
Following rules are strictly oriented:
admit(x,.(u,.(v,.(w(),z)))) = [8] u + [8] v + [8] z + [24]
> [1] u + [1] v + [8] z + [4]
= cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
Following rules are (at-least) weakly oriented:
admit(x,nil()) = [16]
>= [2]
= nil()
cond(true(),y) = [1] y + [1]
>= [1] y + [0]
= y
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
cond(true(),y) -> y
- Signature:
{admit/2,cond/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {admit,cond} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))