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Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270367
details
property
value
status
complete
benchmark
h02.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n168.star.cs.uiowa.edu
space
Hamana_Kikuchi_18
run statistics
property
value
solver
Wanda 2.1c
configuration
default
runtime (wallclock)
0.30069 seconds
cpu usage
0.300945
user time
0.269933
system time
0.031012
max virtual memory
113176.0
max residence set size
10084.0
stage attributes
key
value
starexec-result
YES
output
0.00/0.28 YES 0.00/0.29 We consider the system theBenchmark. 0.00/0.29 0.00/0.29 Alphabet: 0.00/0.29 0.00/0.29 0 : [] --> c 0.00/0.29 1 : [] --> c 0.00/0.29 add : [] --> c -> a -> c 0.00/0.29 cons : [] --> a -> b -> b 0.00/0.29 fold : [] --> (c -> a -> c) -> b -> c -> c 0.00/0.29 mul : [] --> c -> a -> c 0.00/0.29 nil : [] --> b 0.00/0.29 prod : [] --> b -> c 0.00/0.29 sum : [] --> b -> c 0.00/0.29 0.00/0.29 Rules: 0.00/0.29 0.00/0.29 fold (/\x./\y.f x y) nil z => z 0.00/0.29 fold (/\x./\y.f x y) (cons z u) v => fold (/\w./\x'.f w x') u (f v z) 0.00/0.29 sum x => fold (/\y./\z.add y z) x 0 0.00/0.29 fold (/\x./\y.mul x y) z 1 => prod z 0.00/0.29 0.00/0.29 Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term. 0.00/0.29 0.00/0.29 We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0). This leads to the following AFSM: 0.00/0.29 0.00/0.29 Alphabet: 0.00/0.29 0.00/0.29 0 : [] --> c 0.00/0.29 1 : [] --> c 0.00/0.29 add : [] --> c -> a -> c 0.00/0.29 cons : [a * b] --> b 0.00/0.29 fold : [c -> a -> c * b * c] --> c 0.00/0.29 mul : [c * a] --> c 0.00/0.29 nil : [] --> b 0.00/0.29 prod : [b] --> c 0.00/0.29 sum : [b] --> c 0.00/0.29 ~AP1 : [c -> a -> c * c] --> a -> c 0.00/0.29 0.00/0.29 Rules: 0.00/0.29 0.00/0.29 fold(/\x./\y.~AP1(F, x) y, nil, X) => X 0.00/0.29 fold(/\x./\y.~AP1(F, x) y, cons(X, Y), Z) => fold(/\z./\u.~AP1(F, z) u, Y, ~AP1(F, Z) X) 0.00/0.29 sum(X) => fold(/\x./\y.~AP1(add, x) y, X, 0) 0.00/0.29 fold(/\x./\y.mul(x, y), X, 1) => prod(X) 0.00/0.29 fold(/\x./\y.mul(x, y), nil, X) => X 0.00/0.29 fold(/\x./\y.mul(x, y), cons(X, Y), Z) => fold(/\z./\u.mul(z, u), Y, mul(Z, X)) 0.00/0.29 ~AP1(F, X) => F X 0.00/0.29 0.00/0.29 Symbol ~AP1 is an encoding for application that is only used in innocuous ways. We can simplify the program (without losing non-termination) by removing it. Additionally, we can remove some (now-)redundant rules. This gives: 0.00/0.29 0.00/0.29 Alphabet: 0.00/0.29 0.00/0.29 0 : [] --> c 0.00/0.29 1 : [] --> c 0.00/0.29 add : [c * a] --> c 0.00/0.29 cons : [a * b] --> b 0.00/0.29 fold : [c -> a -> c * b * c] --> c 0.00/0.29 mul : [c * a] --> c 0.00/0.29 nil : [] --> b 0.00/0.29 prod : [b] --> c 0.00/0.29 sum : [b] --> c 0.00/0.29 0.00/0.29 Rules: 0.00/0.29 0.00/0.29 fold(/\x./\y.X(x, y), nil, Y) => Y 0.00/0.29 fold(/\x./\y.X(x, y), cons(Y, Z), U) => fold(/\z./\u.X(z, u), Z, X(U, Y)) 0.00/0.29 sum(X) => fold(/\x./\y.add(x, y), X, 0) 0.00/0.29 fold(/\x./\y.mul(x, y), X, 1) => prod(X) 0.00/0.29 0.00/0.29 We use rule removal, following [Kop12, Theorem 2.23]. 0.00/0.29 0.00/0.29 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 0.00/0.29 0.00/0.29 fold(/\x./\y.X(x, y), nil, Y) >? Y 0.00/0.29 fold(/\x./\y.X(x, y), cons(Y, Z), U) >? fold(/\z./\u.X(z, u), Z, X(U, Y)) 0.00/0.29 sum(X) >? fold(/\x./\y.add(x, y), X, 0) 0.00/0.29 fold(/\x./\y.mul(x, y), X, 1) >? prod(X) 0.00/0.29 0.00/0.29 We use a recursive path ordering as defined in [Kop12, Chapter 5]. 0.00/0.29 0.00/0.29 Argument functions: 0.00/0.29 0.00/0.29 [[0]] = _|_ 0.00/0.29 [[prod(x_1)]] = x_1 0.00/0.29 0.00/0.29 We choose Lex = {fold} and Mul = {1, add, cons, mul, nil, sum}, and the following precedence: 1 > cons > mul > nil > sum > add > fold 0.00/0.29 0.00/0.29 Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: 0.00/0.29 0.00/0.29 fold(/\x./\y.X(x, y), nil, Y) >= Y 0.00/0.29 fold(/\x./\y.X(x, y), cons(Y, Z), U) > fold(/\x./\y.X(x, y), Z, X(U, Y)) 0.00/0.29 sum(X) >= fold(/\x./\y.add(x, y), X, _|_) 0.00/0.29 fold(/\x./\y.mul(x, y), X, 1) >= X 0.00/0.29 0.00/0.29 With these choices, we have: 0.00/0.29 0.00/0.29 1] fold(/\x./\y.X(x, y), nil, Y) >= Y because [2], by (Star) 0.00/0.29 2] fold*(/\x./\y.X(x, y), nil, Y) >= Y because [3], by (Select) 0.00/0.29 3] Y >= Y by (Meta)
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