details

property	value
status	complete
benchmark	process.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n149.star.cs.uiowa.edu
space	Mixed_HO_10

run statistics

property	value
solver	Wanda 2.2a
configuration	default
runtime (wallclock)	4.55645 seconds
cpu usage	4.55749
user time	4.25545
system time	0.302038
max virtual memory	242328.0
max residence set size	127144.0

stage attributes

key	value
starexec-result	YES

output

YES We consider the system theBenchmark. Alphabet: !plus : [proc * proc] --> proc !times : [proc * proc] --> proc delta : [] --> proc sigma : [data -> proc] --> proc Rules: !plus(x, x) => x !times(!plus(x, y), z) => !plus(!times(x, z), !times(y, z)) !times(!times(x, y), z) => !times(x, !times(y, z)) !plus(x, delta) => x !times(delta, x) => delta sigma(/\x.y) => y !plus(sigma(/\x.f x), f y) => sigma(/\z.f z) sigma(/\x.!plus(f x, g x)) => !plus(sigma(/\y.f y), sigma(/\z.g z)) !times(sigma(/\x.f x), y) => sigma(/\z.!times(f z, y)) 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. 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: Alphabet: !plus : [proc * proc] --> proc !times : [proc * proc] --> proc delta : [] --> proc sigma : [data -> proc] --> proc ~AP1 : [data -> proc * data] --> proc Rules: !plus(X, X) => X !times(!plus(X, Y), Z) => !plus(!times(X, Z), !times(Y, Z)) !times(!times(X, Y), Z) => !times(X, !times(Y, Z)) !plus(X, delta) => X !times(delta, X) => delta sigma(/\x.X) => X !plus(sigma(/\x.~AP1(F, x)), ~AP1(F, X)) => sigma(/\y.~AP1(F, y)) sigma(/\x.!plus(~AP1(F, x), ~AP1(G, x))) => !plus(sigma(/\y.~AP1(F, y)), sigma(/\z.~AP1(G, z))) !times(sigma(/\x.~AP1(F, x)), X) => sigma(/\y.!times(~AP1(F, y), X)) ~AP1(F, X) => F X We use the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs. After applying [Kop12, Thm. 7.22] to denote collapsing dependency pairs in an extended form, we thus obtain the following dependency pair problem (P_0, R_0, minimal, all): Dependency Pairs P_0: 0] !times#(!plus(X, Y), Z) =#> !plus#(!times(X, Z), !times(Y, Z)) 1] !times#(!plus(X, Y), Z) =#> !times#(X, Z) 2] !times#(!plus(X, Y), Z) =#> !times#(Y, Z) 3] !times#(!times(X, Y), Z) =#> !times#(X, !times(Y, Z)) 4] !times#(!times(X, Y), Z) =#> !times#(Y, Z) 5] !plus#(sigma(/\x.~AP1(F, x)), ~AP1(F, X)) =#> sigma#(/\y.~AP1(F, y)) 6] !plus#(sigma(/\x.~AP1(F, x)), ~AP1(F, X)) =#> ~AP1#(F, y) 7] sigma#(/\x.!plus(~AP1(F, x), ~AP1(G, x))) =#> !plus#(sigma(/\y.~AP1(F, y)), sigma(/\z.~AP1(G, z))) 8] sigma#(/\x.!plus(~AP1(F, x), ~AP1(G, x))) =#> sigma#(/\y.~AP1(F, y)) 9] sigma#(/\x.!plus(~AP1(F, x), ~AP1(G, x))) =#> ~AP1#(F, y) 10] sigma#(/\x.!plus(~AP1(F, x), ~AP1(G, x))) =#> sigma#(/\y.~AP1(G, y)) 11] sigma#(/\x.!plus(~AP1(F, x), ~AP1(G, x))) =#> ~AP1#(G, y) 12] !times#(sigma(/\x.~AP1(F, x)), X) =#> sigma#(/\y.!times(~AP1(F, y), X)) 13] !times#(sigma(/\x.~AP1(F, x)), X) =#> !times#(~AP1(F, y), X) 14] !times#(sigma(/\x.~AP1(F, x)), X) =#> ~AP1#(F, y) 15] ~AP1#(F, X) =#> F(X) Rules R_0: !plus(X, X) => X !times(!plus(X, Y), Z) => !plus(!times(X, Z), !times(Y, Z)) !times(!times(X, Y), Z) => !times(X, !times(Y, Z)) !plus(X, delta) => X !times(delta, X) => delta sigma(/\x.X) => X !plus(sigma(/\x.~AP1(F, x)), ~AP1(F, X)) => sigma(/\y.~AP1(F, y)) sigma(/\x.!plus(~AP1(F, x), ~AP1(G, x))) => !plus(sigma(/\y.~AP1(F, y)), sigma(/\z.~AP1(G, z))) !times(sigma(/\x.~AP1(F, x)), X) => sigma(/\y.!times(~AP1(F, y), X)) ~AP1(F, X) => F X Thus, the original system is terminating if (P_0, R_0, minimal, all) is finite. We consider the dependency pair problem (P_0, R_0, minimal, all). We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows: * 0 : 5, 6 * 1 : 0, 1, 2, 3, 4, 12, 13, 14 * 2 : 0, 1, 2, 3, 4, 12, 13, 14 * 3 : 0, 1, 2, 3, 4, 12, 13, 14 * 4 : 0, 1, 2, 3, 4, 12, 13, 14 * 5 : 7, 8, 9, 10, 11 * 6 : * 7 : 5, 6 * 8 : 7, 8, 9, 10, 11 * 9 : * 10 : 7, 8, 9, 10, 11 popout

output may be truncated. 'popout' for the full output.

job log

popout

actions

all output return to Higher Order Rewriting Union Beta