fixpoints | Fixpoints | Storage library
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kandi X-RAY | fixpoints Summary
Fixpoints enables saving, restoring and comparing the database state before & after tests.
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QUESTION
I have realised that the following program
...ANSWER
Answered 2020-Jul-04 at 19:11The key idea is to share work. In the naive version, fib (n-2)
is computed twice from scratch (fib n = fib (n-1) + fib (n-2) = fib (n-2) + fib (n-2) + fib (n-3)
). In the list version, the argument xs
represents the recursive call, which is evaluated once, and used twice.
QUESTION
For a research project, I would like to form two groups (e.g. left/right) within a panel data set of a highly fragmented party system, based on a string variable containing the coalitions the parties form, as in column (3) of the following simplified example:
...ANSWER
Answered 2020-Feb-06 at 12:55library(tidyverse)
d %>%
separate_rows(coalitions) %>%
filter(parties != coalitions) %>%
count(parties, coalitions)
# # A tibble: 6 x 3
# parties coalitions n
#
# 1 A C 2
# 2 A D 1
# 3 A E 1
# 4 B F 1
# 5 B G 2
# 6 B H 1
QUESTION
I want to create a data structure for storing various possible paths through a plane with polygons scattered across it. I decided on using nested, multi-level dictionaries to save the various possible paths splitting at fixed points.
A possible instance of such a dictionary would be:
...ANSWER
Answered 2019-Apr-21 at 23:13You can certainly write accessors for such a nested dictionary:
QUESTION
I am trying to define a recursive predicate using well-founded fixpoints with the obligation to show F_ext when rewriting with Fix_eq. The CPDT says that most such obligations are dischargeable with straightforward proof automation, but unhappily this does not appear to be so for my predicate.
I have reduced the problem to the following lemma (from Proper (pointwise_relation A eq ==> eq) (@all A)
). Is it provable in Coq without additional axioms?
ANSWER
Answered 2018-Jul-25 at 21:08You might be interested in this code I wrote a while ago, which includes variants of Fix_eq
for various numbers of arguments, and don't depend on function extensionality. Note that you don't need to change Fix_F
, and can instead just prove variants of Fix_eq
.
To answer the question you asked, rather than solve your context, the lemma you state is called "forall extensionality".
It is present in Coq.Logic.FunctionalExtensionality, where the axiom of function extensionality is used to prove it. The fact that the standard library version uses an axiom to prove this lemma is, at the very least, strong evidence that it is not provable without axioms in Coq.
Here is a proof sketch of that fact. Since Coq is strongly normalizing*, every proof of x = y
in the empty context is judgmentally equal to eq_refl
. That is, if you can prove x = y
in the empty context, then x
and y
are convertible. Let f x := inhabited (Vector.t (x + 1))
and let g x := inhabited (Vector.t (1 + x))
. It is straightforward to prove forall x, f x = g x
by induction on x
. Therefore, if your lemma were true without axioms, we could get a proof of
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