Most of the proofs suggested by Sledgehammer use this notation of number inside of parentheses:
by (smt (z3) ApplyAllResult.distinct(1)
ApplyResult.case(1)
ApplyResult.case(2)
ApplyResult.exhaust
applyInput.simps(1))
What does it mean for the fact to have such number?
Isabelle permits the use of fact lists indexed by a natural number starting with 1. Given a fact list fs and an index i, you can access an individual fact from the list by using the syntax fs(i). You can also select multiple facts from the list using multiple indexes (e.g., fs(1,3)), ranges (e.g. fs(2-5), fs(3-)) or a combination of both (e.g., fs(2,4-6)).
Examples of predefined fact lists are assms (which contains the assumptions of a theorem) and f.simps (which contains the equations defining function f).
Related
I understand canonicalization and normalization to mean removing any non-meaningful or ambiguous parts of of a data's presentation, turning effectively identical data into actually identical data.
For example, if you want to get the hash of some input data and it's important that anyone else hashing the canonically same data gets the same hash, you don't want one file indenting with tabs and the other using spaces (and no other difference) to cause two very different hashes.
In the case of JSON:
object properties would be placed in a standard order (perhaps alphabetically)
unnecessary white spaces would be stripped
indenting either standardized or stripped
the data may even be re-modeled in an entirely new syntax, to enforce the above
Is my definition correct, and the terms are interchangeable? Or is there a well-defined and specific difference between canonicalization and normalization of input data?
"Canonicalize" & "normalize" (from "canonical (form)" & "normal form") are two related general mathematical terms that also have particular uses in particular contexts per some exact meaning given there. It is reasonable to label a particular process by one of those terms when the general meaning applies.
Your characterizations of those specific uses are fuzzy. The formal meanings for general & particular cases are more useful.
Sometimes given a bunch of things we partition them (all) into (disjoint) groups, aka equivalence classes, of ones that we consider to be in some particular sense similar or the same, aka equivalent. The members of a group/class are the same/equivalent according to some particular equivalence relation.
We pick a particular member as the representative thing from each group/class & call it the canonical form for that group & its members. Two things are equivalent exactly when they are in the same equivalence class. Two things are equivalent exactly when their canonical forms are equal.
A normal form might be a canonical form or just one of several distinguished members.
To canonicalize/normalize is to find or use a canonical/normal form of a thing.
Canonical form.
The distinction between "canonical" and "normal" forms varies by subfield. In most fields, a canonical form specifies a unique representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness.
Applying the definition to your example: Have you a bunch of values that you are partitioning & are you picking some member(s) per each class instead of the other members of that class? Well you have JSON values and short of re-modeling them you are partitioning them per what same-class member they map to under a function. So you can reasonably call the result JSON values canonical forms of the inputs. If you characterize re-modeling as applicable to all inputs then you can also reasonably call the post-re-modeling form of those canonical values canonical forms of re-modeled input values. But if not then people probably won't complain that you call the re-modeled values canonical forms of the input values even though technically they wouldn't be.
Consider a set of objects, each of which can have multiple representations. From your example, that would be the set of JSON objects and the fact that each object has multiple valid representations, e.g., each with different permutations of its members, less white spaces, etc.
Canonicalization is the process of converting any representation of a given object to one and only one, unique per object, representation (a.k.a, canonical form). To test whether two representations are of the same object, it suffices to test equality on their canonical forms, see also wikipedia's definition.
Normalization is the process of converting any representation of a given object to a set of representations (a.k.a., "normal forms") that is unique per object. In such case, equality between two representations is achieved by "subtracting" their normal forms and comparing the result with a normal form of "zero" (typically a trivial comparison). Normalization may be a better option when canonical forms are difficult to implement consistently, e.g., because they depend on arbitrary choices (like ordering of variables).
Section 1.2 from the "A=B" book, has some really good examples for both concepts.
My professor is having us create a series of functions relating to approximating pi and e based on continuing fractions. In order to set this up, he is having us create a function that takes an integer and maps that many odd numbers squared, starting from 1. For instance, here is the desired behavior:
oddSquares 6;;
val it : int list = [1.0; 9.0; 25.0; 49.0; 81.0; 121.0]
I can see that one mapping will likely be used to square all the values in the list, but I can't figure out a way to map the number to a list of numbers. I don't want to ask anybody to write code for me, but methodically, what am I trying to do when I'm assembling the base list?
It feels like the best method is to work backwards, starting from the base number of 6 terms. We then evaluate the 6th odd term (11, or 2x-1 practically), but then require some method of recursion to continue to evaluate smaller values of oddSquares. I also think this is against the spirit of trying to map the number into these values. Can someone give me some guidance as to the first translation from the number into the list form?
F# offers special neat syntax for creating lists of successive numbers:
let oneToSix = [1..6]
This is a special case of something called "list comprehension". They can be more complex than just successive numbers - they can include multiple generators, filters, projections, Cartesian products, etc. In particular, your whole task of generating first N odd numbers can be expressed as one list comprehension. However, since you explicitly asked not to write the code for you, I won't.
Even though loops are kind of a logical concept in X12 (not directly physically represented in the text), every transaction set defines a set of loops that it can contain, including identifiers for the loops and an ordering for them. My question is, what is the rule for sorting loops, generically? Is there a concise set of rules that can be expressed in some code that should be able to take a collection of loops (with known identifiers such as 1000A, 2300BB, etc) and properly sort them?
The context of my question is that I'm working on a general-purpose library that applications will use to construct a model of an X12 document/transaction-set (and write out the text such a model represents). It has objects to represent Elements, Segments, and Loops. Ordering of Segments in a particular Loop is easy, they're dictated by the Implementation Guides. But I'm trying to get Loop ordering (within a Transaction Set) to work generically; that's what I'm asking about
It seems that the general rule is that Loops are ordered based on their identifiers using the numeric portion as the primary sort key, with the alpha portion as the secondary sort key. Of course hierarchical loops contained in others will be placed before and loops following the parent in that sort order (eg: 1000A, 2000A, 2010A, 2010B, 2100, 2300 - where 2010A and 2010B are children of 2000A).
I understand that the spec and Implementation Guides contain all of this info; I'm looking for the all-encompassing rule about loop ordering (not Segment ordering). Is there any concise way to express the rule algorithmically? Is there even a hard-and-fast rule at all?
As I mentioned in my comments, the standard has a loop value. Take a look at my screenshot of the Liaison Dictionary Viewer. The CLM segment has a LOOP value of 100. The segments underneath are children of the CLM segment (extended tag). Any "order" can be defined arbitrarily by the partner, or can be in any (undefined) order provided the data is qualified. But that loop can occur 100 times max and can have repeating segments inside the loop value.
The implementation guide will give you the correct order your partner wants them in. It seems like you're writing your own syntax validation engine though.
given an expression x'=x+1, I wish to rename x' to y. How to do using z3?
There are a number of API functions that let you modify terms and substitute new subterms for old ones. They are described under the Modifiers section that contain the modifiers Z3_update_term, z3_substitute, and Z3_substitute_vars (there is also Z3_translate to port terms between two contexts).
Here is the link:
http://research.microsoft.com/en-us/um/redmond/projects/z3/group__capi.html#gaa7497c70a827db2d61ba98889fe657b5
You can also traverse terms directly and write utilities to modify terms.
The display_ast example shows the main cases for recursively traversing terms:
http://research.microsoft.com/en-us/um/redmond/projects/z3/group__capi__ex.html#ga807b5fe0e26acdec09e52a77318208d0
List Comprehension is a very useful code mechanism that is found in several languages, such as Haskell, Python, and Ruby (just to name a few off the top of my head). I'm familiar with the construct.
I find myself working on an Open Office Spreadsheet and I need to do something fairly common: I want to count all of the values in a range of cells that fall between a high and low bounds. I instantly thought that list comprehension would do the trick, but I can't find anything analogous in Open Office. There is a function called "COUNTIF", and it something similar, but not quite what I need.
Is there a construct in Open Office that could be used for list comprehension?
CountIf can count values equal to one chosen. Unfortunately it seems that there is no good candidate for such function. Alternatively you can use additional column with If to display 1 or 0 if the value fits in range or not accordingly:
=If(AND({list_cell}>=MinVal; {list_cell}<=MaxVal); 1; 0)
Then only thing left is to sum up this additional column.
Assuming:
your range is A1:A10
your lower bound is at B1
your upper bound is at B2
then what you want can be achieved by:
=COUNTIFS(A1:A10, ">" & B1, A1:A10, "<" & B2)
(you might need to change commas into semicolons, depending on your language preference for decimal point)
Quoting from the installed OpenOffice documentation:
The logical relation between criteria can be defined as logical AND (conjunction). In other words, if and only if all given criteria are met, a value from the corresponding cell of the given Func_Range is taken into calculation.
This function is part of the Open Document Format for Office Applications (OpenDocument) standard Version 1.2. (ISO/IEC 26300:2-2015)