isabelle

Post Answer Follow Up Question Brian provided an answer with a suggested solution being to use lifting and transfer. However, I can't find enough tutorial information on lifting and transfer to know how to tweak his answer to finish off what I would need to do. Here, I work in the dark, and use the answer given as a plug'n'play template to ask this follow up question. The command in my initial code, typedef trivAlg = "{x::sT. x = emS}" gives me a new type that is a subset of the mother type sT . I have my membership operator consts inP :: "sT => sT => bool" , and so in my naive view of lifting

How can one define a function in Isabelle that has a different definition depending on either the type of its argument, or the type of the context it is used in? For example, I might want to define a functions is_default with type 'a ⇒ bool , where each different type 'a has a potentially different "default value". (I am also assuming, for the sake of argument, that existing concepts such as zero are not suitable.) This kind of overloading looks like a perfect fit for type classes. First you define a type class for your desired function is_default : class is_default = fixes is_default :: "'a ⇒

Every goal that I have encountered in Isabelle so far that could be solved using arith could also be solved by presburger and vice versa, for example lemma "odd (n::nat) ⟹ Suc (2 * (n div 2)) = n" by presburger (* or arith *) What's the difference between the two solvers? Examples of goals that one can solve but the other can't would be nice. Edit: I managed to come up with a lemma proved by arith that presburger can't handle. It seems like this has something to do with real numbers: lemma "max i (i + 1) > (i::nat)" by arith -- ✔ lemma "max i (i + 1) > (i::nat)" by presburger -- ✔ lemma "max i

let's assume I want to show the following lemma lemma "⟦ A; B; C ⟧ ⟹ D" I get the goal 1. A ⟹ B ⟹ C ⟹ D However, I don't need B . How can I transfer my goal to something like 1. A ⟹ C ⟹ D I don't want to alter the original lemma statement, just the current goal in apply style. What you want is apply (thin_tac B) . However, the last time I did this, Peter Lammich shouted "Oh god, why are you doing this!" in disgust and rewrote my proof in order to get rid of the thin_tac. So using this tactic doesn't exactly seem to be encouraged anymore. Normally it is better to avoid unwanted stuff in a goal

There is a matrix multiplication definition in Cartesian_Euclidean_Space (in directory HOL/Multivariate_Analysis"): definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m ⇒ 'a ^'p^'n ⇒ 'a ^ 'p ^'m" (infixl "**" 70) where "m ** m' == (χ i j. setsum (λk. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m" Now the the squared matrix would be A ** A and A^3 would be A ** A ** A . I couldn't find a powerfunction, to define A^n (i.e., A ** A ** ... ** A n times). Is there a power function in the library? Is a manual definition needed? I have found the following definition in HOL/Power.thy :

I would like to find the maximum in a vector of natural numbers. Vector (i.e., ‘vec’), however, is a different type than Set or List. I thought about several ideas that did not work, like leveling or lifting the type of vec or the definition of a recursive function. What solution do you suggest to get the maximum value in a vector? (* IMPORTS: "~~/src/HOL/Algebra/Ring" "~~/src/HOL/Library/Numeral_Type" "~~/src/HOL/Library/Permutations" "~~/src/HOL/Library/Polynomial" "~~/src/HOL/Big_Operators" vec (VECTOR) is from Finite_Cartesian_Product degree is from Polynomial Max is from Big_Operators *)

I'm reading the Isabelle tutorial and been trying to clear my concept on use of primrec and fun. With what I've searched so far, including the answer here ; I understand that constructor inside primrec can have only one equation and primrec has [simp] by default whereas fun can have multiple equations and the automation tactics need to be specified explicitly. However, I still struggle to understand it clearly. Anyone kind enough to explain with some examples? Manuel Eberl primrec does primitive recursion on an algebraic datatype (or something that has been set up to look like one, like the

How can I use Isabelle/HOL to automatically generate LaTeX from my source theory files? Isabelle/HOL's tutorial.pdf is very beautiful. I'm going to write a paper in LaTeX with a lot of Isabelle code in it. You should first have a look at the existing documentation and come back with more specific questions afterwards (if there remain any; but I'm sure there will ;)). What you want to do is called document preparation in Isabelle. The first place to look is Chapter 4 Presenting Theories of the Isabelle System Manual . (Actually it is also a good idea to first read the previous chapter on