coq

In Coq, sig is defined as Inductive sig (A:Type) (P:A -> Prop) : Type := exist : forall x:A, P x -> sig P. Which I read as "A sig P is a type, where P is a function taking an A and returning a Prop. The type is defined such that an element x of type A is of type sig P if P x holds." proj1_sig is defined as Definition proj1_sig (e:sig P) := match e with | exist _ a b => a end. I'm not sure what to make of that. Could somebody provide a more intuitive understanding? Non-dependent pairs vs. sig The type is defined such that an element x of type A is of type sig P if P x holds. That is not

I'm trying to go through the Software Foundations Coq book ( http://www.cis.upenn.edu/~bcpierce/sf/current/toc.html ), but when I compile Induction.v (which looks like http://www.cs.uml.edu/~rhenniga/coq/sf_induction.html ), I get the error message "Error: The reference evenb was not found in the current environment." -- even after compilation of Basics.v. Any ideas why? I can confirm that opening CoqIDE from the same directory works on macOS: cd <sf-dir>; /Applications/CoqIDE_8.5.app/Contents/MacOS/coqide from: The reference "X" was not found in the current environment Try to erase every

I'm attempting to do that as follows: λ (A : *) -> λ (B : (A -> *)) -> λ (t : (∀ (r : *) -> (∀ (x : a) -> (B x) -> r)) -> r) -> (t (B (t A (λ (x : A) -> λ (y : (B x)) -> x))) (λ (x : A) -> λ (y : (B x)) -> y)) Notice that, since the value returned by that function depends on a value inside the sigma itself, I need to extract that value. This code doesn't check, because, I believe, it fails to unify the type extracted from Sigma with the type inside it. Is there any workaround? 来源： https://stackoverflow.com/questions/43957592/how-to-extract-the-second-element-of-sigma-on-the-calculus-of

I'm quite new at Coq and trying to develop a framework based on my research. My work is quite definition-heavy and I'm having trouble encoding it because of how Coq seems to treat sets. There are Type and Set , which they call 'sorts', and I can use them to define a new set: Variable X: Type. And then there's a library encoding (sub)sets as ' Ensembles ', which are functions from some Type to a Prop . In other words, they are predicates on a Type : Variable Y: Ensemble X. Ensemble s feel more like proper mathematical sets. Plus, they are built upon by many other libraries. I've tried focussing

Coq, unlike many others, accepts an optional explicit parameter,which can be used to indicate the decreasing structure of a fixpoint definition. From Gallina specification, 1.3.4, Fixpoint ident params {struct ident0 } : type0 := term0 defines the syntax. but from it, we've known that it must be an identifier, instead of a general measure. However, in general, there are recursive functions, that the termination is not quite obvious,or it in fact is, but just difficult for the termination checker to find a decreasing structure. For example, following program interleaves two lists, Fixpoint