Coq induction by two

Author: zxsw

August undefined, 2024

WebMay 4, 2024 · You're saying that you need to prove this by induction? Why? The classical way of proving this is simply by stating: n = 2a + 1, m = 2b therefore n + m = 2a + 1 + 2b => n + m = 2 (a + b) + 1. I cannot find an apparent way of doing this proof by definition/substitution in coq. WebJan 1, 2024 · That is fine for nat, but for some complex inductive types, the default induction principle is too weak and a handwritten fix is the only way. Finally, going back …

Coq in a Hurry - CEL

WebIntroducing ev with an Inductive declaration tells Coq not only that the constructors ev_0 and ev_SS are valid ways to build evidence that some number is ev, but also that these two constructors are the only ways to build evidence that numbers are ev. WebDec 22, 2024 · The following two are axially definitions from CoLoR library: ... You need simplification of Vnth_vmap and a generalized induction to achieve this: From Coq … craigslist surrey rentals

Tactics: More Basic Tactics - University of Pennsylvania

WebPUMPKIN-PATCH/plugin/coq/Induction.v Go to file Cannot retrieve contributors at this time 633 lines (547 sloc) 16.9 KB Raw Blame (* Case study for porting a library from one Coq definition to another *) (* Original problems are from Software Foundations (Inductive.v), solutions are from students *) http://flint.cs.yale.edu/cs430/coq/sf/Induction.html#:~:text=In%20Coq%2C%20the%20steps%20are%20the%20same%20but,P%28O%29%20and%20then%20showing%20P%28n%27%29%20%E2%86%92%20P%28S%20n%27%29. WebIn this case, the proof of A /\ B begins with that formula as the goal. This can be transformed into two subgoals, A and B, followed by the proofs of A and B. Coq and its tactics use backward reasoning. A tactic may fully prove a goal, in … diy ice roller for face

UseAuto: Theory and Practice of Automation in Coq Proofs

MoreCoq: More About Coq - Brown University

Webinduction proof, the programmer may prefer overriding the proof entirely, as we observe in our case studies (Section7). Overriding is potentially incompatible with having equalities on late bound names, however. Coq distinguishes opaque definitions fromtransparent ones. fpopsupports the overriding of opaque definitions, which include most proofs. WebIn Coq, the steps are the same: we begin with the goal of proving P(n) for all n and break it down (by applying the induction tactic) into two separate subgoals: one where we must … craigslist support phone numberWebJul 12, 2024 · The function firstn is provided in the Coq standard library, and it is recursive on its two inputs. An analysis of g_value (and the feedback message from Coq) tells us that g_value is recursive on its first argument, this suggests that value_greater should be proved by induction on n. Here is an example of how this proof can be done: diy ice shanty plans

"http://flint.cs.yale.edu/cs428/coq/pdf/RecTutorial.pdf " - Coq induction by two

Coq induction by two

Induction: Proof by Induction - flint.cs.yale.edu

WebThis definition introduces a new predicate le : nat -> nat -> Prop, and the two constructors le_n and le_S, which are the defining clauses of le.That is, we get not only the “axioms” le_n and le_S, but also the converse property, that (le n m) if and only if this statement can be … Coq has a rich ecosystem of external packages (libraries and plugins) that … The Coq development team maintains the following reference documents: the … Coq implements a program specification and mathematical higher-level language … The Coq user community has contributed a large ecosystem of formalization works … WebCoq Tactics Index Stage 1: Proving Easy Goals reflexivity assumption discriminate constructor Stage 2: Transforming Your Goal apply subst rewrite simpl cut unfold Stage 3: Breaking Apart Your Goal destruct inversion induction Stage 4: Powerful Automatic Tactics auto intuition omega reflexivity

Did you know?

WebAs in a fix expression, induction hypotheses must be used on structurally smaller arguments. The verification that inductive proof arguments are correct is done only when registering …

WebThe chapter is organized in two parts. The first part focuses on a general mechanism called "proof search." ... derive from the induction hypothesis. So, rather than stating this conclusion explicitly, we are going to ask Coq to instantiate the induction hypothesis, using automation to figure out how to instantiate it. The tactic ... WebJun 23, 2024 · Theorem leb_correct. Now the implication is reversed: we’re proving that the inductive proposition definition le implies the boolean definition leb for any two numbers …

WebThus, there are two cases for a natural number, either it is 0 or it is the successor of another number. Expressing the distinction between the two cases is written as in the following function, which returns a boolean value. This boolean value is true exactly when the argument is 0. Definition is_zero (n:nat) := match n with 0 => true S p ... WebJul 17, 2013 · In Coq, the steps are the same but the order is backwards: we begin with the goal of proving P (n) for all n and break it down (by applying the induction tactic) into …

WebJan 23, 2024 · UseTactics: Tactic Library for Coq. (* Chapter written and maintained by Arthur Chargueraud *) Coq comes with a set of builtin tactics, such as reflexivity , intros, inversion and so on. While it is possible to conduct proofs using only those tactics, you can significantly increase your productivity by working with a set of more powerful ...

WebCoq generates induction principles depending on the universe that the inductive type belongs to. The induction principles are named ident_rect, ident_ind , ident_rec and ident_sind, which respectively correspond to on Type, Prop, Set and SProp. Their types expresses structural induction/recursion principles over objects of type ident. craigslist surround sound speakersWebFirst create a file named _CoqProject containing the following line (if you obtained the whole volume "Logical Foundations" as a single archive, a _CoqProject should already exist … diy ice shack plansWebJan 30, 2013 · In the previous chapter, we noticed the importance of controlling the exact form of the induction hypothesis when carrying out inductive proofs in Coq. In particular, we need to be careful about which of the assumptions we move (using intros ) from the goal to the context before invoking the induction tactic. craigslist sutherlin oregon rentalsWebIn Coq, the steps are the same: we begin with the goal of proving P(n) for all n and break it down (by applying the induction tactic) into two separate subgoals: one where we must show P(O) and another where we must show P(n') → P(S n'). Here's how this works for the theorem at hand: Theorem plus_n_O : ∀ n: nat, n = n + 0. Proof. craigslist surround sound systemWebJun 13, 2024 · 1 Answer. A classic solution is to define a stronger property, which you prove by induction. Lemma strong_ind (P : nat -> Prop) : (forall m, (forall k : nat, k < m -> P k) -> P m) -> forall n, P n. Proof. intros H n; enough (H0: forall p, p <= n -> P p). - apply H0, le_n. - … craigslist sutherlin oregonWebOne is to use destruct m to break the proof into two cases, each focusing on a more concrete choice of m ( O vs S _ ). In each case, the match inside of bar can now make … diy ice sled runnersWebJun 13, 2024 · 1 Answer. A classic solution is to define a stronger property, which you prove by induction. Lemma strong_ind (P : nat -> Prop) : (forall m, (forall k : nat, k < m -> P k) … craigslist suv for sale bay area