Structural induction example
WebExample structures: \(Σ^*\) is defined by \(x ∈ Σ^* ::= ε \mid xa\). To prove \(∀x \in Σ^*, P(x)\), you must prove (1) \(P(ε)\), and (2) \(P(xa)\); but in the proof of (2) you may … WebStructural Induction To prove P(S)holds for any list S, prove two implications Base Case: prove P(nil) –use any known facts and definitions Inductive Hypothesis: assume P(L)is true –use this in the inductive step, but not anywhere else Inductive Step: prove P(cons(x, L))for any x : ℤ, L : List –direct proof
Structural induction example
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WebStructural induction exploits the fact that recursive data types are ultimately built up using applications of constructors — to construct a list, we "cons" the elements onto the "nil" element. The structural induction principle for a data type can be obtained as follows. WebStructural induction is used to prove that some proposition P(x) holds for all x of some sort of recursively defined structure, such as formulas, lists, or trees. A well-founded partial …
WebStructural Induction The set of natural numbers N has a particular structure that allows us to de ne it using the following recursive de nition: 0 2N if n 2N, then n+ 1 2N N contains … WebJul 1, 2024 · A structural induction proof has two parts corresponding to the recursive definition: Prove that each base case element has the property. Prove that each constructor case element has the property, when the constructor is applied to elements that have the …
Web5 Structural induction Inductive proofs on trees can also be written using “structural induction.” In structural induction, there is no explicit induction variable. Rather, the outline of the proof follows the structure of a recursive definition. This is slightly simpler for trees and a lot simpler for some other types of examples WebStructural Induction - Example Example Consider the following property of lists: length (L ++ M) = length L + length M Here ++ denotes the list concatenation operation, and L and M are lists. In order to prove this, we need definitions for …
WebFor some other examples of structural induction, including a proof of correctness for a very simple compiler (one that just compiles individual expressions), see Proving properties of programs by structural induction by R. M. Burstall. Below is an outline of how Burstall's compiler works, and what the proof of correctness involves.
WebInduction and Recursion. 6.8. Structural Induction. So far we’ve proved the correctness of recursive functions on natural numbers. We can do correctness proofs about recursive functions on variant types, too. That requires us to figure out how induction works on variants. We’ll do that, next, starting with a variant type for representing ... hots butcher guideWebStructural Induction Example - Binary Trees CMPT 384 Lecture Notes Robert D. Cameron September 24, 1999. As a further example of structural induction, we consider an example worked out in detail on the domain of full binary trees. First consider the following GRAIL grammar for full binary trees. hots c4WebStructural induction is a very complicated way to define something. You generally want to avoid it. There are only 2 ways (in sequential logic) to define an inductive set: define it using an inductive axiom define it in terms of some other set that itself is inductive, such as integers or lists or trees linear sclerosis on xrayWebFormalize the structural induction principle for formulas that follows from the de nition. Give an example of the use of the induction principle to prove a property ... So here’s our example induction. Proposition 2 Suppose that Sis a set of atoms, and M(A) = M0(A) for every X2S. For all formulas , if uses only atoms in S, then linear science graphWebStructural Induction To prove P(S)holds for any list S, prove two implications Base Case: prove P(nil) –use any known facts and definitions Inductive Hypothesis: assume P(L)is … hots build tyrandeWebAn Example Structural Induction Proof These notes include a skeleton framework for an example structural induction proof, a proof that all propositional logic expressions … linear sclerosisWebWe will see examples of structural induction and induction on derivations throughout the course. The intuition for why the inductive reasoning principle works is that same as the intuition for why mathematical induction works, i.e., for why the inductive reasoning principle for natural numbers works. 2.3 Example inductive reasoning principles hots c6