Proof by induction

Other concepts we will use freely include parhal derivatives, trigonometric identities, the natural numbers N = 1, 2, 3,. .., basic properties of integra-hon and proof by induction. Interested readers will hnd a nice introduction to proof by induction in [Sp, Chapter 2]. 

Exercise 7.1 In Section 7.1 there is an informal proof that the Laplacian restricted to P is surjective onto Pf for any nonnegative integer . Turn this informal proof into a formal proof by induction. 

Since, by definition, it is also valid for n = 1 from Rule 1, the proof by induction is complete. 

There is an obvious interest in proofs-by-induction, because these allow the synthesis of recursive programs. Note that there often is a similar proof content in transformational synthesis and proofs-as-programs synthesis, and it seems that the same proof construction techniques should be applicable to both. This may suggest that these approaches are probably two facets of the same process. For instance, the work of [Neugebauer 93] shows how the specification forms of these approaches may be reconciled. 

Attention Inductive inference is not to be confused with deductive proofs-by-induction. 

A completely different approach is advocated by [Hagiya 90]. He re-formulates Summers recurrence relation detection mechanism in a logic framework, using higher-order unification in a type theory with a recursion operator. The method is extended to synthesizing deductive proofs-by-induction from concrete sample proofs. 

A more sophisticated extension would be the handling of proofs-by-induction. This requires additional rules of inference, such as those of [Kanamori and Fujita 86] or [Fribourg 93]. Such proofs are sometimes necessary, as shown in [Flener 93]. 

The proof is by induction. It is clearly true for two factors since then it reduces to the definition of the contraction symbol. Furthermore, it is sufficient to prove the theorem under the assumption that Z is a creation operator and that all the operators UV XY are destruction operators. If UV- - -XY are all destruction operators and Z is a creation operator, we may then add any number of creation operators to the left of all factors on both sides of Eq. (10-196) within the N product, without impairing the validity of our theorem, since the contraction between two creation operators gives zero. If on the other hand Z is a destruction operator and UV - - - are creation operators, then Eq. (10-196) reduces to a trivial identity... 

The proof is carried out by induction. Assuming Qi < 1 we will show that jai+il < 1. Since aj = < 1, the same propertry will cover all... 

Proof We proceed to prove this assertion by induction on n. Let n = 2. Then... 

The proof proceeds by induction on n. First consider n = 0. As we mentioned when considering tie nenredundaney of W, there must be an input (e,y y2 v ) from D which enters W with tail y2 v and goes around the loop at least k+1 times. If the tail were unchanged by a cycle around W, then the computation could never leave W and halt as it must hence each of these k+1 times around W decreases the length of the tail by at least one symbol and yet cannot decrease it beyond the or the output would never be correct. So we must... 

The answer is that if the tree is a full binary tree of depth N, then N+l markers are necessary and sufficient. The proof is by induction on N. The case N - 1 is obvious, for then there are just two leaves and both must be covered before the root can be covered. Suppose this is true for depth N-l. The root has two sons each of which can be regarded as the root of a full binary subtree of depth N-l call these nodes n and n. The root can be covered when and only... 

The physical meaning of this theorem is very intuitive and its proof will not be reported here (loc. cit.) it can easily be verified from the example of Fig. 12 and the general demonstration is obtained by induction. 

Proof. We will prove the first conclusion of this proposition by induction on the dimension of VF. We start with the subspace of dimension zero, i.e., the trivial subspace 0. It is easy to check that the linear transformation taking every vector of V to the zero vector is an orthogonal projection onto 0. ... 

Proof. We proceed by induction on n. For the base case (n = 1), consider P(C). By Exercise 10.1, P(C) consists of a single point. So 5 must be the identity function, in which case the desired unitary transformation of V is the identity linear operator (or any modulus-one complex multiple of the identity operator) and the function k is the identity. 

The derivations presented below illustrate the logical technique of proof by contradiction. In this method of proof, we begin by assuming that Carnot s principle is untrue, then demonstrate that we could easily produce crazy consequences that contradict experience if this assumption were valid. That is, we conclude that Carnot s principle must be true, because the contrary assumption leads to inconsistencies with inductive experience. 

Proof. This follows from Lemma 3.1.l(i) by induction. 

Proof. First, this is a fixed point theorem because it is enough to show some d, 0 in fc" is fixed by all g in G. Indeed, G then acts by unipotent maps on By induction on the dimension there is a basis [t>2], [r ] of the... 

An upper bound for (i) is substituted into equation (54) in order to obtain a lower bound for A. If this value of A and the upper bound for (t) are substituted into the right-hand side of equation (60), then, since the integral in equation (60) will clearly assume a lower bound, the right-hand side of equation (60) will assume an upper bound, and hence a new upper bound for g (x) will be given by equation (60). The fact that the resulting new upper bound for (t) is lower than the previous upper bound requires proof by mathematical induction. ... 

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