We prove the theorem by structural induction on the set representing the game.

(This is easy to prove by structural induction on P'.)

The first one is easy to prove (in the denotational semantics) by structural induction.

Coinduction is the mathematical dual to structural induction.

Then the relation of semantic entailment is defined by structural induction on :

Early publications about structural induction include:

Other laws can be deduced from these laws, either by elementary manipulation, or by structural induction on normal forms.

When the well-founded set is a set of recursively-defined data structures, the technique is called structural induction.

Structural recursion is usually proved correct by structural induction; in particularly easy cases, the inductive step is often left out.

The second lemma is easy to prove using a combination of structural induction and mathematical induction (the latter for WHILE loops).

Just as standard mathematical induction is equivalent to the well-ordering principle, structural induction is also equivalent to a well-ordering principle.

The language introduces a special construct in which such polytypic functions can be defined via structural induction over the structure of the pattern functor of a regular datatype.

The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science.

Structural recursion is a recursion method bearing the same relationship to structural induction as ordinary recursion bears to ordinary mathematical induction.

Structural induction is a proof method that is used in mathematical logic (e.g., in the proof of Łoś' theorem), computer science, graph theory, and some other mathematical fields.

All structurally recursive functions on finite (inductively defined) data structures can easily be shown to terminate, via structural induction: intuitively, each recursive call receives a smaller piece of input data, until a base case is reached.

The structural induction proof is a proof that the proposition holds for all the minimal structures, and that if it holds for the immediate substructures of a certain structure S, then it must hold for S also.

As for the case of Alfred Tarski's satisfiability relation for first-order formulas, the positive and negative satisfiability relations of the team semantics for dependence logic are defined by structural induction over the formulas of the language.

Applying the standard technique of proof by cases to recursively-defined sets or functions as in the preceding sections yields structural induction, a powerful generalization of mathematical induction which is widely used to derive proofs in mathematical logic and computer science.

Our proposition P(l) is that EQ is true for all lists M when L is l. We want to show that P(l) is true for all lists l. We will prove this by structural induction on lists.