# Framework of Inductive Definitions

This chapter is the development of basic framework of inductive definitions.

An inductive definition consists of a set of rules for deriving judgments.

Judgments are statements about one or more abt’s of some sort.

The rules specify necessary and sufficient conditions for the validity of a judgment, hence fully determine its meaning.

# Judgments

Examples:

Notations Explanations
$n$ nat $n$ is a natural number
$n_1 + n_2 = n$ $n$ is the sum of $n_1$ and $n_2$
$\tau$ type $\tau$ is a type
$e : \tau$ expression $e$ has type $\tau$
$e \Downarrow v$ expression $e$ has value $v$

## Judgment Form

A judgment states that one or more abt’s have a property or stand in some relation to one another.

This kind of property or relation is called a judgment form.

## Instance of Judgment Form

The judgment that an object or objects have the property or stand in the relation is said to be an instance of that judgment form.

## Predicate and Subjects

A judgment form is also called a predicate, the objects constituting an instance are its subjects.

## Some Notations

We write $a\space\mathrm{J}$ or $\mathrm J \space a$ for the judgment asserting $\mathrm J$ holds for abt $a$.

The notation of the judgment form $\mathrm J$ sometimes can be $-\space\mathrm J$ or $\mathrm J\space-$ to indicate that argument is absent.

If it’s not important to stress the subject of the judgment, we write $J$ to stand for unspecified judgment or an instance of some judgment form.

Freely use the prefix/infix/mix-fix notation to enhance the readability.

# Inference Rules

## Inductive Definition

An inductive definition of a judgment form consists of a collection of rules of form:

$$\frac {J_1 \space\dots\space J_k}J$$

in which $J, J_1, \dots, J_k$ are all judgments of the form being defined.

## Premises and Conclusion

The judgments above the horizontal line are premises of the rule.

The judgment below the horizontal line is conclusion.

## Axiom and Proper Rule

If there’s no premises in a rule, the rule is an axiom, otherwise it is an proper rule.

## Sufficient Relationship

An inference rule can be read as: “premises are sufficient for conclusion”: to show $J$, it is enough to show $\forall 1 \le i \le k, J_i$.

If $k = 0$, the rule is axiom, a rule states that its conclusion holds unconditionally.

Since many rules with same conclusion and each specifying sufficient conditions for conclusion.

If the conclusion holds, it is not necessary that premises hold.

## Rule Scheme

A finite number of patterns or rule schemes can specify an infinite family of rules by making use of a notational convention.

Example: $$\frac{a \space\text{nat}}{\text{succ} (a) \space \text{nat}}$$ is a rule scheme that determines one rule, called an instance of the rule scheme for each choice of object $a$ in the rule.

We rely context to determine whether a rule is stated for:

• A specific object
• As a rule scheme specifying a rule for each choice of objects in a rule.

## “Closed Under” and “Strongest”

A collection of rules is considered to define the strongest judgment form that is closed under or respects the rules.

### “Closed Under”

To be closed under the rules means the rules are sufficient to show the validity of a judgment:

$J$ holds if there’s a way to obtain it with given rules.

### “Strongest”

To be the strongest judgment form closed under the rules means that the rules are also necessary:

$J$ holds only if there’s a way to obtain it with given rules.

### Sufficiency and Necessity

Sufficiency means we may show that $J$ holds by deriving it with composing rules.

Necessity means we may reason about it with rule induction.

# Derivation

A derivation of a judgment is a finite composition of rules, starting with axioms and ending with that judgment.

We sometimes say a derivation of $J$ is evidence for the validity of an inductively defined judgment $J$.

## As a Tree

We depict derivations as trees with conclusion at bottom and with the children of a node corresponding to a rule appearing above as evidence for the premises of that rule.

If $$\frac {J_1 \space\dots\space J_k}J$$ is an inference rule and $\bigtriangledown_1,\dots,\bigtriangledown_k$ are derivations of its premises, then $$\frac {\bigtriangledown_1\space\dots\space\bigtriangledown_k}J$$ is a derivation of its conclusion.

## Find Derivations

Two methods to find derivations:

• forward chaining, or bottom-up construction
• backward chaining, or top-down construction

Check Page 16 for these ideas.

# Rule Induction

The principle of rule induction states that: to show property $a\space P$ holds whenever $a\space\mathrm{J}$ is derivable, and more precisely:

• It is enough to show $P$ is closed under the rules defining the judgment form $\textrm{J}$.
• The property $P$ respects the rule $$\frac{a_1\space\textrm{J}\space\dots\space a_k\space\textrm{J}}{a\space \textrm J}$$ if $P(a)$ holds whenever $\forall 1 \le i \le k, P(a_i)$ do.

The assumptions $\forall 1 \le i \le k, P(a_i)$ are called inductive hypotheses and $P(a)$ is called inductive conclusion of the inference.

The principle of rule induction is simply the expression of the definition of an inductively defined judgment form as the strongest judgment form closed under the rules comprising the definition.

So the requirement of the judgment form defined by a set of rules is simply:

• closed under those rules.
• sufficient for any other property also closed under those rules.

# Iterated

Inductive definitions are often iterated, meaning that one inductive definition builds on top of another.

Theoretical Example: in an iterate inductive definition $$\frac {J_1 \space\dots\space J_k}J$$ the premises of a rule may be instances of:

• a previously defined judgment form
• judgment form being defined

# Simultaneous Inductive Definitions

Frequently two or more judgments are defined at once by a simultaneous inductive definition.

A simultaneous inductive definition consists of a set of rules deriving instances of several different judgment forms, any of which may appear as premise of any rule.

None of the judgment forms are being defined prior to the others.

# Defining Functions by Rules

A common use of inductive definitions is to define a function by giving an inductive definition of its “graph” relating inputs and outputs. Then show that the relation uniquely determines the outputs for given inputs.

# Exercises

## 2.1

$$\frac{}{\mathrm{max}(q; \mathrm{zero}; q)}$$

$$\frac{}{\mathrm{max}(\mathrm{zero}; \mathrm{succ}(p); \mathrm{succ}(p))}$$

$$\frac{p\space \mathrm{nat}; q\space \mathrm{nat}; \mathrm{max}(p; q; r)}{\mathrm{max}(\mathrm{succ}(p); \mathrm{succ}(q); \mathrm{succ}(r))}$$

If we want to prove that for $\mathrm{max}(m;n;p)$ $\forall m,n$ are related to a unique $p$, then we must prove the existence and uniqueness.

• existence:
• According to the first and second rule, we can see that $P(q;\text{zero})$ and $P(\text{zero};\text{succ}(p))$ holds.
• According to the final rule, we see that if $P(p; q)$ then $P(\text{succ}(p); \text{succ}(q))$ holds.
• Thus the existence part proved.
• uniqueness:
• We try to prove that if $\mathrm{max}(m;n;p)$ and $\mathrm{max}(m;n;q)$, then $p$ is $q$.
• Since the first two judgment form guarantee that if $m$ is $n$, then there’s only one unique solution, then these rule guarantee that the uniqueness proved.

The proof establishes that the three-place relation defines a total function of its first two arguments.

## 2.2

According to the 2.1 solution, we can see that every tree can be mapped to a $n$ nat.

Then by rule induction:

• If $t$ is $\mathrm{empty}$, then $n$ nat since $n$ is $\mathrm{zero}$.
• If $t$ is $\mathrm{node}(t_1; t_2)$, then we can see $n_1$ nat, $n_2$ nat, $n$ nat by 2.1.
• Thus it is proved that hgt is a function from trees to nat.

## 2.3

$$\frac{}{\text{nil}\space\mathrm{forest}}$$

$$\frac{f \space\mathrm{forest}}{\text{node}(f)\space\mathrm{tree}}$$

$$\frac{t\space\mathrm{tree}\space\space f\space\mathrm{forest}}{\mathrm{cons}(t;f)\space\mathrm{forest}}$$

## 2.4

$$\frac{}{\mathrm{hgtF}(\mathrm{nil};\mathrm{zero})}$$

$$\frac{\mathrm{hgtF}(f; h)}{\mathrm{hgtH}(\mathrm{node}(f);\mathrm{succ}(h))}$$

$$\frac{\mathrm{hgtF}(f;h_1)\space\space\mathrm{hgtT}(t;h_2)\space\space\mathrm{max}(h_1;h_2;h)}{\mathrm{hgtF}(\mathrm{cons}(f;t);h)}$$

## 2.5

$$\frac{}{\mathrm{zero}\space\mathrm{bin}}$$

$$\frac{n\space\mathrm{bin}}{\mathrm{twice}(n)\space\mathrm{bin}}$$

$$\frac{n\space\mathrm{bin}}{\mathrm{twiceplusone}(n)\space\mathrm{bin}}$$

## 2.6

$$\frac{}{\mathrm{sum}(\mathrm{zero};\mathrm{zero};\mathrm{zero})}$$

$$\frac{\mathrm{sum}(m;n;p)}{\mathrm{sum}(\mathrm{twice}(m);\mathrm{twice}(n);\mathrm{twice}(p))}$$

$$\frac{\mathrm{sum}(m;n;p)}{\mathrm{sum}(\mathrm{twiceplusone}(m);\mathrm{twice}(n);\mathrm{twiceplusone}(p))}$$

$$\frac{\mathrm{sum}(m;n;p)}{\mathrm{sum}(\mathrm{twice}(m);\mathrm{twiceplusone}(n);\mathrm{twiceplusone}(p))}$$

$$\frac{\mathrm{sum}(m;n;p)\space\mathrm{succ}(p; q)}{\mathrm{sum}(\mathrm{twiceplusone}(m);\mathrm{twiceplusone}(n);\mathrm{twiceplusone}(q))}$$

With auxiliary function for succ in bin:

$$\frac{}{\mathrm{succ}(\mathrm{zero}; \mathrm{twiceplusone}(\mathrm{zero}))}$$

$$\frac{}{\mathrm{succ}(\mathrm{twice}(p); \mathrm{twiceplusone}(p))}$$

$$\frac{\mathrm{succ}(p; q)}{\mathrm{succ}(\mathrm{twiceplusone}(p); \mathrm{twice}(q))}$$

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