Induction-induction

Key takeaways

Induction-induction belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Induction-induction to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Induction-induction from memory before moving on to harder problems.

Reference excerpt

In intuitionistic type theory (ITT), a discipline within mathematical logic, induction-induction is for simultaneously declaring some inductive type and some inductive predicate over this type. An inductive definition is given by rules for generating elements of some type. One can then define some predicate on that type by providing constructors for forming the elements of the predicate, such inductively on the way the elements of the type are generated. Induction-induction generalizes this situation since one can simultaneously define the type and the predicate, because the rules for generating elements of the type A : T y p e {\displaystyle A:{\mathsf {Type}}} are allowed to refer to the predicate B : A → T y p e {\displaystyle B:A\to {\mathsf {Type}}} . Induction-induction can be used to define larger types including various universe constructions in type theory. and limit constructions in category/topos theory.

Example 1 Present the type A {\displaystyle A} as having the following constructors, note the early reference to the predicate B {\displaystyle B} :

a a : A {\displaystyle aa:A}

ℓ ℓ : ∑ x : A B ( x ) → A ; {\displaystyle \ell \ell :\sum _{x:A}B(x)\to A;}

and-simultaneously present the predicate B {\displaystyle B} as having the following constructors :

T r u : B ( a a ) {\displaystyle {\mathsf {Tru}}:B(aa)}

F a l : B ( a a ) {\displaystyle {\mathsf {Fal}}:B(aa)}

if x : A {\displaystyle x:A} and y : B ( x ) {\displaystyle y:B(x)} then Z e r : B ( ℓ ℓ ( x , y ) ) {\displaystyle {\mathsf {Zer}}:B(\ell \ell (x,y))}

if x : A {\displaystyle x:A} and y : B ( x ) {\displaystyle y:B(x)} and z : B ( ℓ ℓ ( x , y ) ) {\displaystyle z:B(\ell \ell (x,y))} then S u c ( z ) : B ( ℓ ℓ ( x , y ) ) {\displaystyle {\mathsf {Suc}}(z):B(\ell \ell (x,y))} .

Example 2 A simple common example is the Universe à la Tarski type former. It creates some inductive type U : T y p e {\displaystyle U:{\mathsf {Type}}} and some inductive predicate T : U → T y p e {\displaystyle T:U\to {\mathsf {Type}}} . For every type in the type theory (except U {\displaystyle U} itself!), there will be some element of U {\displaystyle U} which may be seen as some code for this corresponding type; The predicate T {\displaystyle T} inductively encodes each possible type to the corresponding element of U {\displaystyle U} ; and constructing new codes in U {\displaystyle U} will require referring to the decoding-as-type of earlier codes, via the predicate T {\displaystyle T} .

See also Induction-recursion – for simultaneously declaring some inductive type and some recursive function on this type.

References

External links A list of Peter Dybjer's publications on induction and induction-recursion

Worked examples

Example 1 — a first encounter with Induction-induction

Start with the simplest possible case. Write down what Induction-induction claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Induction-induction before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Induction-induction ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Induction-induction

In research: Induction-induction appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Induction-induction in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Induction-induction is common in secondary-school and first-year university syllabi. It links to neighbouring topics Type theory, so understanding it makes those chapters shorter.
In everyday life: Look for Induction-induction outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Induction-induction in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Induction-induction means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Induction-induction out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Induction-induction in simple terms?

In intuitionistic type theory (ITT), a discipline within mathematical logic, induction-induction is for simultaneously declaring some inductive type and some inductive predicate over this type. An inductive definition is given by rules for generating elements of some type.

Why does Induction-induction matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Induction-induction?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Induction-induction.