Calculus of constructions #142

Chandanra65 · 2024-11-25T01:26:10Z

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• Tis a term (also called type);

• P is a term (also called prop, the type of all propositions);

• Variables (æ,y,...) are terms;
• If A and B are terms, then so is (AB):

• If A and B are terms and z is a variable, then the following are also terms:
• (Az: A. B),
o (Ve:A.B).

In other words, the term syntax, in Backus-Naur form, is then:

e =T|P |z |ee|Az:e. e | Ve:e. e
The calculus of constructions has five kinds of objects:

proofs, which are terms whose types are propositions;
propositions, which are also known as small types;
predicates, which are functions that return propositions;
large types, which are the types of predicates (P is an example of a large type);
5.T itself, which is the type of large types.

Judgments

The calculus of constructions allows proving typing judgments:
æ1 : A1, z? : Az,...Ft:B,

which can be read as the implication

If variables T1,T2,... have, respectively, types A1, A2,..., then term t has type B.
The valid judgments for the calculus of constructions are derivable from a set of inference rules. In
the following, we use to meana sequence of type assignments a1 : A1,2 : Az,...;

A, B, C, D to mean terms; andK, L to mean either P or T. We shall write B[r;= N to mean
the result of substituting the term N for the free variable in the term B.

An inference rule is written in the form

TEA: B
VEC: D'

which means

ifIEA: Bis a valid judgment, then so is TEC:D.

Inference rules for the calculus of constructions

1 TEP:T
2. T,a:A, T'Ea:A

TEA:K I,z:AE B:L

TE (e: A. B) : L
TEA:K T,z:AFN:B

4 TE (Az:A. N) : (Ve :A. B)

TEM:(Ve:A. B) TEN: A
5 TE MN: B = N]

TE M: A A=8 B TEB:K

6

TEM: B

Defining logical operators

The calouue of hac vory fa bocio

tho ony logiogl onors

The calculus of constructions has very few basic operators: the only logical operator for forming
propositions is . However, this one operator is sufficient to define all the other logical operators:

A→ B = V :A. B (z¢ B)
A^B = VC:P. (A → B→ C) →C
AVB = VO:P. (A→ C) (B→ C) →0

= VC:P. (A → )
Bz:A. B = VO:P. (Vz: A. (B → C) →0

Defining data types

The basic data types used in computer science can be defined within the calculus of constructions:
Booleans

VA:P.A → A→ A

Naturals

VA: P. (A → A) → A→A
Product Ax B

AAB
Disjoint union A + B
AVB

Note that Booleans and Naturals are defined in the same way as in Church encoding. However,
additional problems arise from propositional extensionality and proof irrelevance. 2)

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Calculus of constructions #142

Calculus of constructions #142

Chandanra65 commented Nov 25, 2024 •

edited

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Calculus of constructions #142

Calculus of constructions #142

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Chandanra65 commented Nov 25, 2024 • edited Loading

Chandanra65 commented Nov 25, 2024 •

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