We have seen all the ingredients of the foundations of mathematics as implemented in Lean. Our goal now is to:
- Systematically review them.
- Put them in the context of what we mean by foundations of mathematics.
- Separate the foundations from the syntax parsing and elaboration which help us work efficiently with Lean.
Foundations of foundations #
The starting assumption is that we are in agreement with the meaning of syntax, including pattern matching and substitution. For example, we can understand the following definition is valid in Lean, assuming we know what
def cube : ℕ → ℕ := fun n → n * n * n
and that we can deduce from the above that
cube 3 = 3 * 3 * 3.
We have rules for foundations based on syntax. We also assume that the rules can be checked mechanically, i.e., by a computer program, and we are convinced that some such program is correct. For a flexible syntax, this is a non-trivial task. So we may want to restrict to a simpler syntax (as we will when we consider classical foundations).
We have to also check that we correctly represent the corresponding "real world" concept.
Context, Rules, Terms #
We assume that at any point we have a Context. In Lean this means that we have terms that we can refer to by name, and that these terms have a type, though this may be an expression.
In any context, we have Rules to
- form expressions from terms and expressions, with a well-formed expression representing a term.
- make two kinds of judgments:
e : tmeans that
eis an expression of type
e = e'means that
e'are equal expressions.
Generative grammars #
Expressions and Rules are based on a generative grammar. Thus, we do not specify whether an expression or judgement is valid or not. Instead, we specify how we can make well-formed expressions and valid judgements.
Local Contexts #
In addition to the names in the context, we can have some names that are only valid in a particular expression. These are called local variables.
Initial Context #
Even an empty Lean file does not have an empty context. Instead the context has Foundational Types.
We can add axioms to the context. This means that we introduce a term and give an expression for its type. This should be done with care so that the context does not become inconsistent.
Rules for Lean #
The rules for Lean let us introduce expressions and at the same time make judgements about them.
- Function types
- Function application
- Function definitions by λ-abstraction.
- Dependent function definition and application
- Inductive types
- Indexed inductive types
Propositions as Types #
Remarkably, the same set of rules let us encode both objects of mathematics and statements about them. The statements (called propositions) are encoded as types with a proof of the proposition being a term of that type.
Logical connectives are encoded as function types, Π-types, and inductive types. The rules for deduction follow from the rules for Lean.
In Lean, there is an additional rule for a proposition
α : Prop, namely if
p : α and
q : α then
p = q.
We use various rules:
- Given types
β, we can form the type of functions from
α → β.
- We can form a function application
f xfrom a function
fand an argument
- We can form a function definition
fun x ↦ efrom a variable
xand an expression
eis defined in the local context with the additional expression
- When defining a function using
x ↦ e, we also introduce a rule that the function applied to
ais the result of substituting
- In the definition below, types should match.
- Often Lean can infer some of the types.
theorem is just like a definition, except
- The type must be a proposition, i.e. the type of the type must be
- The type must be fully inferred from the left-hand side.
- The value is then a proof of the proposition.
one plus two is three
These can be viewed as:
- products of a family of types indexed by a type.
- a type corresponding to functions whose codomain depends on the argument.
- these are sections of a bundle.
We consider an example
We define a dependent function associating to a natural number
n the tuple of
n natural numbers
(n, n-1, ..., 0).
- this has type
(n: ℕ) → NatTuple (n + 1).
- descending 0 = (0, ())
- descending (Nat.succ n) = (n + 1, descending n)