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schema
simply typed lambda-calculus
logical verification lecture 2
2015 04 02
simply typed λ-calculus
Curry-Howard-de Bruijn isomorphism
Curry-Howard-De Bruijn isomorphism
some remarks
further reading
from logic to type theory
schema
simply typed lambda-calculus
logic
type theory
proofs
λ-terms
on paper
×
today
some remarks
in Coq
×
today
further reading
Curry-Howard-De Bruijn isomorphism
programming styles
functional programming
imperative programming
C
object-oriented programming
C++ Java
functions are first class objects
functions can be anonymous
logic programming
prolog
functions may return functional values
functional programming
lisp
ML
Haskell
currying
simply typed λ-calculus: types
f :A×B →C
• atomic type (type variable)
may admit partial evaluation:
f (a, ·) : B → C
a b c ...
• function type
(A → B)
curried version:
f :A→B→C
simply typed λ-calculus: terms
parentheses conventions
• variable
x
function types associate to the right
• lambda abstraction
(λx : A. M)
the function that maps the variable x of type A to M
• function application
application associates to the left
(f a) b for f : A → (B → C )
is written as
f a b for f : A → B → C
(F M)
the application of the function F to the argument M
lambda-terms: examples
lambda-terms: more examples
term:
type:
λx : A. x
A→A
term:
type:
λx : R. x
R→R
λx : A. λy : B. x : A → B → A
term:
type:
λx : (A → B) → C → D. λy : C . λz : B. x (λw : A. z) y
((A → B) → C → D) → C → B → D
λx : A. λy : A. y : A → A → A
λx : A. x : A → A
λx : A. λy : A. x : A → A → A
judgement
typing rules
variable rule:
Γ`x :A
x1 : A1 , . . . , xn : An ` M : A
in an environment x1 : A1 , . . . , xn : An the term M has type A
if x : A ∈ Γ
abstraction rule:
Γ, x : A ` M : B
Γ ` (λx : A. M) : A → B
application rule:
Γ`F :A→B
Γ`M:A
Γ ` (F M) : B
lambda-terms: examples reconsidered
x :A`x :A
` λx : A. x : A → A
schema
simply typed lambda-calculus
Curry-Howard-De Bruijn isomorphism
x : A, y : B ` x : A
x : A ` λy : B. x : B → A
` λx : A. λy : B. x : A → B → A
consider also λx : (A → B) → C → D. λy : C . λz : B. x (λw : A. z) y
some remarks
further reading
Curry-Howard-De Bruijn isomorphism
Curry- Howard- De Bruijn- isomorphism:
logic ∼ typed λ-calculus
formulas as types
propositional variable a
∼
type variable a
implication A → B
∼
function space A → B
today more in particular:
minimal propositional logic ∼ simply typed λ-calculus
proofs as terms: introduction
proofs as terms: elimination
implication introduction and abstraction
implication elimination and application
[Ax ]
..
.
B
I [x] →
A→B
Γ, x : A ` M : B
Γ ` (λx : A. M) : A → B
A→B
B
A
E→
Γ`F :A→B
Γ`M:A
Γ ` (F M) : B
Curry-Howard-De Bruijn isomorphism
provability is inhabitation
minimal prop1
∼
λ→
formula
propositional variable
connective →
∼
∼
∼
type
type variable
type constructor →
inhabitation
do we have a closed term P of type A?
proof
assumption
implication introduction
implication elimination
∼
∼
∼
∼
term
term variable
abstraction
application
provability
do we have a proof P of the formula A?
provability
proof checking
∼
∼
inhabitation
type checking
proof checking is type checking
type checking:
is the typing derivation Γ ` P : A correct?
corresponds to
proof checking
is the proof P of the formula A using assumptions Γ correct?
and is decidable for λ→ and ML
corresponds to
decidable for λ→ and ML
(but for instance not for first-order predicate logic)
Brouwer-Heyting-Kolmogorov interpretation
proof of A → B
∼
proof of A ∧ B
proof of A ∨ B
proof of ⊥
∼
∼
function that maps
proofs of A to proofs of B
pair of a proof of A and a proof of B
either a proof of A or a proof of B
does not exist
example: proof objects
• λx : A. x : A → A
the function type A → A represents a proposition
the term λx : A. x represents a proof of that proposition
example: proof and proof object
[Ax ]
I [y ] →
B→A
I [x] →
A→B→A
• λx : A. λy : B. x : A → B → A
• λx : A. λy : A. x : A → A → A
:
A→B→A
∼
λx : A. λy : B. x
:
A→B→A
• λx : A. λy : A. y : A → A → A
schema
alpha equivalence
simply typed lambda-calculus
Curry-Howard-De Bruijn isomorphism
we identify expressions
that are equal
some remarks
up to a renaming of bound variables
further reading
alpha equivalence: examples
same terms: λx : A. x and λy : A. y
same proofs:
[Ax ]
I [x] →
A→A
and
[Ay ]
I [y ] →
A→A
Coq term syntax
x
alpha equivalence: non-examples
different terms: λx : A. λy : A. x and λx : A. λy : A. y
different proofs:
[Ax ]
I [y ] →
A→A
I [x] →
A→A→A
and
[Ay ]
I [y ] →
A→A
I [x] →
A→A→A
Coq commands
Check
prints a term with its type
fun x:A => M
Print
prints the term for a symbol with its type
M N
Definition
binds a term to an identifier
Coq example
fun x:A => x : A -> A
further reading
• intuitionism
• ten questions about intuitionism
arrow is implication for Coq as proof checker
arrow is function type for Coq as programming language
• interactive theorem proving