Introduction to the Calculus of Inductive Constructions Workshop X - PowerPoint PPT Presentation

Introduction to the Calculus of Inductive Constructions Workshop ∀ X . X π , Vienna Summer of Logic Christine Paulin-Mohring e Paris Sud & INRIA Saclay - ˆ Universit´ Ile-de-France July 18, 2014 C. Paulin (Paris-Sud) Calculus of Inductive Constructions Jul. 2014 1 / 33

Introduction Motivations ◮ Limitations of first-order logic ◮ need to work in axiomatic theories ◮ consistency issue ◮ infinite models of any cardinality ◮ equational reasoning but no computation on terms ◮ Calculus of Inductive Constructions ◮ a powerful functional basis ◮ types to automatically classify objects ◮ a general scheme to declare data-structures and relations ◮ rules for recursion / induction ◮ logical foundation of type theory (Agda, Coq) C. Paulin (Paris-Sud) Calculus of Inductive Constructions Jul. 2014 2 / 33

Introduction Examples of inductive data-types Emphasis is on constructors Inductive bool := true | false. O | S : nat → nat. Inductive nat := Inductive min := At : nat → min | Imp : min → min → min. Inductive tree A := leaf | node : A → tree A → tree A → tree A. Inductive BDT := T | F | var : nat → (bool → BDT) → BDT. Inductive W A B := node : ∀ (a:A),(B a → W A B) → W A B. C. Paulin (Paris-Sud) Calculus of Inductive Constructions Jul. 2014 3 / 33

Introduction Examples of inductive data-types Emphasis is on constructors Inductive bool := true | false. O | S : nat → nat. Inductive nat := Inductive min := At : nat → min | Imp : min → min → min. Inductive tree A := leaf | node : A → tree A → tree A → tree A. Inductive BDT := T | F | var : nat → (bool → BDT) → BDT. Inductive W A B := node : ∀ (a:A),(B a → W A B) → W A B. C. Paulin (Paris-Sud) Calculus of Inductive Constructions Jul. 2014 3 / 33

Introduction Examples of inductive data-types Emphasis is on constructors Inductive bool := true | false. O | S : nat → nat. Inductive nat := Inductive min := At : nat → min | Imp : min → min → min. Inductive tree A := leaf | node : A → tree A → tree A → tree A. Inductive BDT := T | F | var : nat → (bool → BDT) → BDT. Inductive W A B := node : ∀ (a:A),(B a → W A B) → W A B. C. Paulin (Paris-Sud) Calculus of Inductive Constructions Jul. 2014 3 / 33

Introduction Expected properties Inductive min := At : nat → min | Imp : min → min → min. Initiality – iterator (fold) Variable X : Type. Variable Xat : nat → X. Variable Ximp : X → X → X. Definition It (A : min) : X := ... Lemma Itat : ∀ n, It (At n) = Xat n. Lemma Itint : ∀ A B, It (Imp A B) = Ximp (It A) (It B). Non confusion Lemma diff : ∀ n A B, (At n) � = (Imp A B). Induction Variable P (A : min) : Prop. Variable Pat : ∀ (n:nat),P (At n). Variable Pimp : ∀ A B : min,P A → P B → P (Imp A B) Definition min_ind : ∀ A : min,P A := ... C. Paulin (Paris-Sud) Calculus of Inductive Constructions Jul. 2014 4 / 33

Introduction Examples of Inductive Relations Relations defined by inference rules, clauses A , l ⊢ B , r l ⊢ A , r B , l ⊢ r A , l ⊢ A , r l ⊢ A ⇒ B , r A ⇒ B , l ⊢ r Inductive proof : set min → set min → Prop := ax : ∀ (A:min)(l r:set min), proof (A :: l) (A :: r) | right : ∀ (A B:min)(l r:set min) proof (A :: l) (B :: r) → proof l (Imp(A,B) :: r) | left : ∀ (A B:min)(l r:set min) proof l (A :: r) → proof (B :: l) r → proof (Imp(A,B) :: l) r C. Paulin (Paris-Sud) Calculus of Inductive Constructions Jul. 2014 5 / 33

Introduction Examples of Inductive Relations Relations defined by inference rules, clauses A , l ⊢ B , r l ⊢ A , r B , l ⊢ r A , l ⊢ A , r l ⊢ A ⇒ B , r A ⇒ B , l ⊢ r Inductive proof : set min → set min → Prop := ax : ∀ (A:min)(l r:set min), proof (A :: l) (A :: r) | right : ∀ (A B:min)(l r:set min) proof (A :: l) (B :: r) → proof l (Imp(A,B) :: r) | left : ∀ (A B:min)(l r:set min) proof l (A :: r) → proof (B :: l) r → proof (Imp(A,B) :: l) r C. Paulin (Paris-Sud) Calculus of Inductive Constructions Jul. 2014 5 / 33

Introduction Expected properties Minimality Variable P (l r : set min) : Prop. Variable Pax : ∀ (A:min)(l r:set min),P (A :: l) (A :: r). Variable Pright : ∀ (A B:min)(l r:set min) P (A :: l) (B :: r) → P l (Imp(A,B) :: r). Variable Pleft : ∀ (A B:min)(l r:set min) P l (A :: r) → P (B :: l) r → P (Imp(A,B) :: l) r. Lemma proof_ind : ∀ l r, proof l r → P l r. C. Paulin (Paris-Sud) Calculus of Inductive Constructions Jul. 2014 6 / 33

Introduction Mathematical justification Intersection of a non-empty family of sets � � G = X = { x ∈ X 0 |∀ X ∈ G , x ∈ X } X ∈ G Fixpoints of monotonic operators ∃ A l ′ r ′ , l = A :: l ′ ∧ r = A :: r ′ F ( P )( l , r ) def = ∨∃ A B r ′ , r = Imp ( A , B ) :: r ′ ∧ P ( A :: l , B :: r ′ ) ∨∃ A B l ′ , l = Imp ( A , B ) :: l ′ ∧ P ( l ′ , A :: r ) ∧ P ( B :: l ′ , r ) proof def = � { P ∈ ℘ ( M × M ) |∀ l r , F ( P )( l , r ) ⇒ P ( l , r ) } ⇔ F ( proof ) Minimality ( ∀ l r , F ( P )( l , r ) ⇒ P ( l , r )) ⇒ ∀ l r . proof ( l , r ) ⇒ P ( l , r ) C. Paulin (Paris-Sud) Calculus of Inductive Constructions Jul. 2014 7 / 33

Introduction Inductive data-types ◮ start with an infinite set (usually N ) ◮ find a physical representation ◮ support type : words, trees, . . . (large enough, infinite branching) ◮ map “nodes” to index of constructors ◮ inductive definition of the set of terms ◮ (abstract) type of well-formed terms C. Paulin (Paris-Sud) Calculus of Inductive Constructions Jul. 2014 8 / 33

Introduction Calculus of Inductive Constructions ◮ calculus and logic behind the C OQ proof assistant ◮ can be used as higher-order logic ◮ propositions as types, proofs as objects ◮ dependent types ◮ internal computation fib ( 7 ) ≡ 13 ◮ constructive, intensional logic ◮ λ x ⇒ ¬¬ x � = λ x ⇒ x ◮ bool � = Prop ◮ �⊢ ¬¬ A → A C. Paulin (Paris-Sud) Calculus of Inductive Constructions Jul. 2014 9 / 33

Language and Rules Outline Introduction Language and Rules Properties C. Paulin (Paris-Sud) Calculus of Inductive Constructions Jul. 2014 10 / 33

Language and Rules Logical Rules : functional part ◮ Same language for terms and types: t ::= S | V | ∀ x : t , t | λ x : t ⇒ t | t t ◮ Sorts: S = { Prop , ( Type i ) i ∈ N } t → u def ◮ Notation: = ∀ : t , u ◮ Judgements: Γ ⊢ t : t ′ ◮ every object is typed ◮ a type is a term of type a sort ◮ everything (signature, hypothesis) is declared and named in the context in a consistent way A : Type , R : A → A → Prop , x : A , y : A , p : R x y ⊢ C. Paulin (Paris-Sud) Calculus of Inductive Constructions Jul. 2014 11 / 33

Language and Rules (Informal) Rules ◮ Sorts: Prop , Type i : Type i + 1 Γ ⊢ A : s x �∈ Γ ( x : A ) ∈ Γ ◮ Variables: Γ , x : A ⊢ Γ ⊢ x : A Γ , x : A ⊢ B : s Γ ⊢ A : s ′ s = s ′ or s = Prop ◮ Product: Γ ⊢ ∀ x : A , B : s ◮ Abstraction/Application: Γ , x : A ⊢ t : B Γ ⊢ t : ∀ x : A , B Γ ⊢ u : A Γ ⊢ λ x : A ⇒ t : ∀ x : A , B Γ ⊢ t u : B [ x ← u ] Γ ⊢ t : A Γ ⊢ B : s A ≡ B ◮ Computation ( β ) Γ ⊢ t : B C. Paulin (Paris-Sud) Calculus of Inductive Constructions Jul. 2014 12 / 33

Language and Rules Examples Γ , x : A ⊢ t : B Γ ⊢ t : ∀ x : A , B Γ ⊢ u : A Γ ⊢ λ x : A ⇒ t : ∀ x : A , B Γ ⊢ t u : B [ x ← u ] ◮ functional application Γ , x : A ⊢ t : B Γ ⊢ t : A → B Γ ⊢ u : A Γ ⊢ λ x : A ⇒ t : A → B Γ ⊢ t u : B ◮ natural deduction for implication Γ , A ⊢ B Γ ⊢ A ⇒ B Γ ⊢ A Γ ⊢ A ⇒ B Γ ⊢ B ◮ natural deduction for universal quantification Γ ⊢ B x �∈ Γ Γ ⊢ ∀ x , B Γ ⊢ ∀ x , B Γ ⊢ B [ x ← u ] C. Paulin (Paris-Sud) Calculus of Inductive Constructions Jul. 2014 13 / 33

Language and Rules Dependent types type of maps parameterized by a domain map : ∀ (A B:Type)(A → Prop) → Type find : ∀ (A B:Type)(d:A → Prop)(m:map A B d)(x:A)d x → B C. Paulin (Paris-Sud) Calculus of Inductive Constructions Jul. 2014 14 / 33

Language and Rules Fundamental (non-)inductive definitions Inductive zero : Type := . Inductive unit : Type := tt : one. Inductive sum (A B:Type) : Type := inl : A → sum A B | inr : B → sum A B. Inductive sig (A:Type) (B:A → Type) : Type := pair : ∀ a:A, B a → sig A B. propositions as types � constructor = introduction rule zero ≡ ⊥ unit ≡ ⊤ sum A B ≡ A ∨ B sig A ( λ x ⇒ B ) ≡ ∃ x : A , B sig A ( λ ⇒ B ) ≡ A ∧ B C. Paulin (Paris-Sud) Calculus of Inductive Constructions Jul. 2014 15 / 33

Language and Rules Elimination by pattern-matching ◮ Any value in an inductive type I starts with a constructor c 1 | . . . | c p ◮ Initiality in the non-recursive case : X : Type f 1 , . . . , f p : . . . match I ( f 1 , . . . , f p ) : I → X ◮ Dependent pattern-matching (proof by case): P : I → Type f 1 , . . . , f p : . . . match I ( f 1 , . . . , f p ) : ∀ i : I , P i ◮ Exactly one branch f i for each constructor c i ◮ when c i : ∀ ( x 1 : A 1 ) . . . ( x n : A n ) I we require f i : ∀ ( x 1 : A 1 ) . . . ( x n : A n ) P ( c i x 1 . . . x n ) ◮ Computation : match I ( f 1 , . . . , f p )( c i t 1 . . . t n ) − → f i t 1 . . . t n C. Paulin (Paris-Sud) Calculus of Inductive Constructions Jul. 2014 16 / 33