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Proofs and Programs

Proofs and Programs. Wei Hu 11/01/2007. Outline. Motivation Theory Lambda calculus Curry-Howard Isomorphism Dependent types Practice Coq. Motivation - Why Learn Coq. Helps understand PL theory better Good for CS615 (sadly, not quals) Coq is becoming popular

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Proofs and Programs

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  1. Proofs and Programs Wei Hu 11/01/2007

  2. Outline • Motivation • Theory • Lambda calculus • Curry-Howard Isomorphism • Dependent types • Practice • Coq Wei Hu

  3. Motivation - Why Learn Coq • Helps understand PL theory better • Good for CS615 (sadly, not quals) • Coq is becoming popular • People rethinking trusted software • Functional programming is gaining attention • Has been used to • Prove mathematical theorems • Specify hardware • Recently, papers being published for • Operating systems • Compilers Wei Hu

  4. Coq is an Interactive Theorem Prover • Two types of theorem provers • Automated TP • Simplify, CVC, … • Interactive TP(aka, Proof Assistants) • Coq, PVS, ACL2, HOL, Isabelle, Twelf, NuPRL, Agda • Coq’s highlights • Higher-order intuitionistic logic • Higher-order type theory • Embedded functional programming language • Goal transformation through tactics • Mainly works interactively, with limited support of automation to discharge trivial propositions Wei Hu

  5. Coq as a Programming Language • ML-like Fixpoint is_even (n:nat) : bool := match n with | 0 => true | 1 => false | S (S n') => is_even n’ end. Eval compute in is_even 3. Let’s try it out! • Restrictions • No side effects • No non-terminating programs (to avoid inconsistency) Wei Hu

  6. Program Extraction Fixpoint is_even (n:nat) : bool := match n with | 0 => true | 1 => false | S (S n') => is_even n’ end. Coq let recis_even = function | O -> True | S n0 -> (match n0 with | O -> False | S n' -> is_even n') OCaml is_even n = case n of O -> True S n0 -> (case n0 of O -> False S n' -> is_even n') Haskell Wei Hu

  7. Nothing Is Built-in! Inductive bool : Set := | true : bool | false : bool. Inductive nat : Set := | O : nat | S : nat -> nat. Coq (theories/Init/Datatypes.v) type bool = | True | False type nat = | O | S of nat OCaml data Bool = True | False data Nat = O | S Nat Haskell Wei Hu

  8. Simply Typed Lambda Calculus • Type • Base types: nat, bool, … • Function types: nat->nat, nat->bool, … • Term • Variables: x, y, z • Abstractions: x: T1.t2 • Applications: M N • Environment () • Gives typing for variables Wei Hu

  9. Typing Rules x : T   |- t1 : T11 -> T12 , x: T1 |- t2 : T2  |- t2 : T11 (Var) (Abs) (App)  |- x: T1.t2 : T1 -> T2  |- x : T  |- t1t2 : T12 The typing rules work equally well in logic! Wei Hu

  10. Curry-Howard Isomorphism • Propositions = Types • Proofs (inhabit Props) = Programs (inhabit Sets) • Proof tactics = Program constructs • So what? • Proof checking now becomes type checking (can be undecidable if non-termination is allowed) • Programs and proofs are organically unified • Why is it hard to prove something? • Unlike general programming, we are doing type-level programming • In other words, we are constructing programs for a given type • Tactics help with the construction Wei Hu

  11. C-H at Work Section example1. Hypothesis A B : Prop. Hypothesis C : B -> A. Lemma Var : B -> A. exact C. Qed. Lemma Abs : A -> A. intros. exact H. Qed. Lemma App : A -> (A -> B) -> B. intros. apply H0. exact H. Qed. x : T   |- t1 : T11 -> T12 , x: T1 |- t2 : T2  |- t2 : T11  |- t1t2 : T12  |- x : T  |- x: T1.t2 : T1 -> T2 function or implication? Wei Hu

  12. Type-centric View • -> is primitive, others not • theories/Init/Logic.v Inductive True : Prop := I : True. Inductive False : Prop :=. Definition not (A:Prop) := A -> False. Inductive and (A B:Prop) : Prop := conj : A -> B -> A /\ B. Inductive or (A B:Prop) : Prop := | or_introl : A -> A \/ B | or_intror : B -> A \/ B. BHK Interpretation (Brouwer-Heyting-Kolmogorov) Wei Hu

  13. Intuitionistic (Constructive) Logic vs. Classical Logic • A set of equivalent axioms missing in IL • Law of excluded middle: P ∨ ~P • Double negation rule: ~~P → P • Peirce’s Law: ((P→Q)→P)→P • The correspondence between call/cc and Pierce’s Law • See Coq FAQ #30 “What axioms can be safely added to Coq?” • CL takes a denotational view • Assigns to every variable a value (true or false) • Builds truth tables for primitive operations /\ ∨ ~ • (P -> Q) ≡ (~P ∨ Q ) Wei Hu

  14. Polymorphism and Universal Quantification • Parametric polymorphism • You have been using it all the time! • Polymorphic functions • Definition id := fun (X : Set) (x : X) => x. • Check id. id : forall X : Set, X -> X. • Polymorphic data types (e.g., lists) • Universal quantification • Lemma refl : forall P : Prop, P -> P. intros. apply H. Qed. • Print refl. refl = fun (P : Prop) (H : P) => H : forall P : Prop, P -> P Wei Hu

  15. Dependent Types • Type systems in ML-like languages are just not expressive enough • We want to specify propositions like ∀ n:nat, n ≤ n The result type (n ≤ n) depends on arguments (n) • We generalize arrows (A -> B, or λx:A. B) to dependent products (∏x:A. B) where B is dependent on x. • In Coq: • forall n:nat, n<=n • forall mn: nat, m + n = n + m Wei Hu

  16. Dependent Types • Compatibility with non-dependent product • The type of a function that builds a pair: forall (A B:Set) (a:A) (b:B), A*B forall A B : Set, A -> B -> A * B • Defining existential • ∀ x, (P x -> ∃ x, P x) • Dependent sums • Dependent types in programming • Epigram (http://www.e-pig.org/) • Ynot (http://www.eecs.harvard.edu/~greg/ynot.html) • Concoqtion (http://www.metaocaml.org/concoqtion/) • Omega (http://web.cecs.pdx.edu/~sheard/) • ATS (http://www.cs.bu.edu/~hwxi/ATS/) Wei Hu

  17. Polymorphism vs. Dependent Types • Polymorphism • Terms can take types as arguments • Dependent types • Types can take terms as arguments • Inductive types and predicates • Generalization of conventional user-defined data types • We are talking about parameterized data types (e.g. vector n) and propositions (e.g. m <= n) Wei Hu

  18. Conclusions • We talked about type theory • Curry-Howard correspondence is cool • Dependent types matter* • We did not cover common proof techniques • Proof by induction • Use of tactics and tacticals * T. Altenkirch, C. McBride, and J. McKinna, “Why Dependent Types Matter” Wei Hu

  19. How to Learn Coq? • Resources (for beginners) • http://www.cs.virginia.edu/~wh5a/coq.html • Hardest parts (IMO) • Design/Formalization • Combination of tactics • Tools are getting better, but still hard • What’s the next breakthrough? Wei Hu

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