Congruence in univalent type theory Luis Scoccola lscoccol@uwo.ca University of Western Ontario June 11, 2019
Goals for the talk ◮ Congruence and congruence closure for propositional equality. ◮ Solution of Selsam & de Moura for a non-univalent type theory. ◮ Proposed approach for congruence in the univalent case. ◮ Issues with congruence closure.
Informal definitions Definition A relation R satisfies congruence if ∀ f x i R y i for 0 ≤ i ≤ n ⇒ f ( x 1 , . . . , x n ) R f ( y 1 , . . . , y n ) . The congruence closure of R is the smallest equivalence relation that satisfies congruence and contains R. Example During a proof, determine whether x = y follows from applying reflexivity , symmetry , transitivity , or congruence lemmas to the equalities in our context.
Congruence in dependent type theory Propositional equality is an equivalence relation: refl : x = x inv : x = y → y = x concat : x = y → y = z → x = z Can also prove congruence lemma (for non-dependent function): congr f : ( f : A → B ) → ( x = A y ) → f ( x ) = B f ( y ) . But if f : ( a : A ) → B ( a ), the above doesn’t type check.
A solution: Heterogeneous equality Definition Heterogeneous equality is the inductive family heq : ( A , A ′ : U ) → A → A ′ → U generated by refl : ( A : U ) → ( a : A ) → heq( a , a ) . Selsam & de Moura implement full congruence closure procedure in Lean 3 using heq. Need to assume an axiom to prove the congruence lemmas: ofheq : ( A : U ) → ( x , y : A ) → heq( x , y ) → x = A y . The axiom ofheq implies that paths in U transport trivially: ( e : A = U A ) → a = transport X �→ X ( e , a ) . U can’t be univalent.
Congruence in univalent type theory? Use pathovers . Definition Given a type B : U and a type family X : B → U , the type family pathover : ( b , b ′ : B ) → ( b = b ′ ) → X ( b ) → X ( b ′ ) → U is defined by path induction. � e � x ′ for pathover( b , b ′ , e , x , x ′ ). We write x = B
The congruence lemma I will describe an inductive algorithm that produces: ◮ A pathover type for each dependent family. ◮ A congruence lemma for each dependent function. This is implemented as a tactic in Lean 3. We have to be careful with one thing: Congruence lemmas depend on previous congruence lemmas.
The congruence lemma (cont.) Example Congruence lemma for cons : ( n : N ) → A → vec A ( n ) → vec A (succ( n )) , should have type congr cons ( n , m : N ) ( x , y : A ) ( xs : vec A ( n )) ( ys : vec A ( m )) ( e 1 : n = m ) ( e 2 : x = y ) ( e 3 : xs = � e 1 � ys ) : cons( n , x , xs ) = vec A � congr succ ( e 1 ) � cons( m , y , ys ) where congr succ : ( n , m : N ) → ( n = m ) → succ( n ) = succ( m ).
The algorithm Given context Γ, and dependent function h : ( x 0 : A 0 ) → ( x 1 : A 1 ( x 0 )) → · · · → ( x n : A n ( x 1 , . . . , x n − 1 )) → A n +1 ( x 1 , . . . , x n ) , in context Γ. (1) Decompose the type of h as type families applied to dependent functions: write A i ( x i − 1 ) ≡ C i ( f i ( x i − 1 )) such that ◮ C i is not an application; i , . . . , f k ( i ) ◮ f i is a sequence f 1 of dependent functions. i
The algorithm (cont.) (2) Define the pathovers for the type families C i : Id C i : ≡ λ x i , x ′ i , e i , c , c ′ . congr C i ( x i , x ′ i , e i ) ∗ c = c ′ (3) Define the congruence for all the functions f k i : Recursively. Caveat: Each function might be a composite, so return the composite of the congruences.
The algorithm (cont.) (4) Define the congruence for h : congr h : ( x 0 : C 0 ) → ( x 1 : C 1 ( f 1 ( x 0 ))) → · · · → ( x n : C n ( f n ( x n − 1 ))) ( x ′ 0 : C 0 ) → ( x ′ 1 : C 1 ( f 1 ( x ′ 0 ))) → · · · → ( x ′ n : C n ( f n ( x ′ n − 1 ))) ( e 0 : Id C 0 ) → ( e 1 : Id C 1 (congr f 1 ( e 0 )) → · · · → ( e n : Id C n (congr f n ( e n − 1 )) → Id C n +1 (congr f n +1 ( e n )) , By path induction on the pathovers e 0 , · · · , e n , using refl.
Congruence lemma Also gives us a useful characterization of the identity types of: ◮ structures; ◮ iterated sigmas.
Congruence closure Work in progress: ◮ Must keep a congruence data structure for each type family. ◮ Coherence problems (e.g. congruence of concatenation is concatenation of congruences). ◮ Some of them can be dealt with by working in a cubical type theory (e.g. composite of congruences is definitionally equal to congruence of composites, inverse of inverse is identity). ◮ Different equalities between the same pair of elements: cannot use union-find data structure for congruence closure. Use graphs instead, but this is inefficient. These are not problems if we limit congruence closure to type families that depend on sets.
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