[Typer] Re: Implement `Quotient` formation

Merci pour les commentaires, je viens de faire les changements nécessaires et c’est déjà poussé sur Git.

James

Le 14 août 2023 à 10:17, Stefan Monnier monnier@iro.umontreal.ca a écrit :

Voici quelques commentaires sur les commits sur cette branche (sous la forme de "# FIXME: ..blabla...", quoique c'est pas tous des "fixme").

   Stefan

commit 61d01f99e741c75520814dea2e643121bca83374 Author: James Tan james_tan97@outlook.com Date: Sun Jul 2 21:15:55 2023 -0400

Implement `Quotient` formation

diff --git a/btl/builtins.typer b/btl/builtins.typer index 228999b..5ec79b0 100644 --- a/btl/builtins.typer +++ b/btl/builtins.typer @@ -531,4 +531,11 @@ Heap_unsafe-store-cell = Built-in "Heap.store-cell"; Heap_unsafe-load-cell : Int -> Int -> Heap ?t; Heap_unsafe-load-cell = Built-in "Heap.load-cell";

+%% +%% Quotient types +%% +Quotient = Built-in "Quotient" : (l1 : TypeLevel) ≡> (l2 : TypeLevel) ≡>

•                             (A : Type_ l1) -> (R : A -> A -> Type_ l2) ->
  

•                             Type_ (_∪_ l1 l2);
  

%%% builtins.typer ends here. diff --git a/samples/quotient.typer b/samples/quotient.typer new file mode 100644 index 0000000..1108133 --- /dev/null +++ b/samples/quotient.typer @@ -0,0 +1,33 @@ +Nat : Type;

+type Nat

• | zero
• | succ Nat;

# # FIXME: Can we take this from elsewhere instead?

+_-_ : Nat -> Nat -> Nat; +_-_ x y = case x

• | zero => y
• | succ m => case y

•            | zero => x
  

•            | succ n => m - n;
  

+NatPair = Pair Nat Nat;

+fst p = case p

• | pair m _ => m;

+snd p = case p

• | pair _ n => n;

+normaliseZ : NatPair -> NatPair; +normaliseZ np = case np

• | pair m n => pair (m - n) (n - m);

+equalZ : NatPair -> NatPair -> Type; +equalZ x1 x2 = Pair (Eq (fst x1) (snd x2)) (Eq (snd x1) (fst x2)); # # FIXME: normaliseZ is not used? # FIXME: Shouldn't we define a generic "Quotient by normalization function" # first and then define Z using that?

+%% FIXME: We shouldn't get this error +%% "Requested Built-in "Quotient" does not exist" +%% Z = Quotient (A := NatPair) equalZ;

+%% TODO: Intro and elim \ No newline at end of file # # FIXME: "No newline at end of file" # diff --git a/src/debruijn.ml b/src/debruijn.ml index c2efce7..255d82b 100644 --- a/src/debruijn.ml +++ b/src/debruijn.ml @@ -145,7 +145,7 @@ let type_eq_type = let type_eq = mkBuiltin ((dloc, "Eq"), type_eq_type)

let builtin_axioms =

• ["Int"; "Elab_Context"; "IO"; "Ref"; "Sexp"; "Array"; "FileHandle";

• ["Int"; "Elab_Context"; "IO"; "Ref"; "Sexp"; "Array"; "FileHandle"; "Quotient"; (* From `healp.ml`. *) "Heap"; "DataconsLabel"]

commit 1282e0bab0833e9267701dbc4bd4619961c11308 Author: James Tan james_tan97@outlook.com Date: Tue Jul 11 02:01:17 2023 -0400

Implement `Quotient` introduction

diff --git a/btl/builtins.typer b/btl/builtins.typer index 5ec79b0..53c0bce 100644 --- a/btl/builtins.typer +++ b/btl/builtins.typer @@ -538,3 +538,6 @@ Quotient = Built-in "Quotient" : (l1 : TypeLevel) ≡> (l2 : TypeLevel) ≡> (A : Type_ l1) -> (R : A -> A -> Type_ l2) -> Type_ (_∪_ l1 l2);

+Quotient_in : ?A -> Quotient ?A ?R; +Quotient_in = Built-in "Quotient.in";

# FIXME: Yay! Tous les arguments implicites sont inférés! # %%% builtins.typer ends here. diff --git a/samples/quotient.typer b/samples/quotient.typer index 1108133..a424bf0 100644 --- a/samples/quotient.typer +++ b/samples/quotient.typer @@ -26,8 +26,21 @@ normaliseZ np = case np equalZ : NatPair -> NatPair -> Type; equalZ x1 x2 = Pair (Eq (fst x1) (snd x2)) (Eq (snd x1) (fst x2));

+%% See definition in `builtins.typer` +%% Quotient : (l1 : TypeLevel) ≡> (l2 : TypeLevel) ≡> (A : Type_ l1) -> +%% (R : A -> A -> Type_ l2) -> Type_ (TypeLevel_⊔ l1 l2) +%% +%% Quotient_in : (l1 : TypeLevel) ≡> (l2 : TypeLevel) ≡> (A : Type_ l1) ≡> +%% A -> (R : A -> A -> Type_ l2) => Quotient A R

%% FIXME: We shouldn't get this error %% "Requested Built-in "Quotient" does not exist" -%% Z = Quotient (A := NatPair) equalZ; +%% Z = Quotient NatPair equalZ;

+Z_To_Int : Quotient NatPair equalZ -> Int; +Z_To_Int q = 5 : Int; # # FIXME: 5, really?

+test : Int; +test = Z_To_Int (Quotient_in (pair zero zero));

-%% TODO: Intro and elim \ No newline at end of file +%% TODO: Elim diff --git a/src/env.ml b/src/env.ml index 2f5f05e..c9781c1 100644 --- a/src/env.ml +++ b/src/env.ml @@ -184,7 +184,12 @@ let value_string_with_type v ltype ctx = (Lexp.to_string left) (Lexp.to_string right) (Lexp.to_string t)

•         | _ -> value_string v)
  

•          | [_; _; (_, t); (_, r)]
  

•              when OL.conv_builtin_p ctx e' "Quotient"
  

•              -> sprintf "(Quotient.in %s %s)"
  

•                         (Lexp.to_string t)
  

•                         (Lexp.to_string r)
  

•          | _ -> value_string v)
  | _ -> value_string v
  

  in get_string ltype ctx

diff --git a/src/eval.ml b/src/eval.ml index 6385605..9d82577 100644 --- a/src/eval.ml +++ b/src/eval.ml @@ -1084,6 +1084,7 @@ let register_builtin_functions () = ("Test.false" , test_false,2); ("Test.eq" , test_eq,3); ("Test.neq" , test_neq,3);

•          ("Quotient.in"   , nop_fun, 1);.
  

# # FIXME: Yay! # ] let _ = register_builtin_functions ()

commit 00341dfd61751b67fe8672a1736285cb4d67ff6c Author: James Tan james_tan97@outlook.com Date: Wed Jul 12 03:21:06 2023 -0400

Implement `Quotient` elimination

diff --git a/btl/builtins.typer b/btl/builtins.typer index 53c0bce..550773d 100644 --- a/btl/builtins.typer +++ b/btl/builtins.typer @@ -541,4 +541,20 @@ Quotient = Built-in "Quotient" : (l1 : TypeLevel) ≡> (l2 : TypeLevel) ≡> Quotient_in : ?A -> Quotient ?A ?R; Quotient_in = Built-in "Quotient.in";

+%% FIXME: We want to be able to say the following +%% Quotient_elim : (f : ?A -> ?B) -> +%% (p : (a : ?) -> (a' : ?) -> ?R a a' -> Eq (f a) (f a')) ≡> +%% (q : Quotient ?A ?R) -> +%% ?B; +%% But we running into the following issue for now: +%% "Bug in the elaboration of a repeated metavar!" +Quotient_elim : (l1 : TypeLevel) ≡> (l2 : TypeLevel) ≡> (l3 : TypeLevel) ≡>

•            (A : Type_ l1) ≡> (B : Type_ l2) ≡>
  

•            (R : A -> A -> Type_ l3) ≡>
  

•            (f : A -> B) ->
  

•            (p : (a : A) -> (a' : A) -> R a a' -> Eq (f a) (f a')) ≡>
  

•            (q : Quotient A R) ->
  

•            B;
  

# # FIXME: Try and cut lines right before infix ops rather than right after. # +Quotient_elim = Built-in "Quotient.elim";

%%% builtins.typer ends here. diff --git a/samples/quotient.typer b/samples/quotient.typer index a424bf0..aca5988 100644 --- a/samples/quotient.typer +++ b/samples/quotient.typer @@ -6,7 +6,7 @@ type Nat

_-_ : Nat -> Nat -> Nat; _-_ x y = case x

• | zero => y

• | zero => zero | succ m => case y | zero => x | succ n => m - n;

# # FIXME: Squash this into the original commit that introduces the function. # @@ -24,23 +24,50 @@ normaliseZ np = case np | pair m n => pair (m - n) (n - m);

equalZ : NatPair -> NatPair -> Type; -equalZ x1 x2 = Pair (Eq (fst x1) (snd x2)) (Eq (snd x1) (fst x2)); +equalZ x1 x2 = Eq (normaliseZ x1) (normaliseZ x2); # # FIXME: I think this one as well #

-%% See definition in `builtins.typer` -%% Quotient : (l1 : TypeLevel) ≡> (l2 : TypeLevel) ≡> (A : Type_ l1) -> -%% (R : A -> A -> Type_ l2) -> Type_ (TypeLevel_⊔ l1 l2) %% -%% Quotient_in : (l1 : TypeLevel) ≡> (l2 : TypeLevel) ≡> (A : Type_ l1) ≡> -%% A -> (R : A -> A -> Type_ l2) => Quotient A R +%% See definitions of `Quotient` in `builtins.typer` +%%

%% FIXME: We shouldn't get this error %% "Requested Built-in "Quotient" does not exist" %% Z = Quotient NatPair equalZ;

+NatToInt : Nat -> Int; +NatToInt n = case n

• | zero => (0 : Int)
• | succ n' => (1 : Int) + NatToInt n' ;

+NatPairToInt' : NatPair -> Int; +NatPairToInt' np = case np

• | pair x y =>
• (case x

•  | zero => (NatToInt y) * (-1 : Int)
  

•  | succ _ => NatToInt x);
  

+NatPairToInt : NatPair -> Int; +NatPairToInt np = NatPairToInt' (normaliseZ np);

+%% Proof that NatPairToInt respects the quotient Z +NatPairToIntCompat : (a : NatPair) -> (a' : NatPair) ->

•                 (p : equalZ a a') ->
  

•                 Eq (NatPairToInt a) (NatPairToInt a');
  

+NatPairToIntCompat _ _ p = Eq_eq (f := lambda i ≡>

•                                NatPairToInt' (Eq_uneq (p := p) (i := i)));
  

# # FIXME: Eq_uneq works nicely again, eh?

+%% FIXME: Explicitly providing a value for R should unnecessary, +%% this should be inferred based on the type of `q`. This is +%% because we do not handle residuals during unification for now. Z_To_Int : Quotient NatPair equalZ -> Int; -Z_To_Int q = 5 : Int; +Z_To_Int q = Quotient_elim (R := equalZ) NatPairToInt (p := NatPairToIntCompat) q;

+neg2_Z : Quotient NatPair equalZ; +neg2_Z = Quotient_in (pair (succ zero) (succ (succ (succ zero))));

-test : Int; -test = Z_To_Int (Quotient_in (pair zero zero)); +neg2_Int : Int; +neg2_Int = Z_To_Int neg2_Z;

-%% TODO: Elim +%% FIXME: This could work if we add a reduction rule +%% neg2_refl : Eq neg2_Int (-2 : Int); +%% neg2_refl = Eq_refl; diff --git a/src/eval.ml b/src/eval.ml index 9d82577..c4c9ac5 100644 --- a/src/eval.ml +++ b/src/eval.ml @@ -1027,6 +1027,13 @@ let typelevel_lub loc (_depth : eval_debug_info) (args_val: value_type list) = | [Vint v1; Vint v2] -> Vint(max v1 v2) | _ -> error loc ("`Typlevel.⊔` expects 2 TypeLevel argument2")

+let quotient_elim loc depth args =

• let trace_dum = (Var ((epsilon (loc), None), -1)) in
• match args with

•  | [(Closure _) as f; q] ->
  

•      eval_call loc trace_dum depth f [q]
  

# # FIXME: Why impose (Closure _)? It could be something else! #

•  | _ -> error loc "Quotient.elim expects 2 arguments"
  

let register_builtin_functions () = List.iter (fun (name, f, arity) -> add_builtin_function name f arity) [ @@ -1085,6 +1092,7 @@ let register_builtin_functions () = ("Test.eq" , test_eq,3); ("Test.neq" , test_neq,3); ("Quotient.in" , nop_fun, 1);

•          ("Quotient.elim" , quotient_elim, 2);
        ]
  

let _ = register_builtin_functions ()

commit 55c38d859f79fc86a22fb723b22faaa3fee18810 Author: James Tan james_tan97@outlook.com Date: Wed Jul 12 21:02:28 2023 -0400

Implement `Eq` constructor for `Quotient`

diff --git a/btl/builtins.typer b/btl/builtins.typer index 550773d..120cffd 100644 --- a/btl/builtins.typer +++ b/btl/builtins.typer @@ -541,13 +541,25 @@ Quotient = Built-in "Quotient" : (l1 : TypeLevel) ≡> (l2 : TypeLevel) ≡> Quotient_in : ?A -> Quotient ?A ?R; Quotient_in = Built-in "Quotient.in";

+%% FIXME: We want to be able to say the following +%% Quotient_eq : (a : ?) ≡> (a' : ?) ≡> (p : ?R a a') ≡> +%% Eq (Quotient_in (R := R?) a) +%% (Quotient_in (R := R?) a'); # # FIXME: Actually, we want to use ?a and ?a' rather than `(a : ?) ≡> (a' : ?)` # +%% But we running into the following issue for now: +%% "Bug in the elaboration of a repeated metavar!" +Quotient_eq : (l1 : TypeLevel) ≡> (l2 : TypeLevel) ≡> (A : Type_ l1) ≡>

•          (R : A -> A -> Type_ l2) ≡>
  

•          (a : A) ≡> (a' : A) ≡> (p : R a a') ->
  

•          Eq (Quotient_in (R := R) a)
  

•             (Quotient_in (R := R) a');
  

+Quotient_eq = Built-in "Quotient.eq";

%% FIXME: We want to be able to say the following %% Quotient_elim : (f : ?A -> ?B) -> %% (p : (a : ?) -> (a' : ?) -> ?R a a' -> Eq (f a) (f a')) ≡> %% (q : Quotient ?A ?R) -> %% ?B; -%% But we running into the following issue for now: -%% "Bug in the elaboration of a repeated metavar!" +%% Same issue as above Quotient_elim : (l1 : TypeLevel) ≡> (l2 : TypeLevel) ≡> (l3 : TypeLevel) ≡> (A : Type_ l1) ≡> (B : Type_ l2) ≡> (R : A -> A -> Type_ l3) ≡> diff --git a/samples/quotient.typer b/samples/quotient.typer index aca5988..156805d 100644 --- a/samples/quotient.typer +++ b/samples/quotient.typer @@ -34,16 +34,40 @@ equalZ x1 x2 = Eq (normaliseZ x1) (normaliseZ x2); %% "Requested Built-in "Quotient" does not exist" %% Z = Quotient NatPair equalZ;

+%% +%% Quotient.eq +%% +%% Proof that quotiented elements are equal when +%% the base elements themselves are related in +%% the underlying type. +𝟙-𝟘 : Quotient NatPair equalZ; +𝟙-𝟘 = Quotient_in (pair (succ zero) zero); # # FIXME: Instead of fancy characters, you could use identifiers like # `q1-0` and `q2-1`. #

+𝟚-𝟙 : Quotient NatPair equalZ; +𝟚-𝟙 = Quotient_in (pair (succ (succ zero)) (succ zero));

+𝟙=𝟙 : Eq (t := Quotient NatPair equalZ) 𝟙-𝟘 𝟚-𝟙; +𝟙=𝟙 = Quotient_eq

•    (R := equalZ)
  

•    (a := pair (succ zero) zero)
  

•    (a' := pair (succ (succ zero)) (succ zero))
  

•    Eq_refl;
  

# # FIXME: Hmm... indeed we can't infer `a` and `a'` from the type because # we're not allowed to extract the content of the quotient :-(

+%% +%% Quotient.elim +%% +%% Elimination of quotients requires a proof that +%% the equality between quotients is respected NatToInt : Nat -> Int; NatToInt n = case n

• | zero => (0 : Int)
• | succ n' => (1 : Int) + NatToInt n' ;

• | zero => 0
• | succ n' => 1 + NatToInt n';

# # FIXME: Squash this into its original commit

NatPairToInt' : NatPair -> Int; NatPairToInt' np = case np | pair x y => (case x

•  | zero => (NatToInt y) * (-1 : Int)
  

•  | zero => (NatToInt y) * -1
  

# # FIXME: Same here # | succ _ => NatToInt x);

NatPairToInt : NatPair -> Int; @@ -71,3 +95,5 @@ neg2_Int = Z_To_Int neg2_Z; %% FIXME: This could work if we add a reduction rule %% neg2_refl : Eq neg2_Int (-2 : Int); %% neg2_refl = Eq_refl;

+%% TODO: Define Quotient NatPair equalZ ≃ Int diff --git a/src/eval.ml b/src/eval.ml index c4c9ac5..1f25236 100644 --- a/src/eval.ml +++ b/src/eval.ml @@ -1034,6 +1034,11 @@ let quotient_elim loc depth args = eval_call loc trace_dum depth f [q] | _ -> error loc "Quotient.elim expects 2 arguments"

+let quotient_eq loc _ args =

• match args with

•  | [_] -> Vundefined
  

•  | _ -> error loc "Quotient.eq expects 1 argument"
  

# # FIXME: Why not return the arg (i.e. use `nop_fun`)?

let register_builtin_functions () = List.iter (fun (name, f, arity) -> add_builtin_function name f arity) [ @@ -1092,6 +1097,7 @@ let register_builtin_functions () = ("Test.eq" , test_eq,3); ("Test.neq" , test_neq,3); ("Quotient.in" , nop_fun, 1);

•          ("Quotient.eq"   , quotient_eq, 1);
          ("Quotient.elim" , quotient_elim, 2);
        ]
  

let _ = register_builtin_functions () diff --git a/src/opslexp.ml b/src/opslexp.ml index 8c1285b..6a1b5ac 100644 --- a/src/opslexp.ml +++ b/src/opslexp.ml @@ -304,7 +304,9 @@ and eq_cast_whnf ctx args = match args with | _l1 :: _l2 :: _t :: _x :: _y :: (_, p) :: _f :: (_, fx) :: rest -> (match lexp'_whnf p ctx with

•    | Call (_, eq, _) when conv_builtin_p ctx eq "Eq.eq"
  

•    | Call (_, eq, _) 
  

•      when conv_builtin_p ctx eq "Eq.eq" ||
  

•           conv_builtin_p ctx eq "Quotient.eq"
  

# # FIXME: :-( Have you tried to make `Quotient.eq` return an `(I ≡> Type)` # rather than an `Eq` so we only need to handle `Eq.eq` here? # I guess it probably just moves the problem elsewhere, tho :-( # -> Some (fx, rest) | _ -> None) | _ -> None diff --git a/tests/elab_test.ml b/tests/elab_test.ml index a3931d7..31e1078 100644 --- a/tests/elab_test.ml +++ b/tests/elab_test.ml @@ -180,7 +180,7 @@ unify (f Z (S Z)) (f (S Z) Z); |}

let _ = add_elab_test_decl

• "WHNF of Eq.cast"

• "WHNF of Eq.cast (applied to Eq.eq)" {|

x = (4 : Int); y = x; @@ -192,6 +192,28 @@ test : Eq (Eq_cast (p := p) (f := lambda _ -> Unit) ()) (); test = Eq_refl; |}

+let _ = add_elab_test_decl

• "WHNF of Eq.cast (applied to Quotient.eq)"
• {|

+totalRel : Unit -> Unit -> Type; +totalRel u1 u2 = Unit;

+unitQ : Quotient Unit totalRel; +unitQ = Quotient_in unit;

+unitQ' = unitQ;

+unitQ=unitQ' : Eq (t := Quotient Unit totalRel) unitQ unitQ'; +unitQ=unitQ' = Quotient_eq

•           (R := totalRel)
  

•           (a := unit)
  

•           (a' := unit)
  

•           unit;
  

+test : Eq (Eq_cast (p := unitQ=unitQ') (f := lambda _ -> Unit) ()) (); +test = Eq_refl;

• |}

let _ = add_elab_test_decl "Decidable at the type level" {|

commit aa04cc68dfb0aab732b0b87d749309c0214b9a6b Author: James Tan james_tan97@outlook.com Date: Tue Jul 18 20:58:15 2023 -0400

Improve unification scheme for SLlub (l1, l2) when l1 ≃ l2

diff --git a/src/unification.ml b/src/unification.ml index b78d940..d3c76d1 100644 --- a/src/unification.ml +++ b/src/unification.ml @@ -561,6 +561,10 @@ and unify_sortlvl (matching : scope_level option) -> (* FIXME: This SLlub representation needs to be * more "canonicalized" otherwise it's too restrictive! *) (unify' l11 l21 ctx vs matching)@(unify' l12 l22 ctx vs matching)

•  | SLlub (l1, l2), other | other, SLlub (l1, l2)
  

•    when OL.conv_p ctx l1 l2
  

•    (* Arbitrarily selected `l1` over `l2` *)
  

•    -> unify' l1 (mkSortLevel other) ctx vs matching
  | _, _ -> [(CKimpossible, ctx, sortlvl, lxp)])
  

  | _, _ -> [(CKimpossible, ctx, sortlvl, lxp)]

commit d00058fa5e90c0c6faa0a633d8183383fde31ecb Author: James Tan james_tan97@outlook.com Date: Wed Jul 26 01:37:49 2023 -0400

Implement `qcase` macro

• Facilitates `Quotient` elimination

diff --git a/btl/pervasive.typer b/btl/pervasive.typer index e28fc8a..7a82db4 100644 --- a/btl/pervasive.typer +++ b/btl/pervasive.typer @@ -690,6 +690,9 @@ depelim = load "btl/depelim.typer"; case_as_return_ = depelim.case_as_return_; case_return_ = depelim.case_return_;

+define-operator "qcase" () 42; +qcase_ = let lib = load "btl/qcase.typer" in lib.qcase_macro;

%%%% Unit tests function for doing file

%% It's hard to do a primitive which execute test file diff --git a/btl/qcase.typer b/btl/qcase.typer new file mode 100644 index 0000000..7bf1bae --- /dev/null +++ b/btl/qcase.typer @@ -0,0 +1,284 @@ +%% Qcase macro +%% Make the syntax cleaner for quotient eliminations +%% +%% qcase (e : A / R) +%% | Quotient_in a => e1 +%% | Quotient_eq a a' r i => e2 +%% +%% `a` is bounded in `e1`, and `a`, `a'`, `r` and `i` are +%% bounded in `e2`. `i` can only be used in an erasable manner. +%% +%% TODO: The annotation is necessary for now, as we need the `R` +%% However, we should make it optional. +is_sym : Sexp -> String -> Bool; +is_sym sexp s =

• let

•  kfalse = K false;
  

• in

•  Sexp_dispatch sexp
  

•                (lambda _ _ -> false)    % Nodes
  

•                (String_eq s)            % Symbol
  

•                kfalse                   % String
  

•                kfalse                   % Integer
  

•                kfalse                   % Float
  

•                kfalse;                  % List of Sexp
  

+%% (build_explicit_arg "name" sexp) yields a +%% (name := sexp) Sexp +build_explicit_arg : String -> Sexp -> Sexp; +build_explicit_arg s sexp = Sexp_node (Sexp_symbol "_:=_")

•                                    (cons (Sexp_symbol s)
  

•                                        (cons sexp nil));
  

+qcase_impl = lambda (sexps : List Sexp) ->

• %% For the same example that was given above, we expect
• %% `sexps` to represent the following:
• %% (_|_ (_:_ e (_/_ A R))
• %% (_=>_ (Quotient_in a) e1)
• %% (_=>_ (Quotient_eq a a' r i) e2))
• %% Node : [(_|_ (_:_ e (_/_ A R)) (_=>_ (Qin a) e1) (_=>_ (Qeq a a' r i) e2))]
• let
• %% (_|_ (_:_ e (_/_ A R)) (_=>_ (Qin a) e1) (_=>_ (Qeq a a' r i) e2))
• head = List_head Sexp_error sexps;
• knil = K nil;
• kerr = K Sexp_error;
• get-list : Sexp -> List Sexp;
• get-list node = Sexp_dispatch node

•                              (lambda op lst -> lst) % Nodes
  

•                              knil                   % Symbol
  

•                              knil                   % String
  

•                              knil                   % Integer
  

•                              knil                   % Float
  

•                              knil;                  % List of Sexp
  

• %% List of:
• %% (_:_ e (_/_ A R))
• %% (_=>_ (Qin a) e1)
• %% (_=>_ (Qeq a a' r i) e2)
• body = get-list head;
• elim_targ_sexp = List_nth (Integer->Int 0) body Sexp_error;
• elim_fn_sexp = List_nth (Integer->Int 1) body Sexp_error;
• elim_compat_sexp = List_nth (Integer->Int 2) body Sexp_error;
• %% Triple of the expression to eliminate, the underlying type A
• %% and the relation R
• %% A and R are optional
• elim_expr_details : Triplet Sexp (Option Sexp) (Option Sexp);
• elim_expr_details =

•  let
  

•    kerr = K (triplet Sexp_error none none);
  

•    extract_from_annotated_e : Sexp -> List Sexp ->
  

•                               Triplet Sexp (Option Sexp) (Option Sexp);
  

•    extract_from_annotated_e _ xs =
  

•      if (Int_eq (List_length xs) (Integer->Int 2))
  

•      then
  

•        let
  

•          e = List_nth (Integer->Int 0) xs Sexp_error;
  

•          e_type = List_nth (Integer->Int 1) xs Sexp_error;
  

•          extract_type sexp sexps =
  

•            if (is_sym sexp "_/_")
  

•            then
  

•              %% (_/_    A     R  )
  

•              %%       |___| |___|
  

•              %%         |     |
  

•              %%         a     r
  

•              let
  

•                a = List_nth (Integer->Int 0) sexps Sexp_error;
  

•                r = List_nth (Integer->Int 1) sexps Sexp_error;
  

•              in
  

•                triplet e (some a) (some r)
  

•            else
  

•              triplet e (some Sexp_error) (some Sexp_error);
  

•          kerr' = K (triplet e (some Sexp_error) (some Sexp_error));
  

•        in
  

•          Sexp_dispatch e_type
  

•              extract_type  % Nodes
  

•              kerr'         % Symbol
  

•              kerr'         % String
  

•              kerr'         % Integer
  

•              kerr'         % Float
  

•              kerr'         % List of Sexp
  

•       else
  

•         triplet Sexp_error none none;
  

•    extract_targ_from_node : Sexp -> List Sexp ->
  

•                             Triplet Sexp (Option Sexp) (Option Sexp);
  

•    extract_targ_from_node x xs =
  

•      %% Check if annotation is present
  

•      if (is_sym x "_:_")
  

•      then
  

•        %% Dissect the sexp to extract, the e, A and R
  

•        extract_from_annotated_e x xs
  

•      else
  

•        %% No annotation was given, return the entire
  

•        %% expresson as the elimination target
  

•        triplet x none none;
  

•  in
  

•    Sexp_dispatch elim_targ_sexp
  

•                  extract_targ_from_node                % Nodes
  

•                  (lambda _ -> triplet elim_targ_sexp
  

•                                       none none)       % Symbol
  

•                  kerr                                  % String
  

•                  kerr                                  % Integer
  

•                  kerr                                  % Float
  

•                  kerr;                                 % List of Sexp
  

• %% The function (`f`) argument
• elim_fn : Sexp;
• elim_fn =

•  let
  

•    extract_fn : Sexp -> List Sexp -> Sexp;
  

•    extract_fn sexp sexps =
  

•      %% Check that branch is well formed
  

•      if (and (is_sym sexp "_=>_") (Int_eq (List_length sexps) (Integer->Int 2)))
  

•          then
  

•            let
  

•              %% (_=>_ (Quotient_in a)     e1     )
  

•              %%       |_____________| |_________|
  

•              %%             |              |
  

•              %%            qin          fn_body
  

•              qin_sexp = List_nth (Integer->Int 0) sexps Sexp_error;
  

•              fn_body_sexp = List_nth (Integer->Int 1) sexps Sexp_error;
  

•              bound_var : Sexp;
  

•              bound_var =
  

•                let
  

•                  extract_var head args = if (and (is_sym head "Quotient_in")
  

•                                                  (Int_eq (List_length args)
  

•                                                          (Integer->Int 1)))
  

•                                          then
  

•                                            List_head Sexp_error args
  

•                                          else
  

•                                            %% FIXME: Might be good to have
  

•                                            %% a better way to report this
  

•                                            Sexp_error;
  

•                in
  

•                  Sexp_dispatch qin_sexp
  

•                                extract_var     % Nodes
  

•                                kerr            % Symbol
  

•                                kerr            % String
  

•                                kerr            % Integer
  

•                                kerr            % Float
  

•                                kerr;           % List of Sexp
  

•            in
  

•              Sexp_node (Sexp_symbol "lambda_->_")
  

•                        (cons bound_var
  

•                              (cons fn_body_sexp nil))
  

•          else Sexp_error;
  

•  in
  

•    Sexp_dispatch elim_fn_sexp
  

•            extract_fn      % Nodes
  

•            kerr            % Symbol
  

•            kerr            % String
  

•            kerr            % Integer
  

•            kerr            % Float
  

•            kerr;           % List of Sexp
  

• %% The proof (`p`) argument
• elim_compat : Sexp;
• elim_compat =

•  let
  

•    extract_proof : Sexp -> List Sexp -> Sexp;
  

•    extract_proof sexp sexps =
  

•      %% Check that branch is well formed
  

•      if (and (is_sym sexp "_=>_") (Int_eq (List_length sexps)
  

•                                           (Integer->Int 2)))
  

•      then
  

•        let
  

•          %% (_=>_ (Quotient_eq a a' r i)      p     )
  

•          %%       |____________________| |__________|
  

•          %%                  |                |
  

•          %%                 qeq           proof_exp
  

•          qeq_sexp = List_nth (Integer->Int 0) sexps Sexp_error;
  

•          proof_exp_sexp = List_nth (Integer->Int 1) sexps Sexp_error;
  

•          is_symbol : Sexp -> Bool;
  

•          is_symbol sexp =
  

•            let
  

•              ktrue = K true;
  

•              kfalse = K false;
  

•            in
  

•              Sexp_dispatch sexp
  

•                            (K kfalse)     % Nodes
  

•                            ktrue          % Symbol
  

•                            kfalse         % String
  

•                            kfalse         % Integer
  

•                            kfalse         % Float
  

•                            kfalse;        % List of Sexp
  

•          build_proof : Sexp -> List Sexp -> Sexp;
  

•          build_proof sexp sexps =
  

•            %% Check that the right identifier is used with exactly
  

•            %% 4 arguments, also ensure that identifier are symbols
  

•            %% TODO: Make it possible to omit the `i`, in which case
  

•            %% we expect p to be an equality proof.
  

•            if (and (is_sym sexp "Quotient_eq")
  

•                    (and %% We allow the last parameter `i` to be omitted
  

•                        (or (Int_eq (List_length sexps) (Integer->Int 3))
  

•                            (Int_eq (List_length sexps) (Integer->Int 4)))
  

•                        (List_foldl (lambda acc sexp ->
  

•                                      and acc (is_symbol sexp))
  

•                                    true sexps)))
  

•            then
  

•              let
  

•                mklambda : Sexp -> Sexp -> Sexp;
  

•                mklambda param body =
  

•                  Sexp_node (Sexp_symbol "lambda_->_")
  

•                            (cons param (cons body nil));
  

•                proof_fn =
  

•                  if (Int_eq (List_length sexps) (Integer->Int 3))
  

•                  then
  

•                    %% `i` is absent, i.e. we expect to be provided
  

•                    %% with an equality proof.
  

•                    %% We want to convert Quotient_eq a a' r => e
  

•                    %% to (lambda a a' r -> e)
  

•                    quote (uquote (List_foldr mklambda
  

•                                              sexps proof_exp_sexp))
  

•                  else
  

•                     %% Handle the case where `i` is present
  

•                     %% We have to construct an equality proof
  

•                     %% from what was given
  

•                     %% We want to convert Quotient_eq a a' r i => e
  

•                     %% to (lambda a a' r -> Eq_eq (f := lambda i ≡> e))
  

•                     let
  

•                       erasable_param = List_nth (Integer->Int 3)
  

•                                                 sexps Sexp_error;
  

•                       proof_fn_params =
  

•                        List_reverse (List_tail (List_reverse sexps nil))
  

•                                     nil;
  

•                       eq = quote (Eq_eq (f :=
  

•                                          lambda (uquote erasable_param) ≡>
  

•                                              (uquote proof_exp_sexp)));
  

•                     in
  

•                       quote (uquote (List_foldr mklambda
  

•                                                 proof_fn_params eq));
  

•              in
  

•                build_explicit_arg "p" proof_fn
  

•            else
  

•              Sexp_error;
  

•        in
  

•          Sexp_dispatch qeq_sexp
  

•                        build_proof   % Nodes
  

•                        kerr          % Symbol
  

•                        kerr          % String
  

•                        kerr          % Integer
  

•                        kerr          % Float
  

•                        kerr          % List of Sexp
  

•      else
  

•        Sexp_error;
  

•  in
  

•    Sexp_dispatch elim_compat_sexp
  

•            extract_proof   % Nodes
  

•            kerr            % Symbol
  

•            kerr            % String
  

•            kerr            % Integer
  

•            kerr            % Float
  

•            kerr;           % List of Sexp
  

• qelim_args : List Sexp;
• qelim_args = case elim_expr_details

•  | triplet e a r =>
  

•      let
  

•        res = (cons elim_fn
  

•                    (cons elim_compat
  

•                          (cons e nil)));
  

•        res' = (case r
  

•                | none => res
  

•                | some r' =>
  

•                    (cons (build_explicit_arg "R" r') res));
  

•      in
  

•        res';
  

• qelim_sexp = Sexp_node (Sexp_symbol "Quotient_elim")

•                       qelim_args;
  

• in
• IO_return qelim_sexp;

+qcase_macro = macro qcase_impl; diff --git a/samples/qcase_test.typer b/samples/qcase_test.typer new file mode 100644 index 0000000..fb4df3c --- /dev/null +++ b/samples/qcase_test.typer @@ -0,0 +1,50 @@ +%% Defining a total relation on Unit +R : Unit -> Unit -> Type; +R u1 u2 = Unit;

+inQ : Quotient Unit R; +inQ = Quotient_in ();

+e1 : Unit; +e1 = qcase (inQ : Unit / R)

• | Quotient_in a => ()
• | Quotient_eq a a' r i => ();

+e2 : Unit; +e2 = qcase (inQ : Unit / R)

• | Quotient_in a => ()
• | Quotient_eq a a' r => Eq_refl;

+e3 : Unit; +e3 = qcase inQ

• | Quotient_in a => ()
• | Quotient_eq a a' r i => ();

+e4 : Unit; +e4 = qcase inQ

• | Quotient_in a => ()
• | Quotient_eq a a' r => Eq_refl;

+test-elim-to-unit = do {

• Test_info "QCASE" "elimination to Unit";
• r0 <- Test_eq "annotated elim to `Unit` with explicit `I`" e1 ();
• r1 <- Test_eq "annotated elim to `Unit` without `I`" e2 ();
• r2 <- Test_eq "unannotated elim to `Unit` with explicit `I`" e3 ();
• r3 <- Test_eq "unannotated elim to `Unit` without `I`" e4 ();
• success <- IO_return (and (and (and r0 r1) r2) r3);
• if success then
• (Test_info "QCASE" "elimination to Unit succeeded")
• else
• (Test_warning "QCASE" "elimination to Unit failed");
• IO_return success;

+};

+exec-test = do {

• b1 <- test-elim-to-unit;
• IO_return b1;

+}; diff --git a/samples/quotient.typer b/samples/quotient.typer index 156805d..2fef29a 100644 --- a/samples/quotient.typer +++ b/samples/quotient.typer @@ -96,4 +96,21 @@ neg2_Int = Z_To_Int neg2_Z; %% neg2_refl : Eq neg2_Int (-2 : Int); %% neg2_refl = Eq_refl;

+%% `qcase` macro to facilitate elimination +Z_To_Int' : Quotient NatPair equalZ -> Int; +Z_To_Int' q =

• %% The annotation is optional, but is necessary in
• %% this case, since the propagation of type
• %% information is insufficient the way things are now.
• qcase (q : NatPair / equalZ)
• | Quotient_in a => NatPairToInt a
• | Quotient_eq a a' r i => NatPairToInt' (Eq_uneq (p := r) (i := i));

+%% Omitting the `i` parameter by providing an equality proof on the RHS +Z_To_Int'' : Quotient NatPair equalZ -> Int; +Z_To_Int'' q =

• qcase (q : NatPair / equalZ)
• | Quotient_in a => NatPairToInt a
• | Quotient_eq a a' r => NatPairToIntCompat a a' r;

%% TODO: Define Quotient NatPair equalZ ≃ Int

commit 489829434030802a9fe918cdfd274340e2cb4bdb Author: James Tan james_tan97@outlook.com Date: Thu Jul 20 23:59:52 2023 -0400

Write some proofs about quotient types

diff --git a/samples/quotient_lib.typer b/samples/quotient_lib.typer new file mode 100644 index 0000000..d6a76cd --- /dev/null +++ b/samples/quotient_lib.typer @@ -0,0 +1,129 @@ +%%%%% Prelude %%%%%%

+%% FIXME : Loading hott.typer doesn't work for some reason +%% due to "[X] Fatal :(internal) lub of two SLsucc" +%% Some definitions will be duplicated for now

+Eq_funext : (f : ? -> ?) => (g : ? -> ?) =>

•        ((x : ?) -> Eq (f x) (g x)) ->
  

•        Eq f g;
  

+Eq_funext p = Eq_eq (f := lambda i ≡> lambda x -> Eq_uneq (p := p x) (i := i));

+HoTT_isProp P = (x : P) -> (y : P) -> Eq x y;

+HoTT_isSet A = (x : A) -> (y : A) -> HoTT_isProp (Eq x y);

+HoTT_isContr = typecons (HoTT_isContr (l ::: TypeLevel)

•                                  (A : Type_ l))
  

•                     (isContr (a : A) ((a' : A) -> Eq a a'));
  

+isContr = datacons HoTT_isContr isContr;

+%%%%% Prelude END %%%%%%

+%% TODO: Prove dependent version of this after we introduce +%% dependent elim, which will be more interesting and more +%% worthwhile +recProp : (A : Type_ ?) ≡>

•      (B : Type_ ?) ≡>
  

•      (R : A -> A -> Type_ ?) ≡>
  

•      (p : HoTT_isProp B) ->
  

•      (f : A -> B) ->
  

•      (x : Quotient A R) -> B;
  

+recProp = lambda _ _ _ _ _ R ≡>

•       lambda p f x ->
  

•         Quotient_elim (R := R) f (p := lambda a a' r -> p (f a) (f a')) x;
  

+%% Again, this is not very interesting, unlike its dependent +%% counterpart. +recContr : (A : Type_ ?) ≡>

•       (B : Type_ ?) ≡>
  

•       (R : A -> A -> Type_ ?) ≡>
  

•       (p : HoTT_isContr B) ->
  

•       (x : Quotient A R) -> B;
  

+recContr = lambda _ _ _ _ _ R ≡>

•        lambda p x -> case p 
  

•            | isContr a f => Quotient_elim (R := R) 
  

•                                           (lambda _ -> a)
  

•                                           (p := lambda a a' r -> Eq_refl)
  

•                                           x;
  

+%% FIXME: Quotient_elim should be named Quotient_rec? +%% rec2 : (A : Type_ ?) ≡> +%% (B : Type_ ?) ≡> +%% (C : Type_ ?) ≡> +%% (R : A -> A -> Type_ ?) ≡> +%% (S : B -> B -> Type_ ?) ≡> +%% (C_isSet : HoTT_isSet C) -> +%% (f : A -> B -> C) -> +%% ((a : A) -> (b : A) -> (c : B) -> R a b -> Eq (f a c) (f b c)) -> +%% ((a : A) -> (b : B) -> (c : B) -> S b c -> Eq (f a b) (f a c)) -> +%% Quotient A R -> Quotient B S -> C; +%% rec2 = lambda _ _ _ _ _ A B C R S ≡> +%% lambda C_isSet f feql feqr -> +%% Quotient_elim (R := R) +%% (lambda a -> +%% lambda b -> Quotient_elim (R := S) (f a) +%% (p := feqr a) b) +%% (p := lambda a a' r -> +%% let +%% eqf : (b : B) -> Eq (f a b) (f a' b); +%% eqf b = feql a a' b r; +%% eqf' : Eq (f a) (f a'); +%% eqf' = Eq_funext (f := f a) (g := f a') eqf; +%% p : (x : Quotient B S) -> +%% HoTT_isProp (Eq (Quotient_elim (R := S) (f a) +%% (p := feqr a) x) +%% (Quotient_elim (R := S) (f a') +%% (p := feqr a') x)); +%% p x = C_isSet (Quotient_elim (R := S) (f a) +%% (p := feqr a) x) +%% (Quotient_elim (R := S) (f a') +%% (p := feqr a') x); +%% res : (x : Quotient B S) -> +%% (Eq (Quotient_elim (R := S) (f a) +%% (p := feqr a) x) +%% (Quotient_elim (R := S) (f a') +%% (p := feqr a') x)); +%% res x = Quotient_elim (A := B) +%% %% FIXME: We need depelim here +%% (B := Eq (f a b) (f a' b)) +%% (R := S) +%% eqf +%% (p := lambda u v s -> +%% Eq_eq (f := lambda i ≡> +%% p (Eq_uneq (p := Quotient_eq (R := S) s) (i := i)) +%% (eqf u) (eqf v))) +%% x; +%% in +%% Eq_funext (f := Quotient_elim (R := S) (f a) +%% (p := feqr a)) +%% (g := Quotient_elim (R := S) (f a') +%% (p := feqr a')) +%% res);

+%% Lemma 6.10.2 in HoTT book, to prove this we need to +%% apply propositional truncation on SurjectiveQuotientProof. +%% type SurjectiveQuotientProof (A : ?) (x : Quotient A ?R) +%% | surjectiveQuotientProof (a : A) (Eq (Quotient_in a) x); +%% Quotient_in_surjective : (x : Quotient ?A ?R) -> ||SurjectiveQuotientProof ?A ?R x||₁;

+%% Given a proof that a unary operation preserves the underlying +%% relation, we can apply the operation to the quotiented type. +quotUnaryOp : (A : Type_ ?) ≡>

•          (R : A -> A -> Type_ ?) ≡>
  

•          (op : A -> A) ->
  

•          ((a : A) -> (a' : A) -> R a a' -> R (op a) (op a')) ->
  

•          Quotient A R -> Quotient A R;
  

+quotUnaryOp = lambda _ _ A R ≡>

•          lambda op h x ->
  

•             let
  

•               opPreservesQuotient : (a : A) -> (a' : A) -> R a a' -> 
  

•                                     Eq (t := Quotient A R)
  

•                                        (Quotient_in (op a))
  

•                                        (Quotient_in (op a'));
  

•               opPreservesQuotient a a' r = Quotient_eq (R := R) (h a a' r);
  

•             in
  

•                Quotient_elim (R := R)
  

•                              (lambda a -> Quotient_in (op a))
  

•                              (p := opPreservesQuotient)
  

•                              x;