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add notation for sequence types & use notations everywhere

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Michael Sippel 2024-09-18 11:15:20 +02:00
parent 633843d9c7
commit 6e5c832db7
Signed by: senvas
GPG key ID: F96CF119C34B64A6
7 changed files with 260 additions and 226 deletions
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148
coq/bbencode.v
View file

@ -4,77 +4,101 @@ Require Import terms.
Require Import subst. Require Import subst.
Require Import smallstep. Require Import smallstep.
Include Terms.
Include Subst.
Include Smallstep.
Open Scope ladder_type_scope. Open Scope ladder_type_scope.
Open Scope ladder_expr_scope. Open Scope ladder_expr_scope.
(* let bb_zero = Λα ↦ λs: α->α ↦ λz: α ↦ z
* ∀α.(α->α)->α->α
*)
Definition bb_zero : expr_term :=
(expr_ty_abs "α"
(expr_abs "s" (type_fun (type_var "α") (type_var "α"))
(expr_abs "z" (type_var "α")
(expr_var "z")))).
(* let bb_one = Λα ↦ λs: α->α ↦ λz: α ↦ s z Definition bb_zero : expr_term := [{
*) (Λ"α" ↦
Definition bb_one : expr_term := λ"s",(%"α"% -> %"α"%) ↦
(expr_ty_abs "α" λ"z",(%"α"%) ↦
(expr_abs "s" (type_fun (type_var "α") (type_var "α")) %"z"%)
(expr_abs "z" (type_var "α") as $"ℕ"$~$"BBNat"$
(expr_app (expr_var "s") (expr_var "z"))))). }].
(* let bb_two = Λα ↦ λs: α->α ↦ λz: α ↦ s (s z) Definition bb_one : expr_term := [{
*) (Λ"α" ↦
Definition bb_two : expr_term := λ"s",(%"α"% -> %"α"%) ↦
(expr_ty_abs "α" λ"z",(%"α"%) ↦
(expr_abs "s" (type_fun (type_var "α") (type_var "α")) (%"s"% %"z"%))
(expr_abs "z" (type_var "α") as $"ℕ"$~$"BBNat"$
(expr_app (expr_var "s") (expr_app (expr_var "s") (expr_var "z")))))). }].
Definition bb_two : expr_term := [{
(Λ"α" ↦
λ"s",(%"α"% -> %"α"%) ↦
λ"z",(%"α"%) ↦
(%"s"% (%"s"% %"z"%)))
as $"ℕ"$~$"BBNat"$
}].
Definition bb_succ : expr_term := Definition bb_succ : expr_term := [{
(expr_abs "n" (type_ladder (type_id "ℕ") λ"n",$"ℕ"$
(type_ladder (type_id "BBNat") ~$"BBNat"$
(type_univ "α" ~(∀"β",(%"β'"%->%"β"%)->%"β"%->%"β"%)
(type_fun (type_fun (type_var "α") (type_var "α")) ↦
(type_fun (type_var "α") (type_var "α")))))) ((Λ"α" ↦
λ"s",((%"α"%->%"α"%)->%"α"%->%"α"%) ↦
λ"z",%"α"% ↦
(%"s"% ((%"n"% # %"α"%) %"s"% %"z"%)))
as $"ℕ"$~$"BBNat"$)
}].
(expr_ascend (type_ladder (type_id "ℕ") (type_id "BBNat")) Example example_reduction :
(expr_ty_abs "α" (expr_app bb_succ bb_one) -->β* bb_two
(expr_abs "s" (type_fun (type_var "α") (type_var "α")) .
(expr_abs "z" (type_var "α")
(expr_app (expr_var "s")
(expr_app (expr_app
(expr_ty_app (expr_var "n") (type_var "α"))
(expr_var "s"))
(expr_var "z")))))))).
Definition e1 : expr_term :=
(expr_let "bb-zero" bb_zero
(expr_app (expr_app (expr_var "+") (expr_var "bb-zero")) (expr_var "bb-zero"))
).
Definition t1 : expr_term := (expr_app (expr_var "x") (expr_var "x")).
Compute (expr_subst "x"
(expr_ty_abs "α" (expr_abs "a" (type_var "α") (expr_var "a")))
bb_one
).
Example example_let_reduction :
e1 -->β (expr_app (expr_app (expr_var "+") bb_zero) bb_zero).
Proof. Proof.
apply E_Let.
apply Multi_Step with (y:=[{
(λ"n",$"ℕ"$
~$"BBNat"$
~(∀"β",(%"β'"%->%"β"%)->%"β"%->%"β"%)
↦
((Λ"α" ↦
λ"s",((%"α"%->%"α"%)->%"α"%->%"α"%) ↦
λ"z",%"α"% ↦
(%"s"% ((%"n"% # %"α"%) %"s"% %"z"%)))
as $"ℕ"$~$"BBNat"$))
(Λ"β" ↦
λ"s",(%"β"% -> %"β"%) ↦
λ"z",(%"β"%) ↦
(%"s"% %"z"%))
as $"ℕ"$~$"BBNat"$
}]).
Admitted.
(*
apply E_Alpha.
apply E_App2.
apply EAlpha_Rename.
apply Multi_Step with (y:=[{
(Λ"α" ↦
λ"s",((%"α"%->%"α"%)->%"α"%->%"α"%) ↦
λ"z",%"α"% ↦
(%"s"% (
(((Λ"β" ↦
λ"s",(%"β"% -> %"β"%) ↦
λ"z",(%"β"%) ↦
(%"s"% %"z"%)) as $"ℕ"$~$"BBNat"$)
# %"α"%) %"s"% %"z"%)))
as $"ℕ"$~$"BBNat"$ }]).
apply E_AppLam.
apply Multi_Step with (y:=[{
(Λ"α" ↦
λ"s",((%"α"%->%"α"%)->%"α"%->%"α"%) ↦
λ"z",%"α"% ↦
(%"s"% (
(((Λ"α'" ↦
λ"s",(%"α'"% -> %"α'"%) ↦
λ"z",(%"α'"%) ↦
(%"s"% %"z"%)) as $"ℕ"$~$"BBNat"$)
# %"α"%) %"s"% %"z"%)))
as $"ℕ"$~$"BBNat"$ }]).
}].
Qed. Qed.
*)
Compute (expr_app bb_succ bb_zero) -->β bb_one. Compute (expr_app bb_succ bb_zero) -->β bb_one.
Example example_succ :
(expr_app bb_succ bb_zero) -->β bb_one.
Proof.
Admitted.

102
coq/equiv.v
View file

@ -38,46 +38,44 @@ From Coq Require Import Strings.String.
Reserved Notation "S '--->α' T" (at level 40). Reserved Notation "S '--->α' T" (at level 40).
Inductive type_conv_alpha : type_term -> type_term -> Prop := Inductive type_conv_alpha : type_term -> type_term -> Prop :=
| TAlpha_Rename : forall x y t t', | TAlpha_Rename : forall x y t,
(type_univ x t) --->α (type_univ y (type_subst x (type_var y) t))
(type_subst1 x (type_var y) t t') ->
(type_univ x t) --->α (type_univ y t')
| TAlpha_SubUniv : forall x τ τ', | TAlpha_SubUniv : forall x τ τ',
(τ --->α τ') -> (τ --->α τ') ->
(type_univ x τ) --->α (type_univ x τ') [< ∀x,τ >] --->α [< ∀x,τ' >]
| TAlpha_SubSpec1 : forall τ1 τ1' τ2, | TAlpha_SubSpec1 : forall τ1 τ1' τ2,
(τ1 --->α τ1') -> (τ1 --->α τ1') ->
(type_spec τ1 τ2) --->α (type_spec τ1' τ2) [< <τ1 τ2> >] --->α [< <τ1' τ2> >]
| TAlpha_SubSpec2 : forall τ1 τ2 τ2', | TAlpha_SubSpec2 : forall τ1 τ2 τ2',
(τ2 --->α τ2') -> (τ2 --->α τ2') ->
(type_spec τ1 τ2) --->α (type_spec τ1 τ2') [< <τ1 τ2> >] --->α [< <τ1 τ2'> >]
| TAlpha_SubFun1 : forall τ1 τ1' τ2, | TAlpha_SubFun1 : forall τ1 τ1' τ2,
(τ1 --->α τ1') -> (τ1 --->α τ1') ->
(type_fun τ1 τ2) --->α (type_fun τ1' τ2) [< τ1 -> τ2 >] --->α [< τ1' -> τ2 >]
| TAlpha_SubFun2 : forall τ1 τ2 τ2', | TAlpha_SubFun2 : forall τ1 τ2 τ2',
(τ2 --->α τ2') -> (τ2 --->α τ2') ->
(type_fun τ1 τ2) --->α (type_fun τ1 τ2') [< τ1 -> τ2 >] --->α [< τ1 -> τ2' >]
| TAlpha_SubMorph1 : forall τ1 τ1' τ2, | TAlpha_SubMorph1 : forall τ1 τ1' τ2,
(τ1 --->α τ1') -> (τ1 --->α τ1') ->
(type_morph τ1 τ2) --->α (type_morph τ1' τ2) [< τ1 ->morph τ2 >] --->α [< τ1' ->morph τ2 >]
| TAlpha_SubMorph2 : forall τ1 τ2 τ2', | TAlpha_SubMorph2 : forall τ1 τ2 τ2',
(τ2 --->α τ2') -> (τ2 --->α τ2') ->
(type_morph τ1 τ2) --->α (type_morph τ1 τ2') [< τ1 ->morph τ2 >] --->α [< τ1 ->morph τ2' >]
| TAlpha_SubLadder1 : forall τ1 τ1' τ2, | TAlpha_SubLadder1 : forall τ1 τ1' τ2,
(τ1 --->α τ1') -> (τ1 --->α τ1') ->
(type_ladder τ1 τ2) --->α (type_ladder τ1' τ2) [< τ1 ~ τ2 >] --->α [< τ1' ~ τ2 >]
| TAlpha_SubLadder2 : forall τ1 τ2 τ2', | TAlpha_SubLadder2 : forall τ1 τ2 τ2',
(τ2 --->α τ2') -> (τ2 --->α τ2') ->
(type_ladder τ1 τ2) --->α (type_ladder τ1 τ2') [< τ1 ~ τ2 >] --->α [< τ1 ~ τ2' >]
where "S '--->α' T" := (type_conv_alpha S T). where "S '--->α' T" := (type_conv_alpha S T).
@ -86,38 +84,48 @@ Lemma type_alpha_symm :
forall σ τ, forall σ τ,
(σ --->α τ) -> (τ --->α σ). (σ --->α τ) -> (τ --->α σ).
Proof. Proof.
(* TODO *) intros.
induction H.
- admit.
- auto using TAlpha_Rename,type_subst,type_subst_symm, TAlpha_SubUniv.
- auto using TAlpha_SubSpec1.
- auto using TAlpha_SubSpec2.
- auto using TAlpha_SubFun1.
- auto using TAlpha_SubFun2.
- auto using TAlpha_SubMorph1.
- auto using TAlpha_SubMorph2.
- auto using TAlpha_SubLadder1.
- auto using TAlpha_SubLadder2.
Admitted. Admitted.
(** Define all rewrite steps $\label{coq:type-dist}$ *) (** Define all rewrite steps $\label{coq:type-dist}$ *)
Reserved Notation "S '-->distribute-ladder' T" (at level 40). Reserved Notation "S '-->distribute-ladder' T" (at level 40).
Inductive type_distribute_ladder : type_term -> type_term -> Prop := Inductive type_distribute_ladder : type_term -> type_term -> Prop :=
| L_DistributeOverSpec1 : forall x x' y, | L_DistributeOverSpec1 : forall x x' y,
(type_spec (type_ladder x x') y) [< <x~x' y> >]
-->distribute-ladder -->distribute-ladder
(type_ladder (type_spec x y) (type_spec x' y)) [< <x y>~<x' y> >]
| L_DistributeOverSpec2 : forall x y y', | L_DistributeOverSpec2 : forall x y y',
(type_spec x (type_ladder y y')) [< <x y~y'> >]
-->distribute-ladder -->distribute-ladder
(type_ladder (type_spec x y) (type_spec x y')) [< <x y>~<x y'> >]
| L_DistributeOverFun1 : forall x x' y, | L_DistributeOverFun1 : forall x x' y,
(type_fun (type_ladder x x') y) [< (x~x' -> y) >]
-->distribute-ladder -->distribute-ladder
(type_ladder (type_fun x y) (type_fun x' y)) [< (x -> y) ~ (x' -> y) >]
| L_DistributeOverFun2 : forall x y y', | L_DistributeOverFun2 : forall x y y',
(type_fun x (type_ladder y y')) [< (x -> y~y') >]
-->distribute-ladder -->distribute-ladder
(type_ladder (type_fun x y) (type_fun x y')) [< (x -> y) ~ (x -> y') >]
| L_DistributeOverMorph1 : forall x x' y, | L_DistributeOverMorph1 : forall x x' y,
(type_morph (type_ladder x x') y) [< (x~x' ->morph y) >]
-->distribute-ladder -->distribute-ladder
(type_ladder (type_morph x y) (type_morph x' y)) [< (x ->morph y) ~ (x' ->morph y) >]
| L_DistributeOverMorph2 : forall x y y', | L_DistributeOverMorph2 : forall x y y',
(type_morph x (type_ladder y y')) (type_morph x (type_ladder y y'))
@ -131,34 +139,34 @@ where "S '-->distribute-ladder' T" := (type_distribute_ladder S T).
Reserved Notation "S '-->condense-ladder' T" (at level 40). Reserved Notation "S '-->condense-ladder' T" (at level 40).
Inductive type_condense_ladder : type_term -> type_term -> Prop := Inductive type_condense_ladder : type_term -> type_term -> Prop :=
| L_CondenseOverSpec1 : forall x x' y, | L_CondenseOverSpec1 : forall x x' y,
(type_ladder (type_spec x y) (type_spec x' y)) [< <x y>~<x' y> >]
-->condense-ladder -->condense-ladder
(type_spec (type_ladder x x') y) [< <x~x' y> >]
| L_CondenseOverSpec2 : forall x y y', | L_CondenseOverSpec2 : forall x y y',
(type_ladder (type_spec x y) (type_spec x y')) [< <x y>~<x y'> >]
-->condense-ladder -->condense-ladder
(type_spec x (type_ladder y y')) [< <x y~y'> >]
| L_CondenseOverFun1 : forall x x' y, | L_CondenseOverFun1 : forall x x' y,
(type_ladder (type_fun x y) (type_fun x' y)) [< (x -> y) ~ (x' -> y) >]
-->condense-ladder -->condense-ladder
(type_fun (type_ladder x x') y) [< (x~x') -> y >]
| L_CondenseOverFun2 : forall x y y', | L_CondenseOverFun2 : forall x y y',
(type_ladder (type_fun x y) (type_fun x y')) [< (x -> y) ~ (x -> y') >]
-->condense-ladder -->condense-ladder
(type_fun x (type_ladder y y')) [< (x -> y~y') >]
| L_CondenseOverMorph1 : forall x x' y, | L_CondenseOverMorph1 : forall x x' y,
(type_ladder (type_morph x y) (type_morph x' y)) [< (x ->morph y) ~ (x' ->morph y) >]
-->condense-ladder -->condense-ladder
(type_morph (type_ladder x x') y) [< (x~x' ->morph y) >]
| L_CondenseOverMorph2 : forall x y y', | L_CondenseOverMorph2 : forall x y y',
(type_ladder (type_morph x y) (type_morph x y')) [< (x ->morph y) ~ (x ->morph y') >]
-->condense-ladder -->condense-ladder
(type_morph x (type_ladder y y')) [< (x ->morph y~y') >]
where "S '-->condense-ladder' T" := (type_condense_ladder S T). where "S '-->condense-ladder' T" := (type_condense_ladder S T).
@ -213,14 +221,14 @@ Inductive type_eq : type_term -> type_term -> Prop :=
x === z x === z
| TEq_LadderAssocLR : forall x y z, | TEq_LadderAssocLR : forall x y z,
(type_ladder (type_ladder x y) z) [< (x~y)~z >]
=== ===
(type_ladder x (type_ladder y z)) [< x~(y~z) >]
| TEq_LadderAssocRL : forall x y z, | TEq_LadderAssocRL : forall x y z,
(type_ladder x (type_ladder y z)) [< x~(y~z) >]
=== ===
(type_ladder (type_ladder x y) z) [< (x~y)~z >]
| TEq_Alpha : forall x y, | TEq_Alpha : forall x y,
x --->α y -> x --->α y ->
@ -378,7 +386,7 @@ Admitted.
*) *)
Example example_flat_type : Example example_flat_type :
(type_is_flat (type_spec (type_id "PosInt") (type_id "10"))). type_is_flat [< <$"PosInt"$ $"10"$> >].
Proof. Proof.
apply FlatApp. apply FlatApp.
apply FlatId. apply FlatId.
@ -386,17 +394,15 @@ Proof.
Qed. Qed.
Example example_lnf_type : Example example_lnf_type :
(type_is_lnf type_is_lnf [< [$"Char"$] ~ [$"Byte"$] >].
(type_ladder (type_spec (type_id "Seq") (type_id "Char"))
(type_spec (type_id "Seq") (type_id "Byte")))).
Proof. Proof.
Admitted. Admitted.
Example example_type_eq : Example example_type_eq :
(type_spec (type_id "Seq") (type_ladder (type_id "Char") (type_id "Byte"))) [< [$"Char"$ ~ $"Byte"$] >]
=== ===
(type_ladder (type_spec (type_id "Seq") (type_id "Char")) [< [$"Char"$] ~ [$"Byte"$] >]
(type_spec (type_id "Seq") (type_id "Byte"))). .
Proof. Proof.
apply TEq_Distribute. apply TEq_Distribute.
apply L_DistributeOverSpec2. apply L_DistributeOverSpec2.

45
coq/morph.v
View file

@ -8,13 +8,15 @@ Require Import context.
(* Given a context, there is a morphism path from τ to τ' *) (* Given a context, there is a morphism path from τ to τ' *)
Reserved Notation "Γ '|-' σ '~>' τ" (at level 101, σ at next level, τ at next level). Reserved Notation "Γ '|-' σ '~>' τ" (at level 101, σ at next level, τ at next level).
Open Scope ladder_expr_scope.
Inductive morphism_path : context -> type_term -> type_term -> Prop := Inductive morphism_path : context -> type_term -> type_term -> Prop :=
| M_Sub : forall Γ τ τ', | M_Sub : forall Γ τ τ',
(τ :<= τ') -> (τ :<= τ') ->
(Γ |- τ ~> τ') (Γ |- τ ~> τ')
| M_Single : forall Γ h τ τ', | M_Single : forall Γ h τ τ',
(context_contains Γ h (type_morph τ τ')) -> (context_contains Γ h [< τ ->morph τ' >]) ->
(Γ |- τ ~> τ') (Γ |- τ ~> τ')
| M_Chain : forall Γ τ τ' τ'', | M_Chain : forall Γ τ τ' τ'',
@ -24,58 +26,57 @@ Inductive morphism_path : context -> type_term -> type_term -> Prop :=
| M_Lift : forall Γ σ τ τ', | M_Lift : forall Γ σ τ τ',
(Γ |- τ ~> τ') -> (Γ |- τ ~> τ') ->
(Γ |- (type_ladder σ τ) ~> (type_ladder σ τ')) (Γ |- [< σ ~ τ >] ~> [< σ ~ τ' >])
| M_MapSeq : forall Γ τ τ', | M_MapSeq : forall Γ τ τ',
(Γ |- τ ~> τ') -> (Γ |- τ ~> τ') ->
(Γ |- (type_spec (type_id "Seq") τ) ~> (type_spec (type_id "Seq") τ')) (Γ |- [< [τ] >] ~> [< [τ'] >])
where "Γ '|-' s '~>' t" := (morphism_path Γ s t). where "Γ '|-' s '~>' t" := (morphism_path Γ s t).
Lemma id_morphism_path : forall Γ τ, Γ |- τ ~> τ.
Proof.
intros.
apply M_Sub, TSubRepr_Refl, TEq_Refl.
Qed.
Inductive translate_morphism_path : context -> type_term -> type_term -> expr_term -> Prop := Inductive translate_morphism_path : context -> type_term -> type_term -> expr_term -> Prop :=
| Translate_Subtype : forall Γ τ τ', | Translate_Descend : forall Γ τ τ',
(τ :<= τ') -> (τ :<= τ') ->
(translate_morphism_path Γ τ τ' (translate_morphism_path Γ τ τ'
(expr_morph "x" τ (expr_var "x"))) (expr_morph "x" τ [{ %"x"% des τ' }]))
| Translate_Lift : forall Γ σ τ τ' m, | Translate_Lift : forall Γ σ τ τ' m,
(Γ |- τ ~> τ') -> (Γ |- τ ~> τ') ->
(translate_morphism_path Γ τ τ' m) -> (translate_morphism_path Γ τ τ' m) ->
(translate_morphism_path Γ (type_ladder σ τ) (type_ladder σ τ') (translate_morphism_path Γ [< σ ~ τ >] [< σ ~ τ' >]
(expr_morph "x" (type_ladder σ τ) (expr_morph "x" [< σ ~ τ >] [{ (m (%"x"% des τ)) as σ }]))
(expr_ascend σ (expr_app m (expr_descend τ (expr_var "x"))))))
| Translate_Single : forall Γ h τ τ', | Translate_Single : forall Γ h τ τ',
(context_contains Γ h (type_morph τ τ')) -> (context_contains Γ h [< τ ->morph τ' >]) ->
(translate_morphism_path Γ τ τ' (expr_var h)) (translate_morphism_path Γ τ τ' [{ %h% }])
| Translate_Chain : forall Γ τ τ' τ'' m1 m2, | Translate_Chain : forall Γ τ τ' τ'' m1 m2,
(translate_morphism_path Γ τ τ' m1) -> (translate_morphism_path Γ τ τ' m1) ->
(translate_morphism_path Γ τ' τ'' m2) -> (translate_morphism_path Γ τ' τ'' m2) ->
(translate_morphism_path Γ τ τ'' (translate_morphism_path Γ τ τ''
(expr_morph "x" τ (expr_app m2 (expr_app m1 (expr_var "x"))))) (expr_morph "x" τ [{ m2 (m1 %"x"%) }]))
| Translate_MapSeq : forall Γ τ τ' m, | Translate_MapSeq : forall Γ τ τ' m,
(translate_morphism_path Γ τ τ' m) -> (translate_morphism_path Γ τ τ' m) ->
(translate_morphism_path Γ (translate_morphism_path Γ [< [τ] >] [< [τ'] >]
(type_spec (type_id "Seq") τ) [{
(type_spec (type_id "Seq") τ') λ"xs",[τ] ↦morph (%"map"% # τ # τ' m %"xs"%)
(expr_morph "xs" }])
(type_spec (type_id "Seq") τ)
(expr_app (expr_app (expr_ty_app (expr_ty_app
(expr_var "map") τ) τ') m)
(expr_var "xs"))))
. .
Example morphism_paths : Example morphism_paths :
(ctx_assign "degrees-to-turns" [< $"Angle"$~$"Degrees"$~$"ℝ"$ ->morph $"Angle"$~$"Turns"$~$"ℝ"$ >] (ctx_assign "degrees-to-turns" [< $"Angle"$~$"Degrees"$~$"ℝ"$ ->morph $"Angle"$~$"Turns"$~$"ℝ"$ >]
(ctx_assign "turns-to-radians" [< $"Angle"$~$"Turns"$~$"ℝ"$ ->morph $"Angle"$~$"Radians"$~$"ℝ"$ >] (ctx_assign "turns-to-radians" [< $"Angle"$~$"Turns"$~$"ℝ"$ ->morph $"Angle"$~$"Radians"$~$"ℝ"$ >]
ctx_empty)) ctx_empty))
|- [< <$"Seq"$ $"Hue"$~$"Angle"$~$"Degrees"$~$"ℝ"$> >] |- [< [ $"Hue"$~$"Angle"$~$"Degrees"$~$"ℝ"$ ] >]
~> [< <$"Seq"$ $"Hue"$~$"Angle"$~$"Radians"$~$"ℝ"$> >] ~> [< [ $"Hue"$~$"Angle"$~$"Radians"$~$"ℝ"$ ] >]
. .
Proof. Proof.
intros. intros.

36
coq/smallstep.v
View file

@ -16,19 +16,19 @@ Inductive expr_alpha : expr_term -> expr_term -> Prop :=
| EAlpha_SubAbs : forall x τ e e', | EAlpha_SubAbs : forall x τ e e',
(e -->α e') -> (e -->α e') ->
(expr_abs x τ e) -->α (expr_abs x τ e') [{ λ x , τ ↦ e }] -->α [{ λ x , τ ↦ e' }]
| EAlpha_SubTyAbs : forall α e e', | EAlpha_SubTyAbs : forall α e e',
(e -->α e') -> (e -->α e') ->
(expr_ty_abs α e) -->α (expr_ty_abs α e') [{ Λ α ↦ e }] -->α [{ Λ α ↦ e' }]
| EAlpha_SubApp1 : forall e1 e1' e2, | EAlpha_SubApp1 : forall e1 e1' e2,
(e1 -->α e1') -> (e1 -->α e1') ->
(expr_app e1 e2) -->α (expr_app e1' e2) [{ e1 e2 }] -->α [{ e1' e2 }]
| EAlpha_SubApp2 : forall e1 e2 e2', | EAlpha_SubApp2 : forall e1 e2 e2',
(e2 -->α e2') -> (e2 -->α e2') ->
(expr_app e1 e2) -->α (expr_app e1 e2') [{ e1 e2 }] -->α [{ e1 e2' }]
where "s '-->α' t" := (expr_alpha s t). where "s '-->α' t" := (expr_alpha s t).
@ -45,45 +45,45 @@ Qed.
Inductive beta_step : expr_term -> expr_term -> Prop := Inductive beta_step : expr_term -> expr_term -> Prop :=
| E_App1 : forall e1 e1' e2, | E_App1 : forall e1 e1' e2,
e1 -->β e1' -> e1 -->β e1' ->
(expr_app e1 e2) -->β (expr_app e1' e2) [{ e1 e2 }] -->β [{ e1' e2 }]
| E_App2 : forall v1 e2 e2', | E_App2 : forall v1 e2 e2',
(is_value v1) -> (is_value v1) ->
e2 -->β e2' -> e2 -->β e2' ->
(expr_app v1 e2) -->β (expr_app v1 e2') [{ v1 e2 }] -->β [{ v1 e2' }]
| E_TypApp : forall e e' τ, | E_TypApp : forall e e' τ,
e -->β e' -> e -->β e' ->
(expr_ty_app e τ) -->β (expr_ty_app e' τ) [{ Λ τ ↦ e }] -->β [{ Λ τ ↦ e' }]
| E_TypAppLam : forall α e τ, | E_TypAppLam : forall α e τ,
(expr_ty_app (expr_ty_abs α e) τ) -->β (expr_specialize α τ e) [{ (Λ α ↦ e) # τ }] -->β (expr_specialize α τ e)
| E_AppLam : forall x τ e a, | E_AppLam : forall x τ e a,
(expr_app (expr_abs x τ e) a) -->β (expr_subst x a e) [{ (λ x , τ ↦ e) a }] -->β (expr_subst x a e)
| E_AppMorph : forall x τ e a, | E_AppMorph : forall x τ e a,
(expr_app (expr_morph x τ e) a) -->β (expr_subst x a e) [{ (λ x , τ ↦morph e) a }] -->β (expr_subst x a e)
| E_Let : forall x e a, | E_Let : forall x e a,
(expr_let x a e) -->β (expr_subst x a e) [{ let x := a in e }] -->β (expr_subst x a e)
| E_StripAscend : forall τ e, | E_StripAscend : forall τ e,
(expr_ascend τ e) -->β e [{ e as τ }] -->β e
| E_StripDescend : forall τ e, | E_StripDescend : forall τ e,
(expr_descend τ e) -->β e [{ e des τ }] -->β e
| E_Ascend : forall τ e e', | E_Ascend : forall τ e e',
(e -->β e') -> (e -->β e') ->
(expr_ascend τ e) -->β (expr_ascend τ e') [{ e as τ }] -->β [{ e' as τ }]
| E_AscendCollapse : forall τ' τ e, | E_AscendCollapse : forall τ' τ e,
(expr_ascend τ' (expr_ascend τ e)) -->β (expr_ascend (type_ladder τ' τ) e) [{ (e as τ) as τ' }] -->β [{ e as (τ'~τ) }]
| E_DescendCollapse : forall τ' τ e, | E_DescendCollapse : forall τ' τ e,
(τ':<=τ) -> (τ':<=τ) ->
(expr_descend τ (expr_descend τ' e)) -->β (expr_descend τ e) [{ (e des τ') des τ }] -->β [{ e des τ }]
where "s '-->β' t" := (beta_step s t). where "s '-->β' t" := (beta_step s t).
@ -102,7 +102,7 @@ Notation " s -->β* t " := (multi beta_step s t) (at level 40).
Example reduce1 : Example reduce1 :
[{ [{
let "deg2turns" := let "deg2turns" :=
(λ"x" $"Angle"$~$"Degrees"$~$"ℝ"$ (λ"x",$"Angle"$~$"Degrees"$~$"ℝ"$
↦morph ((%"/"% (%"x"% des $"ℝ"$) %"360"%) as $"Angle"$~$"Turns"$)) ↦morph ((%"/"% (%"x"% des $"ℝ"$) %"360"%) as $"Angle"$~$"Turns"$))
in ( %"deg2turns"% (%"60"% as $"Angle"$~$"Degrees"$) ) in ( %"deg2turns"% (%"60"% as $"Angle"$~$"Degrees"$) )
}] }]
@ -111,7 +111,7 @@ Example reduce1 :
((%"/"% %"60"%) %"360"%) as $"Angle"$~$"Turns"$ ((%"/"% %"60"%) %"360"%) as $"Angle"$~$"Turns"$
}]. }].
Proof. Proof.
apply Multi_Step with (y:=[{ (λ"x" $"Angle"$~$"Degrees"$~$"ℝ"$ apply Multi_Step with (y:=[{ (λ"x",$"Angle"$~$"Degrees"$~$"ℝ"$
↦morph (((%"/"% (%"x"% des $"ℝ"$)) %"360"%) as $"Angle"$~$"Turns"$)) (%"60"% as $"Angle"$~$"Degrees"$) }]). ↦morph (((%"/"% (%"x"% des $"ℝ"$)) %"360"%) as $"Angle"$~$"Turns"$)) (%"60"% as $"Angle"$~$"Degrees"$) }]).
apply E_Let. apply E_Let.

1
coq/soundness.v
View file

@ -8,7 +8,6 @@ Require Import morph.
Require Import smallstep. Require Import smallstep.
Require Import typing. Require Import typing.
Lemma typing_weakening : forall Γ e τ x σ, Lemma typing_weakening : forall Γ e τ x σ,
(Γ |- e \is τ) -> (Γ |- e \is τ) ->
((ctx_assign x σ Γ) |- e \is τ) ((ctx_assign x σ Γ) |- e \is τ)

44
coq/terms.v
View file

@ -64,58 +64,66 @@ Declare Custom Entry ladder_type.
Declare Custom Entry ladder_expr. Declare Custom Entry ladder_expr.
Notation "[< t >]" := t Notation "[< t >]" := t
(t custom ladder_type at level 80) : ladder_type_scope. (t custom ladder_type at level 99) : ladder_type_scope.
Notation "t" := t Notation "t" := t
(in custom ladder_type at level 0, t ident) : ladder_type_scope. (in custom ladder_type at level 0, t ident) : ladder_type_scope.
Notation "'∀' x ',' t" := (type_univ x t) Notation "'∀' x ',' t" := (type_univ x t)
(t custom ladder_type at level 80, in custom ladder_type at level 80, x constr). (t custom ladder_type at level 80, in custom ladder_type at level 80, x constr).
Notation "'<' σ τ '>'" := (type_spec σ τ) Notation "'<' σ τ '>'" := (type_spec σ τ)
(in custom ladder_type at level 80, left associativity) : ladder_type_scope. (in custom ladder_type at level 80, left associativity) : ladder_type_scope.
Notation "'(' τ ')'" := τ Notation "'[' τ ']'" := (type_spec (type_id "Seq") τ)
(in custom ladder_type at level 70) : ladder_type_scope. (in custom ladder_type at level 70) : ladder_type_scope.
Notation "'(' τ ')'" := τ
(in custom ladder_type at level 5) : ladder_type_scope.
Notation "σ '->' τ" := (type_fun σ τ) Notation "σ '->' τ" := (type_fun σ τ)
(in custom ladder_type at level 75, right associativity) : ladder_type_scope. (in custom ladder_type at level 75, right associativity) : ladder_type_scope.
Notation "σ '->morph' τ" := (type_morph σ τ) Notation "σ '->morph' τ" := (type_morph σ τ)
(in custom ladder_type at level 75, right associativity, τ at level 80) : ladder_type_scope. (in custom ladder_type at level 75, right associativity, τ at level 80) : ladder_type_scope.
Notation "σ '~' τ" := (type_ladder σ τ) Notation "σ '~' τ" := (type_ladder σ τ)
(in custom ladder_type at level 70, right associativity) : ladder_type_scope. (in custom ladder_type at level 20, right associativity) : ladder_type_scope.
Notation "'$' x '$'" := (type_id x%string) Notation "'$' x '$'" := (type_id x%string)
(in custom ladder_type at level 0, x constr) : ladder_type_scope. (in custom ladder_type at level 10, x constr) : ladder_type_scope.
Notation "'%' x '%'" := (type_var x%string) Notation "'%' x '%'" := (type_var x%string)
(in custom ladder_type at level 0, x constr) : ladder_type_scope. (in custom ladder_type at level 10, x constr) : ladder_type_scope.
Notation "[{ e }]" := e Notation "[{ e }]" := e
(e custom ladder_expr at level 80) : ladder_expr_scope. (e custom ladder_expr at level 99) : ladder_expr_scope.
Notation "e" := e Notation "e" := e
(in custom ladder_expr at level 0, e ident) : ladder_expr_scope. (in custom ladder_expr at level 0, e ident) : ladder_expr_scope.
Notation "'%' x '%'" := (expr_var x%string) Notation "'%' x '%'" := (expr_var x%string)
(in custom ladder_expr at level 0, x constr) : ladder_expr_scope. (in custom ladder_expr at level 10, x constr) : ladder_expr_scope.
Notation "'Λ' t '↦' e" := (expr_ty_abs t e) Notation "'Λ' t '↦' e" := (expr_ty_abs t e)
(in custom ladder_expr at level 10, t constr, e custom ladder_expr at level 80) : ladder_expr_scope. (in custom ladder_expr at level 10, t constr, e custom ladder_expr at level 80, right associativity) : ladder_expr_scope.
Notation "'λ' x τ '↦' e" := (expr_abs x τ e) Notation "'λ' x ',' τ '↦' e" := (expr_abs x τ e)
(in custom ladder_expr at level 10, x constr, τ custom ladder_type at level 99, e custom ladder_expr at level 99) :ladder_expr_scope. (in custom ladder_expr at level 70, x constr, τ custom ladder_type at level 90, e custom ladder_expr at level 80, right associativity) :ladder_expr_scope.
Notation "'λ' x τ '↦morph' e" := (expr_morph x τ e) Notation "'λ' x ',' τ '↦morph' e" := (expr_morph x τ e)
(in custom ladder_expr at level 10, x constr, τ custom ladder_type at level 99, e custom ladder_expr at level 99) :ladder_expr_scope. (in custom ladder_expr at level 70, x constr, τ custom ladder_type at level 90, e custom ladder_expr at level 80, right associativity) :ladder_expr_scope.
Notation "'let' x ':=' e 'in' t" := (expr_let x e t) Notation "'let' x ':=' e 'in' t" := (expr_let x e t)
(in custom ladder_expr at level 20, x constr, e custom ladder_expr at level 99, t custom ladder_expr at level 99) : ladder_expr_scope. (in custom ladder_expr at level 60, x constr, e custom ladder_expr at level 80, t custom ladder_expr at level 80, right associativity) : ladder_expr_scope.
Notation "e 'as' τ" := (expr_ascend τ e) Notation "e 'as' τ" := (expr_ascend τ e)
(in custom ladder_expr at level 30, e custom ladder_expr, τ custom ladder_type at level 99) : ladder_expr_scope. (in custom ladder_expr at level 30, e custom ladder_expr, τ custom ladder_type at level 99) : ladder_expr_scope.
Notation "e 'des' τ" := (expr_descend τ e) Notation "e 'des' τ" := (expr_descend τ e)
(in custom ladder_expr at level 30, e custom ladder_expr, τ custom ladder_type at level 99) : ladder_expr_scope. (in custom ladder_expr at level 30, e custom ladder_expr, τ custom ladder_type at level 99) : ladder_expr_scope.
Notation "e1 e2" := (expr_app e1 e2) Notation "e1 e2" := (expr_app e1 e2)
(in custom ladder_expr at level 50) : ladder_expr_scope. (in custom ladder_expr at level 90, e2 custom ladder_expr at next level) : ladder_expr_scope.
Notation "e '#' τ" := (expr_ty_app e τ)
(in custom ladder_expr at level 80, τ custom ladder_type at level 101, left associativity) : ladder_expr_scope.
Notation "'(' e ')'" := e Notation "'(' e ')'" := e
(in custom ladder_expr at level 0) : ladder_expr_scope. (in custom ladder_expr, e custom ladder_expr at next level, left associativity) : ladder_expr_scope.
Print Grammar constr.
(* EXAMPLES *) (* EXAMPLES *)
Open Scope ladder_type_scope. Open Scope ladder_type_scope.
Open Scope ladder_expr_scope. Open Scope ladder_expr_scope.
Check [< ∀"α", (< $"Seq"$ %"α"% > ~ < $"List"$ %"α"% >) >]. Check [< ∀"α", [%"α"%] ~ <$"List"$ %"α"%> >].
Definition polymorphic_identity1 : expr_term := [{ Λ"T" ↦ λ"x"%"T"% ↦ (%"x"%) }].
Definition polymorphic_identity2 : expr_term := [{ Λ"T" ↦ λ"y"%"T"% ↦ %"y"% }]. Definition polymorphic_identity1 : expr_term := [{ Λ"T" ↦ λ"x",%"T"% ↦ (%"x"%) }].
Definition polymorphic_identity2 : expr_term := [{ Λ"T" ↦ λ"y",%"T"% ↦ %"y"% }].
Compute polymorphic_identity1. Compute polymorphic_identity1.

110
coq/typing.v
View file

@ -11,40 +11,41 @@ Require Import morph.
(** Typing Derivation *) (** Typing Derivation *)
Open Scope ladder_expr_scope.
Reserved Notation "Gamma '|-' x '\is' X" (at level 101, x at next level, X at level 0). Reserved Notation "Gamma '|-' x '\is' X" (at level 101, x at next level, X at level 0).
Inductive expr_type : context -> expr_term -> type_term -> Prop := Inductive expr_type : context -> expr_term -> type_term -> Prop :=
| T_Var : forall Γ x τ, | T_Var : forall Γ x τ,
(context_contains Γ x τ) -> context_contains Γ x τ ->
(Γ |- (expr_var x) \is τ) Γ |- [{ %x% }] \is τ
| T_Let : forall Γ s σ t τ x, | T_Let : forall Γ s σ t τ x,
(Γ |- s \is σ) -> Γ |- s \is σ ->
((ctx_assign x σ Γ) |- t \is τ) -> (ctx_assign x σ Γ) |- t \is τ ->
(Γ |- (expr_let x s t) \is τ) Γ |- [{ let x := s in t }] \is τ
| T_TypeAbs : forall Γ e τ α, | T_TypeAbs : forall Γ e τ α,
Γ |- e \is τ -> Γ |- e \is τ ->
Γ |- (expr_ty_abs α e) \is (type_univ α τ) Γ |- [{ Λ α ↦ e }] \is [< ∀α,τ >]
| T_TypeApp : forall Γ α e σ τ τ', | T_TypeApp : forall Γ α e σ τ,
Γ |- e \is (type_univ α τ) -> Γ |- e \is [< ∀α, τ >] ->
(type_subst1 α σ τ τ') -> Γ |- [{ e # σ }] \is (type_subst α σ τ)
Γ |- (expr_ty_app e σ) \is τ'
| T_Abs : forall Γ x σ t τ, | T_Abs : forall Γ x σ t τ,
((ctx_assign x σ Γ) |- t \is τ) -> (ctx_assign x σ Γ) |- t \is τ ->
(Γ |- (expr_abs x σ t) \is (type_fun σ τ)) Γ |- [{ λ x , σ ↦ t }] \is [< σ -> τ >]
| T_MorphAbs : forall Γ x σ e τ, | T_MorphAbs : forall Γ x σ t τ,
((ctx_assign x σ Γ) |- e \is τ) -> (ctx_assign x σ Γ) |- t \is τ ->
Γ |- (expr_morph x σ e) \is (type_morph σ τ) Γ |- [{ λ x , σ ↦morph t }] \is [< σ ->morph τ >]
| T_App : forall Γ f a σ' σ τ, | T_App : forall Γ f a σ' σ τ,
(Γ |- f \is (type_fun σ τ)) -> (Γ |- f \is [< σ -> τ >]) ->
(Γ |- a \is σ') -> (Γ |- a \is σ') ->
(Γ |- σ' ~> σ) -> (Γ |- σ' ~> σ) ->
(Γ |- (expr_app f a) \is τ) (Γ |- [{ (f a) }] \is τ)
| T_MorphFun : forall Γ f σ τ, | T_MorphFun : forall Γ f σ τ,
Γ |- f \is (type_morph σ τ) -> Γ |- f \is (type_morph σ τ) ->
@ -52,7 +53,7 @@ Inductive expr_type : context -> expr_term -> type_term -> Prop :=
| T_Ascend : forall Γ e τ τ', | T_Ascend : forall Γ e τ τ',
(Γ |- e \is τ) -> (Γ |- e \is τ) ->
(Γ |- (expr_ascend τ' e) \is (type_ladder τ' τ)) (Γ |- [{ e as τ' }] \is [< τ' ~ τ >])
| T_DescendImplicit : forall Γ x τ τ', | T_DescendImplicit : forall Γ x τ τ',
Γ |- x \is τ -> Γ |- x \is τ ->
@ -62,7 +63,7 @@ Inductive expr_type : context -> expr_term -> type_term -> Prop :=
| T_Descend : forall Γ x τ τ', | T_Descend : forall Γ x τ τ',
Γ |- x \is τ -> Γ |- x \is τ ->
(τ :<= τ') -> (τ :<= τ') ->
Γ |- (expr_descend τ' x) \is τ' Γ |- [{ x des τ' }] \is τ'
where "Γ '|-' x '\is' τ" := (expr_type Γ x τ). where "Γ '|-' x '\is' τ" := (expr_type Γ x τ).
@ -75,73 +76,85 @@ Definition is_well_typed (e:expr_term) : Prop :=
Inductive translate_typing : context -> expr_term -> type_term -> expr_term -> Prop := Inductive translate_typing : context -> expr_term -> type_term -> expr_term -> Prop :=
| Expand_Var : forall Γ x τ, | Expand_Var : forall Γ x τ,
(Γ |- (expr_var x) \is τ) -> (Γ |- (expr_var x) \is τ) ->
(translate_typing Γ (expr_var x) τ (expr_var x)) (translate_typing Γ [{ %x% }] τ [{ %x% }])
| Expand_Let : forall Γ x e e' t t' σ τ, | Expand_Let : forall Γ x e e' t t' σ τ,
(Γ |- e \is σ) -> (Γ |- e \is σ) ->
((ctx_assign x σ Γ) |- t \is τ) -> ((ctx_assign x σ Γ) |- t \is τ) ->
(translate_typing Γ e σ e') -> (translate_typing Γ e σ e') ->
(translate_typing (ctx_assign x σ Γ) t τ t') -> (translate_typing (ctx_assign x σ Γ) t τ t') ->
(translate_typing Γ (expr_let x e t) τ (expr_let x e' t')) (translate_typing Γ
[{ let x := e in t }]
τ
[{ let x := e' in t' }])
| Expand_TypeAbs : forall Γ α e e' τ, | Expand_TypeAbs : forall Γ α e e' τ,
(Γ |- e \is τ) -> (Γ |- e \is τ) ->
(translate_typing Γ e τ e') -> (translate_typing Γ e τ e') ->
(translate_typing Γ (translate_typing Γ
(expr_ty_abs α e) [{ Λ α ↦ e }]
(type_univ α τ) [< ∀ α,τ >]
(expr_ty_abs α e')) [{ Λ α ↦ e' }])
| Expand_TypeApp : forall Γ e e' σ α τ, | Expand_TypeApp : forall Γ e e' σ α τ,
(Γ |- e \is (type_univ α τ)) -> (Γ |- e \is (type_univ α τ)) ->
(translate_typing Γ e τ e') -> (translate_typing Γ e τ e') ->
(translate_typing Γ (expr_ty_app e τ) (type_subst α σ τ) (expr_ty_app e' τ)) (translate_typing Γ
[{ e # τ }]
(type_subst α σ τ)
[{ e' # τ }])
| Expand_Abs : forall Γ x σ e e' τ, | Expand_Abs : forall Γ x σ e e' τ,
((ctx_assign x σ Γ) |- e \is τ) -> ((ctx_assign x σ Γ) |- e \is τ) ->
(Γ |- (expr_abs x σ e) \is (type_fun σ τ)) -> (Γ |- (expr_abs x σ e) \is (type_fun σ τ)) ->
(translate_typing (ctx_assign x σ Γ) e τ e') -> (translate_typing (ctx_assign x σ Γ) e τ e') ->
(translate_typing Γ (expr_abs x σ e) (type_fun σ τ) (expr_abs x σ e')) (translate_typing Γ
[{ λ x , σ ↦ e }]
[< σ -> τ >]
[{ λ x , σ ↦ e' }])
| Expand_MorphAbs : forall Γ x σ e e' τ, | Expand_MorphAbs : forall Γ x σ e e' τ,
((ctx_assign x σ Γ) |- e \is τ) -> ((ctx_assign x σ Γ) |- e \is τ) ->
(Γ |- (expr_abs x σ e) \is (type_fun σ τ)) -> (Γ |- (expr_abs x σ e) \is (type_fun σ τ)) ->
(translate_typing (ctx_assign x σ Γ) e τ e') -> (translate_typing (ctx_assign x σ Γ) e τ e') ->
(translate_typing Γ (expr_morph x σ e) (type_morph σ τ) (expr_morph x σ e')) (translate_typing Γ
[{ λ x , σ ↦morph e }]
[< σ ->morph τ >]
[{ λ x , σ ↦morph e' }])
| Expand_App : forall Γ f f' a a' m σ τ σ', | Expand_App : forall Γ f f' a a' m σ τ σ',
(Γ |- f \is (type_fun σ τ)) -> (Γ |- f \is (type_fun σ τ)) ->
(Γ |- a \is σ') -> (Γ |- a \is σ') ->
(Γ |- σ' ~> σ) -> (Γ |- σ' ~> σ) ->
(translate_typing Γ f (type_fun σ τ) f') -> (translate_typing Γ f [< σ -> τ >] f') ->
(translate_typing Γ a σ' a') -> (translate_typing Γ a σ' a') ->
(translate_morphism_path Γ σ' σ m) -> (translate_morphism_path Γ σ' σ m) ->
(translate_typing Γ (expr_app f a) τ (expr_app f' (expr_app m a'))) (translate_typing Γ [{ f a }] τ [{ f' (m a') }])
| Expand_MorphFun : forall Γ f f' σ τ, | Expand_MorphFun : forall Γ f f' σ τ,
(Γ |- f \is (type_morph σ τ)) -> (Γ |- f \is (type_morph σ τ)) ->
(Γ |- f \is (type_fun σ τ)) -> (Γ |- f \is (type_fun σ τ)) ->
(translate_typing Γ f (type_morph σ τ) f') -> (translate_typing Γ f [< σ ->morph τ >] f') ->
(translate_typing Γ f (type_fun σ τ) f') (translate_typing Γ f [< σ -> τ >] f')
| Expand_Ascend : forall Γ e e' τ τ', | Expand_Ascend : forall Γ e e' τ τ',
(Γ |- e \is τ) -> (Γ |- e \is τ) ->
(Γ |- (expr_ascend τ' e) \is (type_ladder τ' τ)) -> (Γ |- [{ e as τ' }] \is [< τ' ~ τ >]) ->
(translate_typing Γ e τ e') -> (translate_typing Γ e τ e') ->
(translate_typing Γ (expr_ascend τ' e) (type_ladder τ' τ) (expr_ascend τ' e')) (translate_typing Γ [{ e as τ' }] [< τ' ~ τ >] [{ e' as τ' }])
| Expand_Descend : forall Γ e e' τ τ', | Expand_Descend : forall Γ e e' τ τ',
(Γ |- e \is τ) -> (Γ |- e \is τ) ->
(τ :<= τ') -> (τ :<= τ') ->
(Γ |- e \is τ') -> (Γ |- [{ e des τ' }] \is τ') ->
(translate_typing Γ e τ e') -> (translate_typing Γ e τ e') ->
(translate_typing Γ e τ' e') (translate_typing Γ e τ' [{ e' des τ' }])
. .
(* Examples *) (* Examples *)
Example typing1 : Example typing1 :
ctx_empty |- [{ Λ"T" ↦ λ "x" %"T"% ↦ %"x"% }] \is [< ∀"T", %"T"% -> %"T"% >]. ctx_empty |- [{ Λ"T" ↦ λ "x",%"T"% ↦ %"x"% }] \is [< ∀"T", %"T"% -> %"T"% >].
Proof. Proof.
intros. intros.
apply T_TypeAbs. apply T_TypeAbs.
@ -151,7 +164,7 @@ Proof.
Qed. Qed.
Example typing2 : Example typing2 :
ctx_empty |- [{ Λ"T" ↦ λ "x" %"T"% ↦ %"x"% }] \is [< ∀"U", %"U"% -> %"U"% >]. ctx_empty |- [{ Λ"T" ↦ λ "x",%"T"% ↦ %"x"% }] \is [< ∀"U", %"U"% -> %"U"% >].
Proof. Proof.
intros. intros.
apply T_DescendImplicit with (τ:=[< ∀"T",(%"T"% -> %"T"%) >]). apply T_DescendImplicit with (τ:=[< ∀"T",(%"T"% -> %"T"%) >]).
@ -163,14 +176,11 @@ Proof.
apply TSubRepr_Refl. apply TSubRepr_Refl.
apply TEq_Alpha. apply TEq_Alpha.
apply TAlpha_Rename. apply TAlpha_Rename.
apply TSubst_Fun.
apply TSubst_VarReplace.
apply TSubst_VarReplace.
Qed. Qed.
Example typing3 : Example typing3 :
ctx_empty |- [{ ctx_empty |- [{
Λ"T" ↦ Λ"U" ↦ λ"x" %"T"% ↦ λ"y" %"U"% ↦ %"y"% Λ"T" ↦ Λ"U" ↦ λ"x",%"T"% ↦ λ"y",%"U"% ↦ %"y"%
}] \is [< }] \is [<
∀"S",∀"T",(%"S"%->%"T"%->%"T"%) ∀"S",∀"T",(%"S"%->%"T"%->%"T"%)
>]. >].
@ -186,26 +196,14 @@ Proof.
apply TEq_Trans with (y:= [< ∀"S",∀"U",(%"S"%->%"U"%->%"U"%) >] ). apply TEq_Trans with (y:= [< ∀"S",∀"U",(%"S"%->%"U"%->%"U"%) >] ).
apply TEq_Alpha. apply TEq_Alpha.
apply TAlpha_Rename. apply TAlpha_Rename.
apply TSubst_UnivReplace. discriminate.
easy.
apply TSubst_Fun.
apply TSubst_VarReplace.
apply TSubst_Fun.
apply TSubst_VarKeep. discriminate.
apply TSubst_VarKeep. discriminate.
apply TEq_Alpha. apply TEq_Alpha.
apply TAlpha_SubUniv. apply TAlpha_SubUniv.
apply TAlpha_Rename. apply TAlpha_Rename.
apply TSubst_Fun.
apply TSubst_VarKeep. discriminate.
apply TSubst_Fun.
apply TSubst_VarReplace.
apply TSubst_VarReplace.
Qed. Qed.
Example typing4 : (is_well_typed Example typing4 : (is_well_typed
[{ Λ"T" ↦ Λ"U" ↦ λ"x" %"T"% ↦ λ"y" %"U"% ↦ %"x"% }] [{ Λ"T" ↦ Λ"U" ↦ λ"x",%"T"% ↦ λ"y",%"U"% ↦ %"x"% }]
). ).
Proof. Proof.
unfold is_well_typed. unfold is_well_typed.
@ -218,8 +216,6 @@ Proof.
apply C_shuffle, C_take. apply C_shuffle, C_take.
Qed. Qed.
Open Scope ladder_expr_scope.
Example typing5 : Example typing5 :
(ctx_assign "60" [< $"ℝ"$ >] (ctx_assign "60" [< $"ℝ"$ >]
(ctx_assign "360" [< $"ℝ"$ >] (ctx_assign "360" [< $"ℝ"$ >]
@ -228,7 +224,7 @@ Example typing5 :
|- |-
[{ [{
let "deg2turns" := let "deg2turns" :=
(λ"x" $"Angle"$~$"Degrees"$~$"ℝ"$ (λ"x",$"Angle"$~$"Degrees"$~$"ℝ"$
↦morph (((%"/"% %"x"%) %"360"%) as $"Angle"$~$"Turns"$)) ↦morph (((%"/"% %"x"%) %"360"%) as $"Angle"$~$"Turns"$))
in ( %"deg2turns"% (%"60"% as $"Angle"$~$"Degrees"$) ) in ( %"deg2turns"% (%"60"% as $"Angle"$~$"Degrees"$) )
}] }]