coq: move context into separate module, define morphism-path & redefine typing rules
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5 changed files with 58 additions and 140 deletions
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@ -3,8 +3,9 @@ terms.v
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equiv.v
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subst.v
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subtype.v
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typing.v
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context.v
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morph.v
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typing.v
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smallstep.v
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soundness.v
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bbencode.v
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21
coq/context.v
Normal file
21
coq/context.v
Normal file
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@ -0,0 +1,21 @@
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From Coq Require Import Strings.String.
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Require Import terms.
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Include Terms.
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Module Context.
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Inductive context : Type :=
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| ctx_assign : string -> type_term -> context -> context
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| ctx_empty : context
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.
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Inductive context_contains : context -> string -> type_term -> Prop :=
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| C_take : forall (x:string) (X:type_term) (Γ:context),
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(context_contains (ctx_assign x X Γ) x X)
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| C_shuffle : forall x X y Y Γ,
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(context_contains Γ x X) ->
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(context_contains (ctx_assign y Y Γ) x X).
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End Context.
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65
coq/morph.v
65
coq/morph.v
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@ -3,66 +3,37 @@ Require Import terms.
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Require Import subst.
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Require Import equiv.
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Require Import subtype.
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Require Import typing.
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Require Import context.
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Include Terms.
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Include Subst.
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Include Equiv.
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Include Subtype.
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Include Typing.
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Include Context.
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Module Morph.
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(* Given a context, there is a morphism path from τ to τ' *)
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Reserved Notation "Γ '|-' σ '~>' τ" (at level 101, σ at next level, τ at next level).
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Reserved Notation "Γ '|-' '[[' e ']]' '=' t" (at level 101, e at next level, t at level 0).
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Inductive morphism_path : context -> type_term -> type_term -> Prop :=
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| M_Sub : forall Γ τ τ',
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(τ :<= τ') ->
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(Γ |- τ ~> τ')
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Inductive expand_morphisms : context -> expr_term -> expr_term -> Prop :=
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| Expand_Transform : forall Γ e h τ τ',
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| M_Single : forall Γ h τ τ',
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(context_contains Γ h (type_morph τ τ')) ->
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(Γ |- e \is τ) ->
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(Γ |- [[ e ]] = (expr_app (expr_var h) e))
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(Γ |- τ ~> τ')
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| Expand_Otherwise : forall Γ e,
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(Γ |- [[ e ]] = e)
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| M_Chain : forall Γ τ τ' τ'',
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(Γ |- τ ~> τ') ->
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(Γ |- τ' ~> τ'') ->
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(Γ |- τ ~> τ'')
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| Expand_App : forall Γ f f' a a' σ τ,
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(Γ |- f \compatible (type_fun σ τ)) ->
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(Γ |- a \compatible σ) ->
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(Γ |- [[ f ]] = f') ->
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(Γ |- [[ a ]] = a') ->
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(Γ |- [[ (expr_app f a) ]] = (expr_app f a'))
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| M_MapSeq : forall Γ τ τ',
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(Γ |- τ ~> τ') ->
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(Γ |- (type_spec (type_id "Seq") τ) ~> (type_spec (type_id "Seq") τ'))
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where "Γ '|-' '[[' e ']]' '=' t" := (expand_morphisms Γ e t).
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(* For every well-typed term, there is an expanded exactly typed term *)
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Lemma well_typed_is_expandable :
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forall (e:expr_term),
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(is_well_typed e) ->
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exists Γ e', (expand_morphisms Γ e e') -> (is_exactly_typed e')
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.
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Proof.
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Admitted.
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Example expand1 :
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(ctx_assign "morph-radix"
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[< <%"PosInt"% %"16"%> ->morph <%"PosInt"% %"10"%> >]
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(ctx_assign "fn-decimal"
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[< <%"PosInt"% %"10"%> -> <%"PosInt"% %"10"%> >]
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(ctx_assign "x"
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[< <%"PosInt"% %"16"%> >]
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ctx_empty)))
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|- [[ (expr_app (expr_var "fn-decimal") (expr_var "x")) ]]
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= (expr_app (expr_var "fn-decimal")
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(expr_app (expr_var "morph-radix") (expr_var "x")))
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.
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Proof.
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apply Expand_App with (f':=(expr_var "fn-decimal")) (σ:=[< < %"PosInt"% %"10"% > >]) (τ:=[< < %"PosInt"% %"10"% > >]).
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apply TCompat_Native.
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Admitted.
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where "Γ '|-' s '~>' t" := (morphism_path Γ s t).
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End Morph.
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@ -22,15 +22,6 @@ Definition is_stuck (e:expr_term) : Prop :=
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~(exists e', e -->β e')
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.
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(* every exactly typed term is not stuck *)
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Lemma exact_progress :
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forall (e:expr_term),
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(is_exactly_typed e) -> ~(is_stuck e)
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.
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Proof.
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Admitted.
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(* every well typed term is not stuck *)
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Lemma progress :
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forall (e:expr_term),
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@ -40,32 +31,14 @@ Proof.
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Admitted.
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(* reduction step preserves the type *)
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Lemma exact_preservation :
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(* reduction step preserves well-typedness *)
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Lemma preservation :
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forall Γ e e' τ,
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(Γ |- e \is τ) ->
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(e -->β e') ->
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(Γ |- e' \is τ)
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.
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Proof.
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(*
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intros.
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generalize dependent e'.
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induction H.
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intros e' I.
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inversion I.
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*)
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Admitted.
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(* reduction step preserves well-typedness *)
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Lemma preservation :
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forall Γ e e' τ,
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(Γ |- e \compatible τ) ->
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(e -->β e') ->
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(Γ |- e' \compatible τ)
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.
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Proof.
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Admitted.
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(* every well-typed expression can be reduced to a value *)
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76
coq/typing.v
76
coq/typing.v
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@ -6,64 +6,56 @@ Require Import terms.
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Require Import subst.
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Require Import equiv.
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Require Import subtype.
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Require Import context.
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Require Import morph.
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Include Terms.
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Include Subst.
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Include Equiv.
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Include Subtype.
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Include Context.
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Include Morph.
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Module Typing.
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(** Typing Derivation *)
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Inductive context : Type :=
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| ctx_assign : string -> type_term -> context -> context
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| ctx_empty : context
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.
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Inductive context_contains : context -> string -> type_term -> Prop :=
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| C_take : forall (x:string) (X:type_term) (Γ:context),
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(context_contains (ctx_assign x X Γ) x X)
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| C_shuffle : forall x X y Y Γ,
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(context_contains Γ x X) ->
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(context_contains (ctx_assign y Y Γ) x X).
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Reserved Notation "Gamma '|-' x '\is' X" (at level 101, x at next level, X at level 0).
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Reserved Notation "Gamma '|-' x '\compatible' X" (at level 101, x at next level, X at level 0).
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Inductive expr_type : context -> expr_term -> type_term -> Prop :=
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| T_Var : forall Γ x τ,
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(context_contains Γ x τ) ->
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(Γ |- (expr_var x) \is τ)
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| T_Let : forall Γ s (σ:type_term) t τ x,
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| T_Let : forall Γ s σ t τ x,
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(Γ |- s \is σ) ->
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(Γ |- t \is τ) ->
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(Γ |- (expr_let x σ s t) \is τ)
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| T_TypeAbs : forall Γ (e:expr_term) (τ:type_term) α,
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| T_TypeAbs : forall Γ e τ α,
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Γ |- e \is τ ->
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Γ |- (expr_ty_abs α e) \is (type_univ α τ)
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| T_TypeApp : forall Γ α (e:expr_term) (σ:type_term) (τ:type_term),
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| T_TypeApp : forall Γ α e σ τ,
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Γ |- e \is (type_univ α τ) ->
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Γ |- (expr_ty_app e σ) \is (type_subst α σ τ)
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| T_Abs : forall (Γ:context) (x:string) (σ:type_term) (t:expr_term) (τ:type_term),
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| T_Abs : forall Γ x σ t τ,
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(context_contains Γ x σ) ->
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Γ |- t \is τ ->
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Γ |- (expr_abs x σ t) \is (type_fun σ τ)
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| T_App : forall (Γ:context) (f:expr_term) (a:expr_term) (σ:type_term) (τ:type_term),
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Γ |- f \is (type_fun σ τ) ->
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Γ |- a \is σ ->
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Γ |- (expr_app f a) \is τ
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| T_MorphAbs : forall Γ x σ e τ,
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(context_contains Γ x σ) ->
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Γ |- e \is τ ->
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Γ |- (expr_morph x σ e) \is (type_morph σ τ)
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| T_App : forall Γ f a σ' σ τ,
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(Γ |- f \is (type_fun σ τ)) ->
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(Γ |- a \is σ') ->
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(Γ |- σ' ~> σ) ->
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(Γ |- (expr_app f a) \is τ)
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| T_MorphFun : forall Γ f σ τ,
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Γ |- f \is (type_morph σ τ) ->
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Γ |- f \is (type_fun σ τ)
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@ -80,45 +72,7 @@ Inductive expr_type : context -> expr_term -> type_term -> Prop :=
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where "Γ '|-' x '\is' τ" := (expr_type Γ x τ).
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Inductive expr_type_compatible : context -> expr_term -> type_term -> Prop :=
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| TCompat_Native : forall Γ e τ,
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(Γ |- e \is τ) ->
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(Γ |- e \compatible τ)
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| TCompat_Let : forall Γ s (σ:type_term) t τ x,
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(Γ |- s \is σ) ->
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(context_contains Γ x σ) ->
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(Γ |- t \compatible τ) ->
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(Γ |- (expr_let x σ s t) \compatible τ)
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| T_CompatTypeApp : forall Γ α (e:expr_term) (σ:type_term) (τ:type_term),
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Γ |- e \compatible (type_univ α τ) ->
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Γ |- (expr_ty_app e σ) \compatible (type_subst α σ τ)
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| TCompat_App : forall Γ f a σ τ,
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(Γ |- f \compatible (type_fun σ τ)) ->
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(Γ |- a \compatible σ) ->
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(Γ |- (expr_app f a) \compatible τ)
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| TCompat_Morph : forall Γ h x τ τ',
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(context_contains Γ h (type_morph τ τ')) ->
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(Γ |- x \compatible τ) ->
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(Γ |- x \compatible τ')
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| TCompat_Sub : forall Γ x τ τ',
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(Γ |- x \compatible τ) ->
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(τ ~<= τ') ->
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(Γ |- x \compatible τ')
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where "Γ '|-' x '\compatible' τ" := (expr_type_compatible Γ x τ).
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Definition is_well_typed (e:expr_term) : Prop :=
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exists Γ τ,
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Γ |- e \compatible τ
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.
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Definition is_exactly_typed (e:expr_term) : Prop :=
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exists Γ τ,
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Γ |- e \is τ
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.
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@ -200,7 +154,6 @@ Proof.
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apply TSubst_VarReplace.
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Qed.
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Example typing4 : (is_well_typed
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[{ Λ"T" ↦ Λ"U" ↦ λ"x" %"T"% ↦ λ"y" %"U"% ↦ %"x"% }]
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).
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@ -209,7 +162,6 @@ Proof.
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exists (ctx_assign "x" [< %"T"% >]
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(ctx_assign "y" [< %"U"% >] ctx_empty)).
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exists [< ∀"T",∀"U",%"T"%->%"U"%->%"T"% >].
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apply TCompat_Native.
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apply T_TypeAbs.
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apply T_TypeAbs.
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apply T_Abs.
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