paper: wip add more lemmas

coq/terms.v

@@ -36,10 +36,36 @@ Coercion type_var : string >-> type_term.
 Coercion expr_var : string >-> expr_term.
 
 (*
-Notation "( x )" := x (at level 70).
+Coercion type_var : string >-> type_term.
-Notation "x ~ y" := (type_rung x y) (at level 69, left associativity).
+Coercion expr_var : string >-> expr_term.
-Notation "< x y >" := (type_app x y) (at level 68, left associativity).
-Notation "'$' x" := (type_id x) (at level 66).
 *)
 
+Declare Scope ladder_type_scope.
+Declare Scope ladder_expr_scope.
+Declare Custom Entry ladder_type.
+
+Notation "[ e ]" := e (e custom ladder_type at level 80) : ladder_type_scope.
+
+(* TODO: allow any variable names in notation, not just α,β,γ *)
+Notation "'∀α.' τ" := (type_univ "α" τ) (in custom ladder_type at level 80) : ladder_type_scope.
+Notation "'∀β.' τ" := (type_univ "β" τ) (in custom ladder_type at level 80) : ladder_type_scope.
+Notation "'∀γ.' τ" := (type_univ "γ" τ) (in custom ladder_type at level 80) : ladder_type_scope.
+Notation "'<' σ τ '>'" := (type_spec σ τ) (in custom ladder_type at level 80, left associativity) : ladder_type_scope.
+Notation "'(' τ ')'" := τ (in custom ladder_type at level 70) : ladder_type_scope.
+Notation "σ '->' τ" := (type_fun σ τ) (in custom ladder_type at level 75, right associativity) : ladder_type_scope.
+Notation "σ '->morph' τ" := (type_morph σ τ) (in custom ladder_type at level 75, right associativity) : ladder_type_scope.
+Notation "σ '~' τ" := (type_ladder σ τ) (in custom ladder_type at level 70, right associativity) : ladder_type_scope.
+Notation "'α'" := (type_var "α") (in custom ladder_type at level 60, right associativity) : ladder_type_scope.
+Notation "'β'" := (type_var "β") (in custom ladder_type at level 60, right associativity) : ladder_type_scope.
+Notation "'γ'" := (type_var "γ") (in custom ladder_type at level 60, right associativity) : ladder_type_scope.
+
+Open Scope ladder_type_scope.
+
+Definition t1 : type_term := [ ∀α.∀β.(α~β~γ)->β->(α->α)->β ].
+
+Compute t1.
+Close Scope ladder_type_scope.
+
+
+
 End Terms.

coq/typing.v

@@ -21,35 +21,69 @@ Inductive context_contains : context -> string -> type_term -> Prop :=
     (context_contains Γ x X) ->
     (context_contains (ctx_assign y Y Γ) x X).
 
-Reserved Notation "Gamma '|-' x '\in' 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).
+Reserved Notation "Gamma '|-' x '\compatible' X"  (at level 101, x at next level, X at level 0).
 
-Inductive expr_type : context -> expr -> ladder_type -> Prop :=
+Inductive expr_type : context -> expr_term -> type_term -> Prop :=
-  | T_Var : forall Γ x X,
+  | T_Var : forall Γ x τ,
-    (context_contains Γ x X) ->
+    (context_contains Γ x τ) ->
-    Γ |- x \in X
+    (Γ |- x \is τ)
 
-  | T_Let : forall Γ s (σ:ladder_type) t τ x,
+  | T_Let : forall Γ s (σ:type_term) t τ x,
-    Γ |- s \in σ ->
+    (Γ |- s \is σ) ->
-    Γ |- t \in τ ->
+    (Γ |- t \is τ) ->
-    Γ |- (expr_let x σ s t) \in τ
+    (Γ |- (expr_let x σ s t) \is τ)
 
-  | T_Abs : forall (Γ:context) (x:string) (X:ladder_type) (t:expr) (T:ladder_type),
+  | T_TypeAbs : forall Γ (e:expr_term) (τ:type_term) α,
-    Γ |- t \in T ->
+    Γ |- e \is τ ->
-    Γ |- (expr_tm_abs x X t) \in (type_fun X T)
+    Γ |- (expr_ty_abs α e) \is (type_univ α τ)
 
-  | T_App : forall (Γ:context) (f:expr) (a:expr) (S:ladder_type) (T:ladder_type),
+  | T_TypeApp : forall Γ α (e:expr_term) (σ:type_term) (τ:type_term),
-    Γ |- f \in (type_fun S T) ->
+    Γ |- e \is (type_univ α τ) ->
-    Γ |- a \in S ->
+    Γ |- (expr_ty_app e σ) \is (type_subst α σ τ)
-    Γ |- (expr_tm_app f a) \in T
 
-where "Γ '|-' x '\in' X" := (expr_type Γ x X).
+  | T_Abs : forall (Γ:context) (x:string) (σ:type_term) (t:expr_term) (τ:type_term),
+    (context_contains Γ x σ) ->
+    Γ |- t \is τ ->
+    Γ |- (expr_tm_abs x σ t) \is (type_fun σ τ)
 
+  | T_App : forall (Γ:context) (f:expr_term) (a:expr_term) (σ:type_term) (τ:type_term),
+    Γ |- f \is (type_fun σ τ) ->
+    Γ |- a \is σ ->
+    Γ |- (expr_tm_app f a) \is τ
+
+where "Γ '|-' x '\is' τ" := (expr_type Γ x τ).
+
+
+Inductive expr_type_compatible : context -> expr_term -> type_term -> Prop :=
+
+  | T_Compatible : forall Γ x τ,
+    (Γ |- x \is τ) ->
+    (Γ |- x \compatible τ)
+
+where "Γ '|-' x '\compatible' τ" := (expr_type_compatible Γ x τ).
 
 Example typing1 :
-  ctx_empty |-
+  forall Γ,
-  (expr_ty_abs "T" (expr_tm_abs "x" (type_var "T") (expr_var "x"))) \in
+  (context_contains Γ "x" (type_var "T")) ->
+  Γ |- (expr_ty_abs "T" (expr_tm_abs "x" (type_var "T") (expr_var "x"))) \is
     (type_univ "T" (type_fun (type_var "T") (type_var "T"))).
 Proof.
+  intros.
+  apply T_TypeAbs.
+  apply T_Abs.
+  apply H.
+  apply T_Var.
+  apply H.
+Admitted.
+
+Example typing2 :
+  ctx_empty |- (expr_ty_abs "T" (expr_tm_abs "x" (type_var "T") (expr_var "x"))) \is
+    (type_univ "T" (type_fun (type_var "T") (type_var "T"))).
+Proof.
+  apply T_TypeAbs.
+  apply T_Abs.
+  
 Admitted.
 
 End Typing.

paper/main.tex

@@ -176,11 +176,11 @@ $$\\$$
     \metavariable{x} \quad \valnonterm{\typevars}{\exprvars}
 }{Value Conactenation}
 
-%\otherform{
+\otherform{
-%	\exprterminal{\Lambda} \metavariable{\alpha} \quad
+	\exprterminal{\Lambda} \metavariable{\alpha} \quad
-%	\exprterminal{\mapsto} \quad
+	\exprterminal{\mapsto} \quad
-%	\valnonterm{ \typevars \cup \{ \metavariable{\alpha} \} }
+	\valnonterm{ \typevars \cup \{ \metavariable{\alpha} \} }
-%}{Type-Function Value}
+\{Type-Function Value}
 
 \otherform{
 	\exprterminal{\lambda} \metavariable{x} \quad
@@ -371,8 +371,8 @@ As usual, each rule is composed of premises (above the horizontal line) and a co
 	}
 
 	\inferrule[T-TypeApp]{
-		\Gamma \vdash \metavariable{e} : \metavariable{\tau} \\
+		\metavariable{\tau} \in \typenonterm{\typevars \cup \{\metavariable{\alpha}\}} \\
-		\metavariable{\tau} \in \typenonterm{\typevars \cup \metavariable{\alpha}} \\
+		\Gamma \vdash \metavariable{e} : \typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \metavariable{\tau} \\
 		\metavariable{\sigma} \in \typenonterm{\typevars}
 	}{
 		\Gamma \vdash ( \metavariable{e} \quad \metavariable{\sigma} ) : \{\metavariable{\alpha} \mapsto \metavariable{\sigma}\} \metavariable{\tau}
@@ -520,42 +520,66 @@ which are given in \ref{def:evalrules}.
 
 
 \begin{lemma}[\(\beta\)-reduction preserves \(\delta\)-normalform]
-Assume \metavariable{e} is in \(\delta\)-normalform and \(\metavariable{e} \rightarrow \metavariable{e'}\). Then \(\metavariable{e'}\) is in \(\delta\)-normalform as well.
+\label{lemma:preserve-delta-normalform}
+Assume \metavariable{e} is in \(\delta\)-normalform and \(\metavariable{e} \rightarrow_\beta \metavariable{e'}\). Then \(\metavariable{e'}\) is in \(\delta\)-normalform as well.
 \begin{proof}
 \todo{}
 \end{proof}
 \end{lemma}
 
+\begin{lemma}[\(\delta\)-normalform eliminates compatibility]
+\label{lemma:eliminate-compat}
+Assume \(\emptyset \vdash \metavariable{e} :\approx \metavariable{\tau}\) and \(\metavariable{e} \rightarrow_{\delta}^* \metavariable{e'}\) such that \(\metavariable{e'}\) is in \(\delta\)-normalform.
+Then \(\emptyset \vdash \metavariable{e'} : \metavariable{\tau}\) 
+
+\begin{proof}
+\end{proof}
+
+\end{lemma}
+
 \subsection{Proof of Syntactic Type Soundness}
 
+\begin{lemma}[\(\beta\)-Preservation]
+\label{lemma:beta-preservation}
+Assume the expression \(\metavariable{e}\) is \textbf{syntactically well-typed}, i.e. \(\emptyset \vdash \metavariable{e} : \metavariable{\tau}\) for some type \(\metavariable{\tau}\). Then forall \(\metavariable{e'}\) with \(\metavariable{e} \rightarrow_{\beta} \metavariable{e'}\) it holds that \(\emptyset \vdash \metavariable{e'} : \metavariable{\tau}\) as well.
+
+\begin{proof}
+\todo{}
+\end{proof}
+
+\end{lemma}
+
+\begin{lemma}[\(\delta\)-Preservation]
+\label{lemma:delta-preservation}
+
+\begin{proof}
+\todo{}
+\end{proof}
+\end{lemma}
+
+\begin{lemma}[Preservation]
+\label{lemma:preservation}
+Assume the expression \(\metavariable{e}\) is well typed, i.e. \(\emptyset \vdash \metavariable{e} : \metavariable{\tau}\) for some type \(\metavariable{\tau}\). Then forall \(\metavariable{e'}\) with \(\metavariable{e} \rightarrow_{eval} \metavariable{e'}\) it holds that \(\emptyset \vdash \metavariable{e'} : \metavariable{\tau}\) as well.
+
+\begin{proof}
+\todo{}
+\end{proof}
+\end{lemma}
+
 \begin{lemma}[Progress]
 \label{lemma:progress}
-
 If \(\emptyset \vdash \metavariable{e} : \metavariable{\tau}\), then either \(\metavariable{e}\) is a value or there exists some \(\metavariable{e'}\) such that \(\metavariable{e} \rightarrow_{eval} \metavariable{e'}\)
 
 \begin{proof}
 \todo{}
 \end{proof}
-
 \end{lemma}
 
-
+\begin{theorem}[Soundness]
-\begin{lemma}[Preservation]
+If \(\emptyset \vdash \metavariable{e}:\approx\metavariable{\tau}\), then it never occurs that \(\metavariable{e} \rightarrow_{eval}^{*} \metavariable{e'}\) where \metavariable{e'} is in normal form but not a value.
-\label{lemma:preservation}
-
-\begin{proof}
-\todo{}
-\end{proof}
-
-\end{lemma}
-
-
-
-
-\begin{theorem}[Type Soundness]
-If \(\emptyset \vdash \metavariable{e}:\metavariable{\tau}\), then it never occurs that \(\metavariable{e} \rightarrow_{eval}^{*} \metavariable{e'}\) where \metavariable{e'} is in normal form but not a value.
 
 \begin{proof}
+By \ref{lemma:}
 Follows from \ref{lemma:progress} and \ref{lemma:preservation}.
 \end{proof}
 \end{theorem}