Add some syntactic typing rules

theories/typing.v

@@ -0,0 +1,49 @@
+Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax.
+
+Reserved Notation "Γ ⊢ a ∈ A" (at level 70).
+Reserved Notation "Γ ⊢ a ≡ b ∈ A" (at level 70).
+Reserved Notation "⊢ Γ" (at level 70).
+Inductive Wt {n} : (fin n -> PTm n) -> PTm n -> PTm n -> Prop :=
+| T_Var Γ (i : fin n) :
+  ⊢ Γ ->
+  Γ ⊢ VarPTm i ∈ Γ i
+
+| T_Bind Γ i j p (A : PTm n) (B : PTm (S n)) :
+  Γ ⊢ A ∈ PUniv i ->
+  funcomp (ren_PTm shift) (scons A Γ) ⊢ B ∈ PUniv j ->
+  Γ ⊢ PBind p A B ∈ PUniv (max i j)
+
+| T_Abs Γ (a : PTm (S n)) A B i :
+  Γ ⊢ PBind PPi A B ∈ (PUniv i) ->
+  funcomp (ren_PTm shift) (scons A Γ) ⊢ a ∈ B ->
+  Γ ⊢ PAbs a ∈ PBind PPi A B
+
+| T_App Γ (b a : PTm n) A B :
+  Γ ⊢ b ∈ PBind PPi A B ->
+  Γ ⊢ a ∈ A ->
+  Γ ⊢ PApp b a ∈ subst_PTm (scons a VarPTm) B
+
+| T_Pair Γ (a b : PTm n) A B i :
+  Γ ⊢ PBind PSig A B ∈ (PUniv i) ->
+  Γ ⊢ a ∈ A ->
+  Γ ⊢ b ∈ subst_PTm (scons a VarPTm) B ->
+  Γ ⊢ PPair a b ∈ PBind PSig A B
+
+| T_Proj1 Γ (a : PTm n) A B :
+  Γ ⊢ a ∈ PBind PSig A B ->
+  Γ ⊢ PProj PL a ∈ A
+
+| T_Proj2 Γ (a : PTm n) A B :
+  Γ ⊢ a ∈ PBind PSig A B ->
+  Γ ⊢ PProj PR a ∈ subst_PTm (scons (PProj PL a) VarPTm) B
+
+| T_Conv Γ (a : PTm n) A B i :
+  Γ ⊢ a ∈ A ->
+  Γ ⊢ A ≡ B ∈ PUniv i ->
+  Γ ⊢ a ∈ B
+
+with Eq {n} : (fin n -> PTm n) -> PTm n -> PTm n -> PTm n -> Prop :=
+
+with Wff {n} : (fin n -> PTm n) -> Prop :=
+  where
+  "Γ ⊢ a ∈ A" := (Wt Γ a A) and "⊢ Γ" := (Wff Γ) and "Γ ⊢ a ≡ b ∈ A" := (Eq Γ a b A).