yiyunliu/sp-eta-postpone
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Add beta and eta rules to syntactic typing

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Yiyun Liu 2025-02-09 16:33:43 -05:00
parent 133968dd23
commit 5b925e3fa1
2 changed files with 42 additions and 3 deletions
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37
theories/typing.v
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@ -48,6 +48,7 @@ Inductive Wt : forall {n}, (fin n -> PTm n) -> PTm n -> PTm n -> Prop :=
Γ ⊢ a ∈ B
with Eq : forall {n}, (fin n -> PTm n) -> PTm n -> PTm n -> PTm n -> Prop :=
(* Structural *)
| E_Refl n Γ (a : PTm n) A :
Γ ⊢ a ∈ A ->
Γ ⊢ a ≡ a ∈ A
@ -61,6 +62,7 @@ with Eq : forall {n}, (fin n -> PTm n) -> PTm n -> PTm n -> PTm n -> Prop :=
Γ ⊢ b ≡ c ∈ A ->
Γ ⊢ a ≡ c ∈ A
(* Congruence *)
| E_Bind n Γ i p (A0 A1 : PTm n) B0 B1 :
⊢ Γ ->
Γ ⊢ A0 ∈ PUniv i ->
@ -99,12 +101,45 @@ with Eq : forall {n}, (fin n -> PTm n) -> PTm n -> PTm n -> PTm n -> Prop :=
Γ ⊢ A ≲ B ->
Γ ⊢ a ≡ b ∈ B
(* Beta *)
| E_AppAbs n Γ (a : PTm (S n)) b A B i:
Γ ⊢ PBind PPi A B ∈ PUniv i ->
Γ ⊢ b ∈ A ->
funcomp (ren_PTm shift) (scons A Γ) ⊢ a ∈ B ->
Γ ⊢ PApp (PAbs a) b ≡ subst_PTm (scons b VarPTm) a ∈ subst_PTm (scons b VarPTm ) B
| E_ProjPair1 n 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 ⊢ PProj PL (PPair a b) ≡ a ∈ A
| E_ProjPair2 n 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 ⊢ PProj PR (PPair a b) ≡ b ∈ subst_PTm (scons a VarPTm) B
(* Eta *)
| E_AppEta n Γ (b : PTm n) A B i :
⊢ Γ ->
Γ ⊢ PBind PPi A B ∈ (PUniv i) ->
Γ ⊢ b ∈ PBind PPi A B ->
Γ ⊢ PAbs (PApp (ren_PTm shift b) (VarPTm var_zero)) ≡ b ∈ PBind PPi A B
| E_PairEta n Γ (a : PTm n) A B i :
Γ ⊢ PBind PSig A B ∈ (PUniv i) ->
Γ ⊢ a ∈ PBind PSig A B ->
Γ ⊢ a ≡ PPair (PProj PL a) (PProj PR a) ∈ PBind PSig A B
with LEq : forall {n}, (fin n -> PTm n) -> PTm n -> PTm n -> Prop :=
(* Structural *)
| Su_Transitive n Γ (A B C : PTm n) :
Γ ⊢ A ≲ B ->
Γ ⊢ B ≲ C ->
Γ ⊢ A ≲ C
(* Congruence *)
| Su_Univ n Γ i j :
⊢ Γ ->
i <= j ->
@ -122,10 +157,12 @@ with LEq : forall {n}, (fin n -> PTm n) -> PTm n -> PTm n -> Prop :=
funcomp (ren_PTm shift) (scons A1 Γ) ⊢ B0 ≲ B1 ->
Γ ⊢ PBind PSig A0 B0 ≲ PBind PSig A1 B1
(* Injecting from equalities *)
| Su_Eq n Γ (A : PTm n) B i :
Γ ⊢ A ≡ B ∈ PUniv i ->
Γ ⊢ A ≲ B
(* Projection axioms *)
| Su_Pi_Proj1 n Γ (A0 A1 : PTm n) B0 B1 :
Γ ⊢ PBind PPi A0 B0 ≲ PBind PPi A1 B1 ->
Γ ⊢ A1 ≲ A0