Require Import Autosubst2.core Autosubst2.unscoped Autosubst2.syntax common. Reserved Notation "Γ ⊢ a ∈ A" (at level 70). Reserved Notation "Γ ⊢ a ≡ b ∈ A" (at level 70). Reserved Notation "Γ ⊢ A ≲ B" (at level 70). Reserved Notation "⊢ Γ" (at level 70). Inductive Wt : list PTm -> PTm -> PTm -> Prop := | T_Var i Γ A : ⊢ Γ -> lookup i Γ A -> Γ ⊢ VarPTm i ∈ A | T_Bind Γ i p (A : PTm) (B : PTm) : Γ ⊢ A ∈ PUniv i -> cons A Γ ⊢ B ∈ PUniv i -> Γ ⊢ PBind p A B ∈ PUniv i | T_Abs Γ (a : PTm) A B i : Γ ⊢ PBind PPi A B ∈ (PUniv i) -> (cons A Γ) ⊢ a ∈ B -> Γ ⊢ PAbs a ∈ PBind PPi A B | T_App Γ (b a : PTm) 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) 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) A B : Γ ⊢ a ∈ PBind PSig A B -> Γ ⊢ PProj PL a ∈ A | T_Proj2 Γ (a : PTm) A B : Γ ⊢ a ∈ PBind PSig A B -> Γ ⊢ PProj PR a ∈ subst_PTm (scons (PProj PL a) VarPTm) B | T_Univ Γ i : ⊢ Γ -> Γ ⊢ PUniv i ∈ PUniv (S i) | T_Nat Γ i : ⊢ Γ -> Γ ⊢ PNat ∈ PUniv i | T_Zero Γ : ⊢ Γ -> Γ ⊢ PZero ∈ PNat | T_Suc Γ (a : PTm) : Γ ⊢ a ∈ PNat -> Γ ⊢ PSuc a ∈ PNat | T_Ind Γ P (a : PTm) b c i : cons PNat Γ ⊢ P ∈ PUniv i -> Γ ⊢ a ∈ PNat -> Γ ⊢ b ∈ subst_PTm (scons PZero VarPTm) P -> (cons P (cons PNat Γ)) ⊢ c ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P) -> Γ ⊢ PInd P a b c ∈ subst_PTm (scons a VarPTm) P | T_Conv Γ (a : PTm) A B : Γ ⊢ a ∈ A -> Γ ⊢ A ≲ B -> Γ ⊢ a ∈ B with Eq : list PTm -> PTm -> PTm -> PTm -> Prop := (* Structural *) | E_Refl Γ (a : PTm ) A : Γ ⊢ a ∈ A -> Γ ⊢ a ≡ a ∈ A | E_Symmetric Γ (a b : PTm) A : Γ ⊢ a ≡ b ∈ A -> Γ ⊢ b ≡ a ∈ A | E_Transitive Γ (a b c : PTm) A : Γ ⊢ a ≡ b ∈ A -> Γ ⊢ b ≡ c ∈ A -> Γ ⊢ a ≡ c ∈ A (* Congruence *) | E_Bind Γ i p (A0 A1 : PTm) B0 B1 : Γ ⊢ A0 ∈ PUniv i -> Γ ⊢ A0 ≡ A1 ∈ PUniv i -> (cons A0 Γ) ⊢ B0 ≡ B1 ∈ PUniv i -> Γ ⊢ PBind p A0 B0 ≡ PBind p A1 B1 ∈ PUniv i | E_Abs Γ (a b : PTm) A B i : Γ ⊢ PBind PPi A B ∈ (PUniv i) -> (cons A Γ) ⊢ a ≡ b ∈ B -> Γ ⊢ PAbs a ≡ PAbs b ∈ PBind PPi A B | E_App Γ i (b0 b1 a0 a1 : PTm) A B : Γ ⊢ PBind PPi A B ∈ (PUniv i) -> Γ ⊢ b0 ≡ b1 ∈ PBind PPi A B -> Γ ⊢ a0 ≡ a1 ∈ A -> Γ ⊢ PApp b0 a0 ≡ PApp b1 a1 ∈ subst_PTm (scons a0 VarPTm) B | E_Pair Γ (a0 a1 b0 b1 : PTm) A B i : Γ ⊢ PBind PSig A B ∈ (PUniv i) -> Γ ⊢ a0 ≡ a1 ∈ A -> Γ ⊢ b0 ≡ b1 ∈ subst_PTm (scons a0 VarPTm) B -> Γ ⊢ PPair a0 b0 ≡ PPair a1 b1 ∈ PBind PSig A B | E_Proj1 Γ (a b : PTm) A B : Γ ⊢ a ≡ b ∈ PBind PSig A B -> Γ ⊢ PProj PL a ≡ PProj PL b ∈ A | E_Proj2 Γ i (a b : PTm) A B : Γ ⊢ PBind PSig A B ∈ (PUniv i) -> Γ ⊢ a ≡ b ∈ PBind PSig A B -> Γ ⊢ PProj PR a ≡ PProj PR b ∈ subst_PTm (scons (PProj PL a) VarPTm) B | E_IndCong Γ P0 P1 (a0 a1 : PTm) b0 b1 c0 c1 i : (cons PNat Γ) ⊢ P0 ∈ PUniv i -> (cons PNat Γ) ⊢ P0 ≡ P1 ∈ PUniv i -> Γ ⊢ a0 ≡ a1 ∈ PNat -> Γ ⊢ b0 ≡ b1 ∈ subst_PTm (scons PZero VarPTm) P0 -> (cons P0 ((cons PNat Γ))) ⊢ c0 ≡ c1 ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P0) -> Γ ⊢ PInd P0 a0 b0 c0 ≡ PInd P1 a1 b1 c1 ∈ subst_PTm (scons a0 VarPTm) P0 | E_SucCong Γ (a b : PTm) : Γ ⊢ a ≡ b ∈ PNat -> Γ ⊢ PSuc a ≡ PSuc b ∈ PNat | E_Conv Γ (a b : PTm) A B : Γ ⊢ a ≡ b ∈ A -> Γ ⊢ A ≲ B -> Γ ⊢ a ≡ b ∈ B (* Beta *) | E_AppAbs Γ (a : PTm) b A B i: Γ ⊢ PBind PPi A B ∈ PUniv i -> Γ ⊢ b ∈ A -> (cons A Γ) ⊢ a ∈ B -> Γ ⊢ PApp (PAbs a) b ≡ subst_PTm (scons b VarPTm) a ∈ subst_PTm (scons b VarPTm ) B | E_ProjPair1 Γ (a b : PTm) 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 Γ (a b : PTm) 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 | E_IndZero Γ P i (b : PTm) c : (cons PNat Γ) ⊢ P ∈ PUniv i -> Γ ⊢ b ∈ subst_PTm (scons PZero VarPTm) P -> (cons P (cons PNat Γ)) ⊢ c ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P) -> Γ ⊢ PInd P PZero b c ≡ b ∈ subst_PTm (scons PZero VarPTm) P | E_IndSuc Γ P (a : PTm) b c i : (cons PNat Γ) ⊢ P ∈ PUniv i -> Γ ⊢ a ∈ PNat -> Γ ⊢ b ∈ subst_PTm (scons PZero VarPTm) P -> (cons P (cons PNat Γ)) ⊢ c ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P) -> Γ ⊢ PInd P (PSuc a) b c ≡ (subst_PTm (scons (PInd P a b c) (scons a VarPTm)) c) ∈ subst_PTm (scons (PSuc a) VarPTm) P (* Eta *) | E_AppEta Γ (b : PTm) A B i : 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 Γ (a : PTm ) 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 : list PTm -> PTm -> PTm -> Prop := (* Structural *) | Su_Transitive Γ (A B C : PTm) : Γ ⊢ A ≲ B -> Γ ⊢ B ≲ C -> Γ ⊢ A ≲ C (* Congruence *) | Su_Univ Γ i j : ⊢ Γ -> i <= j -> Γ ⊢ PUniv i ≲ PUniv j | Su_Pi Γ (A0 A1 : PTm) B0 B1 i : Γ ⊢ A0 ∈ PUniv i -> Γ ⊢ A1 ≲ A0 -> (cons A0 Γ) ⊢ B0 ≲ B1 -> Γ ⊢ PBind PPi A0 B0 ≲ PBind PPi A1 B1 | Su_Sig Γ (A0 A1 : PTm) B0 B1 i : Γ ⊢ A1 ∈ PUniv i -> Γ ⊢ A0 ≲ A1 -> (cons A1 Γ) ⊢ B0 ≲ B1 -> Γ ⊢ PBind PSig A0 B0 ≲ PBind PSig A1 B1 (* Injecting from equalities *) | Su_Eq Γ (A : PTm) B i : Γ ⊢ A ≡ B ∈ PUniv i -> Γ ⊢ A ≲ B (* Projection axioms *) | Su_Pi_Proj1 Γ (A0 A1 : PTm) B0 B1 : Γ ⊢ PBind PPi A0 B0 ≲ PBind PPi A1 B1 -> Γ ⊢ A1 ≲ A0 | Su_Sig_Proj1 Γ (A0 A1 : PTm) B0 B1 : Γ ⊢ PBind PSig A0 B0 ≲ PBind PSig A1 B1 -> Γ ⊢ A0 ≲ A1 | Su_Pi_Proj2 Γ (a0 a1 A0 A1 : PTm ) B0 B1 : Γ ⊢ PBind PPi A0 B0 ≲ PBind PPi A1 B1 -> Γ ⊢ a0 ≡ a1 ∈ A1 -> Γ ⊢ subst_PTm (scons a0 VarPTm) B0 ≲ subst_PTm (scons a1 VarPTm) B1 | Su_Sig_Proj2 Γ (a0 a1 A0 A1 : PTm) B0 B1 : Γ ⊢ PBind PSig A0 B0 ≲ PBind PSig A1 B1 -> Γ ⊢ a0 ≡ a1 ∈ A0 -> Γ ⊢ subst_PTm (scons a0 VarPTm) B0 ≲ subst_PTm (scons a1 VarPTm) B1 with Wff : list PTm -> Prop := | Wff_Nil : ⊢ nil | Wff_Cons Γ (A : PTm) i : ⊢ Γ -> Γ ⊢ A ∈ PUniv i -> (* -------------------------------- *) ⊢ (cons A Γ) where "Γ ⊢ a ∈ A" := (Wt Γ a A) and "⊢ Γ" := (Wff Γ) and "Γ ⊢ a ≡ b ∈ A" := (Eq Γ a b A) and "Γ ⊢ A ≲ B" := (LEq Γ A B). Scheme wf_ind := Induction for Wff Sort Prop with wt_ind := Induction for Wt Sort Prop with eq_ind := Induction for Eq Sort Prop with le_ind := Induction for LEq Sort Prop. Combined Scheme wt_mutual from wf_ind, wt_ind, eq_ind, le_ind. (* Lemma lem : *) (* (forall n (Γ : fin n -> PTm n), ⊢ Γ -> ...) /\ *) (* (forall n Γ (a A : PTm n), Γ ⊢ a ∈ A -> ...) /\ *) (* (forall n Γ (a b A : PTm n), Γ ⊢ a ≡ b ∈ A -> ...) /\ *) (* (forall n Γ (A B : PTm n), Γ ⊢ A ≲ B -> ...). *) (* Proof. apply wt_mutual. ... *)