Add unscoped syntax

theories/common.v

@@ -1,41 +1,52 @@
-Require Import Autosubst2.fintype Autosubst2.syntax Autosubst2.core ssreflect.
+Require Import Autosubst2.unscoped Autosubst2.syntax Autosubst2.core ssreflect.
 From Ltac2 Require Ltac2.
 Import Ltac2.Notations.
 Import Ltac2.Control.
 From Hammer Require Import Tactics.
 
-Definition renaming_ok {n m} (Γ : fin n -> PTm n) (Δ : fin m -> PTm m) (ξ : fin m -> fin n) :=
-  forall (i : fin m), ren_PTm ξ (Δ i) = Γ (ξ i).
+Inductive lookup : nat -> list PTm -> PTm -> Prop :=
+| here A Γ : lookup 0 (cons A Γ) (ren_PTm shift A)
+| there i Γ A B :
+  lookup i Γ A ->
+  lookup (S i) (cons B Γ) (ren_PTm shift A).
 
-Lemma up_injective n m (ξ : fin n -> fin m) :
-  (forall i j, ξ i = ξ j -> i = j) ->
-  forall i j, (upRen_PTm_PTm ξ)  i = (upRen_PTm_PTm ξ) j -> i = j.
+Definition renaming_ok (Γ : list PTm) (Δ : list PTm) (ξ : nat -> nat) :=
+  forall i A, lookup i Δ A -> lookup (ξ i) Γ (ren_PTm ξ A).
+
+Definition ren_inj (ξ : nat -> nat) := forall i j, ξ i = ξ j -> i = j.
+
+Lemma up_injective (ξ : nat -> nat) :
+  ren_inj ξ ->
+  ren_inj (upRen_PTm_PTm ξ).
 Proof.
-  sblast inv:option.
+  move => h i j.
+  case : i => //=; case : j => //=.
+  move => i j. rewrite /funcomp. hauto lq:on rew:off unfold:ren_inj.
 Qed.
 
 Local Ltac2 rec solve_anti_ren () :=
   let x := Fresh.in_goal (Option.get (Ident.of_string "x")) in
   intro $x;
   lazy_match! Constr.type (Control.hyp x) with
-  | fin _ -> _ _ => (ltac1:(case;hauto lq:on rew:off use:up_injective))
+  | nat -> nat => (ltac1:(case => *//=; qauto l:on use:up_injective unfold:ren_inj))
   | _ => solve_anti_ren ()
   end.
 
 Local Ltac solve_anti_ren := ltac2:(Control.enter solve_anti_ren).
 
-Lemma ren_injective n m (a b : PTm n) (ξ : fin n -> fin m) :
-  (forall i j, ξ i = ξ j -> i = j) ->
+Lemma ren_injective (a b : PTm) (ξ : nat -> nat) :
+  ren_inj ξ ->
   ren_PTm ξ a = ren_PTm ξ b ->
   a = b.
 Proof.
-  move : m ξ b. elim : n / a => //; try solve_anti_ren.
+  move : ξ b. elim : a => //; try solve_anti_ren.
+  move => p ihp ξ []//=. hauto lq:on inv:PTm, nat ctrs:- use:up_injective.
 Qed.
 
 Inductive HF : Set :=
 | H_Pair | H_Abs | H_Univ | H_Bind (p : BTag) | H_Nat | H_Suc | H_Zero | H_Bot.
 
-Definition ishf {n} (a : PTm n) :=
+Definition ishf (a : PTm) :=
   match a with
   | PPair _ _ => true
   | PAbs _ => true
@@ -47,7 +58,7 @@ Definition ishf {n} (a : PTm n) :=
   | _ => false
   end.
 
-Definition toHF {n} (a : PTm n) :=
+Definition toHF (a : PTm) :=
   match a with
   | PPair _ _ => H_Pair
   | PAbs _ => H_Abs
@@ -59,54 +70,53 @@ Definition toHF {n} (a : PTm n) :=
   | _ => H_Bot
   end.
 
-Fixpoint ishne {n} (a : PTm n) :=
+Fixpoint ishne (a : PTm) :=
   match a with
   | VarPTm _ => true
   | PApp a _ => ishne a
   | PProj _ a => ishne a
-  | PBot => true
   | PInd _ n _ _ => ishne n
   | _ => false
   end.
 
-Definition isbind {n} (a : PTm n) := if a is PBind _ _ _ then true else false.
+Definition isbind (a : PTm) := if a is PBind _ _ _ then true else false.
 
-Definition isuniv {n} (a : PTm n) := if a is PUniv _ then true else false.
+Definition isuniv (a : PTm) := if a is PUniv _ then true else false.
 
-Definition ispair {n} (a : PTm n) :=
+Definition ispair (a : PTm) :=
   match a with
   | PPair _ _ => true
   | _ => false
   end.
 
-Definition isnat {n} (a : PTm n) := if a is PNat then true else false.
+Definition isnat (a : PTm) := if a is PNat then true else false.
 
-Definition iszero {n} (a : PTm n) := if a is PZero then true else false.
+Definition iszero (a : PTm) := if a is PZero then true else false.
 
-Definition issuc {n} (a : PTm n) := if a is PSuc _ then true else false.
+Definition issuc (a : PTm) := if a is PSuc _ then true else false.
 
-Definition isabs {n} (a : PTm n) :=
+Definition isabs (a : PTm) :=
   match a with
   | PAbs _ => true
   | _ => false
   end.
 
-Definition ishf_ren n m (a : PTm n)  (ξ : fin n -> fin m) :
+Definition ishf_ren (a : PTm)  (ξ : nat -> nat) :
   ishf (ren_PTm ξ a) = ishf a.
 Proof. case : a => //=. Qed.
 
-Definition isabs_ren n m (a : PTm n)  (ξ : fin n -> fin m) :
+Definition isabs_ren (a : PTm)  (ξ : nat -> nat) :
   isabs (ren_PTm ξ a) = isabs a.
 Proof. case : a => //=. Qed.
 
-Definition ispair_ren n m (a : PTm n)  (ξ : fin n -> fin m) :
+Definition ispair_ren (a : PTm)  (ξ : nat -> nat) :
   ispair (ren_PTm ξ a) = ispair a.
 Proof. case : a => //=. Qed.
 
-Definition ishne_ren n m (a : PTm n)  (ξ : fin n -> fin m) :
+Definition ishne_ren (a : PTm)  (ξ : nat -> nat) :
   ishne (ren_PTm ξ a) = ishne a.
-Proof. move : m ξ. elim : n / a => //=. Qed.
+Proof. move : ξ. elim : a => //=. Qed.
 
-Lemma renaming_shift n m Γ (ρ : fin n -> PTm m) A :
-  renaming_ok (funcomp (ren_PTm shift) (scons (subst_PTm ρ A) Γ)) Γ shift.
-Proof. sfirstorder. Qed.
+Lemma renaming_shift Γ (ρ : nat -> PTm) A :
+  renaming_ok (cons A Γ) Γ shift.
+Proof. rewrite /renaming_ok. hauto lq:on ctrs:lookup. Qed.