Add nat type definition

theories/common.v

@@ -7,16 +7,6 @@ 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).
 
-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 q:on depth:2))
-  | _ => solve_anti_ren ()
-  end.
-
-Local Ltac solve_anti_ren := ltac2:(Control.enter solve_anti_ren).
-
 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.
@@ -24,26 +14,22 @@ Proof.
   sblast inv:option.
 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))
+  | _ => 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) ->
   ren_PTm ξ a = ren_PTm ξ b ->
   a = b.
 Proof.
-  move : m ξ b.
-  elim : n / a => //; try solve_anti_ren.
-
-  move => n a iha m ξ []//=.
-  move => u hξ [h].
-  apply iha in h. by subst.
-  destruct i, j=>//=.
-  hauto l:on.
-
-  move => n p A ihA B ihB m ξ []//=.
-  move => b A0 B0  hξ [?]. subst.
-  move => ?. have ? : A0 = A by firstorder. subst.
-  move => ?. have : B = B0. apply : ihB; eauto.
-  sauto.
-  congruence.
+  move : m ξ b. elim : n / a => //; try solve_anti_ren.
 Qed.
 
 Definition ishf {n} (a : PTm n) :=
@@ -52,6 +38,9 @@ Definition ishf {n} (a : PTm n) :=
   | PAbs _ => true
   | PUniv _ => true
   | PBind _ _ _ => true
+  | PNat => true
+  | PSuc _ => true
+  | PZero => true
   | _ => false
   end.
 
@@ -61,6 +50,7 @@ Fixpoint ishne {n} (a : PTm n) :=
   | PApp a _ => ishne a
   | PProj _ a => ishne a
   | PBot => true
+  | PInd _ n _ _ => ishne n
   | _ => false
   end.