Add admissible simple rules

theories/common.v

@@ -1,3 +1,47 @@
-Require Import Autosubst2.fintype Autosubst2.syntax.
+Require Import Autosubst2.fintype Autosubst2.syntax 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).
+
+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.
+Proof.
+  sblast inv:option.
+Qed.
+
+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.
+Qed.