diff --git a/theories/fp_red.v b/theories/fp_red.v
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+Require Import ssreflect.
+From Hammer Require Import Tactics.
+Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax.
+
+(* Trying my best to not write C style module_funcname *)
+Module Par.
+  Inductive R {n} : Tm n -> Tm n ->  Prop :=
+  (***************** Beta ***********************)
+  | Var i : R (VarTm i) (VarTm i)
+  | AppAbs a0 a1 b0 b1 :
+    R a0 a1 ->
+    R b0 b1 ->
+    R (App (Abs a0) b0)  (subst_Tm (scons b1 VarTm) a1)
+  | AppPair a0 a1 b0 b1 c0 c1:
+    R a0 a1 ->
+    R b0 b1 ->
+    R c0 c1 ->
+    R (App (Pair a0 b0) c0) (Pair (App a1 c1) (App b1 c1))
+  | Proj1Abs a0 a1 :
+    R a0 a1 ->
+    R (Proj1 (Abs a0)) (Abs (Proj1 a0))
+  | Proj1Pair a0 a1 b :
+    R a0 a1 ->
+    R (Proj1 (Pair a0 b)) a1
+  | Proj2Abs a0 a1 :
+    R a0 a1 ->
+    R (Proj2 (Abs a0)) (Abs (Proj2 a0))
+  | Proj2Pair a0 a1 b :
+    R a0 a1 ->
+    R (Proj2 (Pair a0 b)) a1
+
+  (****************** Eta ***********************)
+  | AppEta a0 a1 :
+    R a0 a1 ->
+    R a0 (Abs (ren_Tm shift a1))
+  | PairEta a0 a1 :
+    R a0 a1 ->
+    R a0 (Pair a1 a1)
+
+  (*************** Congruence ********************)
+  | AppCong a0 a1 b0 b1 :
+    R a0 a1 ->
+    R b0 b1 ->
+    R (App a0 b0) (App a1 b1)
+  | Proj1Cong a0 a1 :
+    R a0 a1 ->
+    R (Proj1 a0) (Proj1 a1)
+  | Proj2Cong a0 a1 :
+    R a0 a1 ->
+    R (Proj2 a0) (Proj2 a1).
+End Par.
+
+(***************** Beta rules only ***********************)
+Module RPar.
+  Inductive R {n} : Tm n -> Tm n ->  Prop :=
+  (***************** Beta ***********************)
+  | Var i : R (VarTm i) (VarTm i)
+  | AppAbs a0 a1 b0 b1 :
+    R a0 a1 ->
+    R b0 b1 ->
+    R (App (Abs a0) b0)  (subst_Tm (scons b1 VarTm) a1)
+  | AppPair a0 a1 b0 b1 c0 c1:
+    R a0 a1 ->
+    R b0 b1 ->
+    R c0 c1 ->
+    R (App (Pair a0 b0) c0) (Pair (App a1 c1) (App b1 c1))
+  | Proj1Abs a0 a1 :
+    R a0 a1 ->
+    R (Proj1 (Abs a0)) (Abs (Proj1 a0))
+  | Proj1Pair a0 a1 b :
+    R a0 a1 ->
+    R (Proj1 (Pair a0 b)) a1
+  | Proj2Abs a0 a1 :
+    R a0 a1 ->
+    R (Proj2 (Abs a0)) (Abs (Proj2 a0))
+  | Proj2Pair a0 a1 b :
+    R a0 a1 ->
+    R (Proj2 (Pair a0 b)) a1
+
+  (*************** Congruence ********************)
+  | AppCong a0 a1 b0 b1 :
+    R a0 a1 ->
+    R b0 b1 ->
+    R (App a0 b0) (App a1 b1)
+  | Proj1Cong a0 a1 :
+    R a0 a1 ->
+    R (Proj1 a0) (Proj1 a1)
+  | Proj2Cong a0 a1 :
+    R a0 a1 ->
+    R (Proj2 a0) (Proj2 a1).
+End RPar.
+
+Module EPar.
+  Inductive R {n} : Tm n -> Tm n ->  Prop :=
+  (****************** Eta ***********************)
+  | AppEta a0 a1 :
+    R a0 a1 ->
+    R a0 (Abs (ren_Tm shift a1))
+  | PairEta a0 a1 :
+    R a0 a1 ->
+    R a0 (Pair a1 a1)
+
+  (*************** Congruence ********************)
+  | AppCong a0 a1 b0 b1 :
+    R a0 a1 ->
+    R b0 b1 ->
+    R (App a0 b0) (App a1 b1)
+  | Proj1Cong a0 a1 :
+    R a0 a1 ->
+    R (Proj1 a0) (Proj1 a1)
+  | Proj2Cong a0 a1 :
+    R a0 a1 ->
+    R (Proj2 a0) (Proj2 a1).
+End EPar.
+
+Lemma EPar_Par n (a b : Tm n) : EPar.R a b -> Par.R a b.
+Proof. induction 1; hauto lq:on ctrs:Par.R. Qed.
+
+Lemma EPar_