--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "lambda/ext_lambda.ma".
+
+(* STRONGLY NORMALIZING TERMS *************************************************)
+
+(* SN(t) holds when t is strongly normalizing *)
+(* FG: we axiomatize it for now because we dont have reduction yet *)
+axiom SN: T → Prop.
+
+(* lists of strongly normalizing terms *)
+definition SNl ≝ all ? SN.
+
+(* saturation conditions ******************************************************)
+
+definition CR1 ≝ λ(P:?→Prop). ∀M. P M → SN M.
+
+definition SAT0 ≝ λ(P:?→Prop). ∀n,l. SNl l → P (Appl (Sort n) l).
+
+definition SAT1 ≝ λ(P:?->Prop). ∀i,l. SNl l → P (Appl (Rel i) l).
+
+definition SAT2 ≝ λ(P:?→Prop). ∀N,L,M,l. SN N → SN L →
+ P (Appl M[0:=L] l) → P (Appl (Lambda N M) (L::l)).
+
+definition SAT3 ≝ λ(P:?→Prop). ∀M,N,l. P (Appl (D (App M N)) l) →
+ P (Appl (D M) (N::l)).
+
+definition SAT4 ≝ λ(P:?→Prop). ∀M. P M → P (D M).
+
+lemma SAT0_sort: ∀P,n. SAT0 P → P (Sort n).
+#P #n #HP @(HP n (nil ?) …) //
+qed.
+
+lemma SAT1_rel: ∀P,i. SAT1 P → P (Rel i).
+#P #i #HP @(HP i (nil ?) …) //
+qed.
+
+lemma SAT3_1: ∀P,M,N. SAT3 P → P (D (App M N)) → P (App (D M) N).
+#P #M #N #HP #H @(HP … ([])) @H
+qed.
+
+(* axiomatization *************************************************************)
+
+axiom sn_sort: SAT0 SN.
+
+axiom sn_rel: SAT1 SN.
+
+axiom sn_beta: SAT2 SN.
+
+axiom sn_dapp: SAT3 SN.
+
+axiom sn_dummy: SAT4 SN.
+
+axiom sn_lambda: ∀N,M. SN N → SN M → SN (Lambda N M).
+
+axiom sn_prod: ∀N,M. SN N → SN M → SN (Prod N M).
+
+axiom sn_inv_app_1: ∀M,N. SN (App M N) → SN M.