+(**************************************************************************)
+(* ___ *)
+(* ||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 "Basic_2/grammar/thom.ma".
include "Basic_2/reducibility/tpr.ma".
(* CONTEXT-FREE WEAK HEAD NORMAL TERMS **************************************)
-definition twhnf: term → Prop ≝
- NF … tpr (thom …).
+definition twhnf: predicate term ≝ NF … tpr thom.
interpretation
"context-free weak head normality (term)"
'WHdNormal T = (twhnf T).
+(* Basic inversion lemmas ***************************************************)
+
+lemma twhnf_inv_thom: ∀T. 𝕎ℍℕ[T] → T ≈ T.
+normalize /2 width=1/
+qed-.
+
(* Basic properties *********************************************************)
+
+lemma tpr_thom: ∀T1,T2. T1 ⇒ T2 → T1 ≈ T1 → T1 ≈ T2.
+#T1 #T2 #H elim H -T1 -T2 //
+[ #I #V1 #V2 #T1 #T2 #_ #_ #_ #IHT12 #H
+ elim (thom_inv_flat1 … H) -H #W2 #U2 #HT1U2 #HT1 #_ #H1 #H2 destruct
+ lapply (IHT12 HT1U2) -IHT12 -HT1U2 #HUT2
+ lapply (simple_thom_repl_dx … HUT2 HT1) /2 width=1/
+| #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #H
+ elim (thom_inv_flat1 … H) -H #W2 #U2 #_ #H
+ elim (simple_inv_bind … H)
+| #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #_ #_ #H
+ elim (thom_inv_bind1 … H) -H #W2 #U2 #H destruct //
+| #V2 #V1 #V #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H
+ elim (thom_inv_flat1 … H) -H #U1 #U2 #_ #H
+ elim (simple_inv_bind … H)
+| #V #T #T1 #T2 #_ #_ #_ #H
+ elim (thom_inv_bind1 … H) -H #W2 #U2 #H destruct
+| #V #T1 #T2 #_ #_ #H
+ elim (thom_inv_flat1 … H) -H #W2 #U2 #_ #_ #_ #H destruct
+]
+qed.
+
+lemma twhnf_thom: ∀T. T ≈ T → 𝕎ℍℕ[T].
+/2 width=1/ qed.