(* *)
(**************************************************************************)
-include "ground_2/notation/functions/append_2.ma".
include "basic_2/notation/functions/snitem_2.ma".
include "basic_2/notation/functions/snbind1_2.ma".
include "basic_2/notation/functions/snbind2_3.ma".
| LBind K I ⇒ (append L K).ⓘ{I}
].
-interpretation "append (local environment)" 'Append L1 L2 = (append L1 L2).
+interpretation "append (local environment)" 'plus L1 L2 = (append L1 L2).
interpretation "local environment tail binding construction (generic)"
'SnItem I L = (append (LBind LAtom I) L).
'SnAbst L T = (append (LBind LAtom (BPair Abst T)) L).
definition d_appendable_sn: predicate (lenv→relation term) ≝ λR.
- ∀K,T1,T2. R K T1 T2 → ∀L. R (L@@K) T1 T2.
+ ∀K,T1,T2. R K T1 T2 → ∀L. R (L+K) T1 T2.
(* Basic properties *********************************************************)
-lemma append_atom: ∀L. L @@ ⋆ = L.
+lemma append_atom: ∀L. (L + ⋆) = L. (**) (* () should be redundant *)
// qed.
(* Basic_2A1: uses: append_pair *)
-lemma append_bind: ∀I,L,K. L@@(K.ⓘ{I}) = (L@@K).ⓘ{I}.
+lemma append_bind: ∀I,L,K. L+(K.ⓘ{I}) = (L+K).ⓘ{I}.
// qed.
-lemma append_atom_sn: ∀L. ⋆@@L = L.
+lemma append_atom_sn: ∀L. ⋆ + L = L.
#L elim L -L //
#L #I >append_bind //
qed.
#L1 #L2 #L3 elim L3 -L3 //
qed.
-lemma append_shift: ∀L,K,I. L@@(ⓘ{I}.K) = (L.ⓘ{I})@@K.
+lemma append_shift: ∀L,K,I. L+(ⓘ{I}.K) = (L.ⓘ{I})+K.
#L #K #I <append_assoc //
qed.
(* Basic inversion lemmas ***************************************************)
-lemma append_inv_atom3_sn: ∀L,K. ⋆ = L @@ K → ∧∧ ⋆ = L & ⋆ = K.
+lemma append_inv_atom3_sn: ∀L,K. ⋆ = L + K → ∧∧ ⋆ = L & ⋆ = K.
#L * /2 width=1 by conj/
#K #I >append_bind #H destruct
qed-.
-lemma append_inv_bind3_sn: ∀I0,L,L0,K. L0.ⓘ{I0} = L @@ K →
+lemma append_inv_bind3_sn: ∀I0,L,L0,K. L0.ⓘ{I0} = L + K →
∨∨ ∧∧ L0.ⓘ{I0} = L & ⋆ = K
- | ∃∃K0. K = K0.ⓘ{I0} & L0 = L @@ K0.
+ | ∃∃K0. K = K0.ⓘ{I0} & L0 = L + K0.
#I0 #L #L0 * /3 width=1 by or_introl, conj/
#K #I >append_bind #H destruct /3 width=3 by ex2_intro, or_intror/
qed-.
-lemma append_inj_sn: ∀K,L1,L2. L1@@K = L2@@K → L1 = L2.
+lemma append_inj_sn: ∀K,L1,L2. L1+K = L2+K → L1 = L2.
#K elim K -K //
#K #I #IH #L1 #L2 >append_bind #H
elim (destruct_lbind_lbind_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)