(**************************************************************************)
include "ground_2/notation/functions/append_2.ma".
+include "basic_2/notation/functions/snbind2_3.ma".
+include "basic_2/notation/functions/snabbr_2.ma".
+include "basic_2/notation/functions/snabst_2.ma".
include "basic_2/grammar/lenv_length.ma".
(* LOCAL ENVIRONMENTS *******************************************************)
interpretation "append (local environment)" 'Append L1 L2 = (append L1 L2).
-definition l_appendable_sn: predicate (lenv→relation term) ≝ λR.
+interpretation "local environment tail binding construction (binary)"
+ 'SnBind2 I T L = (append (LPair LAtom I T) L).
+
+interpretation "tail abbreviation (local environment)"
+ 'SnAbbr T L = (append (LPair LAtom Abbr T) L).
+
+interpretation "tail abstraction (local environment)"
+ 'SnAbst L T = (append (LPair LAtom 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.
(* Basic properties *********************************************************)
+lemma append_atom: ∀L. L @@ ⋆ = L.
+// qed.
+
+lemma append_pair: ∀I,L,K,V. L @@ (K.ⓑ{I}V) = (L @@ K).ⓑ{I}V.
+// qed.
+
lemma append_atom_sn: ∀L. ⋆ @@ L = L.
-#L elim L -L normalize //
+#L elim L -L //
+#L #I #V >append_pair //
qed.
lemma append_assoc: associative … append.
-#L1 #L2 #L3 elim L3 -L3 normalize //
+#L1 #L2 #L3 elim L3 -L3 //
qed.
lemma append_length: ∀L1,L2. |L1 @@ L2| = |L1| + |L2|.
-#L1 #L2 elim L2 -L2 normalize //
+#L1 #L2 elim L2 -L2 //
+#L2 #I #V2 >append_pair >length_pair >length_pair //
+qed.
+
+lemma ltail_length: ∀I,L,V. |ⓑ{I}V.L| = ⫯|L|.
+#I #L #V >append_length //
qed.
+(* Basic_1: was just: chead_ctail *)
+lemma lpair_ltail: ∀L,I,V. ∃∃J,K,W. L.ⓑ{I}V = ⓑ{J}W.K & |L| = |K|.
+#L elim L -L /2 width=5 by ex2_3_intro/
+#L #Z #X #IHL #I #V elim (IHL Z X) -IHL
+#J #K #W #H #_ >H -H >ltail_length
+@(ex2_3_intro … J (K.ⓑ{I}V) W) /2 width=1 by length_pair/
+qed-.
+
(* Basic inversion lemmas ***************************************************)
lemma append_inj_sn: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |K1| = |K2| →
L1 = L2 ∧ K1 = K2.
#K1 elim K1 -K1
-[ * normalize /2 width=1/
- #K2 #I2 #V2 #L1 #L2 #_ <plus_n_Sm #H destruct
-| #K1 #I1 #V1 #IH * normalize
- [ #L1 #L2 #_ <plus_n_Sm #H destruct
+[ * /2 width=1 by conj/
+ #K2 #I2 #V2 #L1 #L2 #_ >length_atom >length_pair
+ #H destruct
+| #K1 #I1 #V1 #IH *
+ [ #L1 #L2 #_ >length_atom >length_pair
+ #H destruct
| #K2 #I2 #V2 #L1 #L2 #H1 #H2
- elim (destruct_lpair_lpair … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
- elim (IH … H1) -IH -H1 // -H2 /2 width=1/
+ elim (destruct_lpair_lpair_aux … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
+ elim (IH … H1) -IH -H1 /2 width=1 by conj/
]
]
qed-.
lemma append_inj_dx: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |L1| = |L2| →
L1 = L2 ∧ K1 = K2.
#K1 elim K1 -K1
-[ * normalize /2 width=1/
- #K2 #I2 #V2 #L1 #L2 #H1 #H2 destruct
- normalize in H2; >append_length in H2; #H
- elim (plus_xySz_x_false … H)
-| #K1 #I1 #V1 #IH * normalize
- [ #L1 #L2 #H1 #H2 destruct
- normalize in H2; >append_length in H2; #H
- elim (plus_xySz_x_false … (sym_eq … H))
- | #K2 #I2 #V2 #L1 #L2 #H1 #H2
- elim (destruct_lpair_lpair … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
- elim (IH … H1) -IH -H1 // -H2 /2 width=1/
+[ * /2 width=1 by conj/
+ #K2 #I2 #V2 #L1 #L2 >append_atom >append_pair #H destruct
+ >length_pair >append_length >plus_n_Sm
+ #H elim (plus_xSy_x_false … H)
+| #K1 #I1 #V1 #IH *
+ [ #L1 #L2 >append_pair >append_atom #H destruct
+ >length_pair >append_length >plus_n_Sm #H
+ lapply (discr_plus_x_xy … H) -H #H destruct
+ | #K2 #I2 #V2 #L1 #L2 >append_pair >append_pair #H1 #H2
+ elim (destruct_lpair_lpair_aux … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
+ elim (IH … H1) -IH -H1 /2 width=1 by conj/
]
]
qed-.
lemma append_inv_refl_dx: ∀L,K. L @@ K = L → K = ⋆.
-#L #K #H
-elim (append_inj_dx … (⋆) … H) //
+#L #K #H elim (append_inj_dx … (⋆) … H) //
qed-.
lemma append_inv_pair_dx: ∀I,L,K,V. L @@ K = L.ⓑ{I}V → K = ⋆.ⓑ{I}V.
-#I #L #K #V #H
-elim (append_inj_dx … (⋆.ⓑ{I}V) … H) //
+#I #L #K #V #H elim (append_inj_dx … (⋆.ⓑ{I}V) … H) //
qed-.
-lemma length_inv_pos_dx_append: ∀d,L. |L| = d + 1 →
- ∃∃I,K,V. |K| = d & L = ⋆.ⓑ{I}V @@ K.
-#d @(nat_ind_plus … d) -d
-[ #L #H
- elim (length_inv_pos_dx … H) -H #I #K #V #H
- >(length_inv_zero_dx … H) -H #H destruct
- @ex2_3_intro [4: /2 width=2/ |5: // |1,2,3: skip ] (**) (* /3/ does not work *)
-| #d #IHd #L #H
- elim (length_inv_pos_dx … H) -H #I #K #V #H
- elim (IHd … H) -IHd -H #I0 #K0 #V0 #H1 #H2 #H3 destruct
- @(ex2_3_intro … (K0.ⓑ{I}V)) //
-]
+lemma length_inv_pos_dx_ltail: ∀L,l. |L| = ⫯l →
+ ∃∃I,K,V. |K| = l & L = ⓑ{I}V.K.
+#Y #l #H elim (length_inv_succ_dx … H) -H #I #L #V #Hl #HLK destruct
+elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
qed-.
-(* Basic_eliminators ********************************************************)
-
-fact lenv_ind_dx_aux: ∀R:predicate lenv. R (⋆) →
- (∀I,L,V. R L → R (⋆.ⓑ{I}V @@ L)) →
- ∀d,L. |L| = d → R L.
-#R #Hatom #Hpair #d @(nat_ind_plus … d) -d
-[ #L #H >(length_inv_zero_dx … H) -H //
-| #d #IH #L #H
- elim (length_inv_pos_dx_append … H) -H #I #K #V #H1 #H2 destruct /3 width=1/
-]
+lemma length_inv_pos_sn_ltail: ∀L,l. ⫯l = |L| →
+ ∃∃I,K,V. l = |K| & L = ⓑ{I}V.K.
+#Y #l #H elim (length_inv_succ_sn … H) -H #I #L #V #Hl #HLK destruct
+elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
qed-.
-lemma lenv_ind_dx: ∀R:predicate lenv. R (⋆) →
- (∀I,L,V. R L → R (⋆.ⓑ{I}V @@ L)) →
- ∀L. R L.
-/3 width=2 by lenv_ind_dx_aux/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma length_inv_pos_sn_append: ∀d,L. 1 + d = |L| →
- ∃∃I,K,V. d = |K| & L = ⋆. ⓑ{I}V @@ K.
-#d >commutative_plus @(nat_ind_plus … d) -d
-[ #L #H elim (length_inv_pos_sn … H) -H #I #K #V #H1 #H2 destruct
- >(length_inv_zero_sn … H1) -K
- @(ex2_3_intro … (⋆)) // (**) (* explicit constructor *)
-| #d #IHd #L #H elim (length_inv_pos_sn … H) -H #I #K #V #H1 #H2 destruct
- >H1 in IHd; -H1 #IHd
- elim (IHd K) -IHd // #J #L #W #H1 #H2 destruct
- @(ex2_3_intro … (L.ⓑ{I}V)) // (**) (* explicit constructor *)
- >append_length /2 width=1/
-]
+(* Basic eliminators ********************************************************)
+
+(* Basic_1: was: c_tail_ind *)
+lemma lenv_ind_alt: ∀R:predicate lenv.
+ R (⋆) → (∀I,L,T. R L → R (ⓑ{I}T.L)) →
+ ∀L. R L.
+#R #IH1 #IH2 #L @(f_ind … length … L) -L #x #IHx * // -IH1
+#L #I #V #H destruct elim (lpair_ltail L I V) /4 width=1 by/
qed-.
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