(* *)
(**************************************************************************)
+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).
+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 l_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_sn: ∀L. ⋆ @@ L = L.
#L1 #L2 elim L2 -L2 normalize //
qed.
+lemma ltail_length: ∀I,L,V. |ⓑ{I}V.L| = |L| + 1.
+#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) //
+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/
+[ * normalize /2 width=1 by conj/
#K2 #I2 #V2 #L1 #L2 #_ <plus_n_Sm #H destruct
| #K1 #I1 #V1 #IH * normalize
[ #L1 #L2 #_ <plus_n_Sm #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 (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/
+[ * normalize /2 width=1 by conj/
#K2 #I2 #V2 #L1 #L2 #H1 #H2 destruct
normalize in H2; >append_length in H2; #H
elim (plus_xySz_x_false … 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/
+ 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,d. |L| = d + 1 →
+ ∃∃I,K,V. |K| = d & L = ⓑ{I}V.K.
+#Y #d #H elim (length_inv_pos_dx … H) -H #I #L #V #Hd #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,d. d + 1 = |L| →
+ ∃∃I,K,V. d = |K| & L = ⓑ{I}V.K.
+#Y #d #H elim (length_inv_pos_sn … H) -H #I #L #V #Hd #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 #n #IHn * // -IH1
+#L #I #V normalize #H destruct elim (lpair_ltail L I V) /3 width=1 by/
qed-.
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