(* DROPPING *****************************************************************)
(* Basic-1: includes: drop_skip_bind *)
-inductive drop: lenv → nat → nat → lenv → Prop ≝
-| drop_sort: ∀d,e. drop (⋆) d e (⋆)
-| drop_comp: ∀L1,L2,I,V. drop L1 0 0 L2 → drop (L1. 𝕓{I} V) 0 0 (L2. 𝕓{I} V)
-| drop_drop: ∀L1,L2,I,V,e. drop L1 0 e L2 → drop (L1. 𝕓{I} V) 0 (e + 1) L2
+inductive drop: nat → nat → relation lenv ≝
+| drop_atom: ∀d,e. drop d e (⋆) (⋆)
+| drop_pair: ∀L,I,V. drop 0 0 (L. 𝕓{I} V) (L. 𝕓{I} V)
+| drop_drop: ∀L1,L2,I,V,e. drop 0 e L1 L2 → drop 0 (e + 1) (L1. 𝕓{I} V) L2
| drop_skip: ∀L1,L2,I,V1,V2,d,e.
- drop L1 d e L2 → ↑[d,e] V2 ≡ V1 →
- drop (L1. 𝕓{I} V1) (d + 1) e (L2. 𝕓{I} V2)
+ drop d e L1 L2 → ↑[d,e] V2 ≡ V1 →
+ drop (d + 1) e (L1. 𝕓{I} V1) (L2. 𝕓{I} V2)
.
-interpretation "dropping" 'RDrop L1 d e L2 = (drop L1 d e L2).
+interpretation "dropping" 'RDrop d e L1 L2 = (drop d e L1 L2).
(* Basic inversion lemmas ***************************************************)
fact drop_inv_refl_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
-#d #e #L1 #L2 #H elim H -H d e L1 L2
+#d #e #L1 #L2 * -d e L1 L2
[ //
-| #L1 #L2 #I #V #_ #IHL12 #H1 #H2
- >(IHL12 H1 H2) -IHL12 H1 H2 L1 //
-| #L1 #L2 #I #V #e #_ #_ #_ #H
+| //
+| #L1 #L2 #I #V #e #_ #_ #H
elim (plus_S_eq_O_false … H)
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #_ #H
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H
elim (plus_S_eq_O_false … H)
]
qed.
lemma drop_inv_refl: ∀L1,L2. ↓[0, 0] L1 ≡ L2 → L1 = L2.
/2 width=5/ qed.
-fact drop_inv_sort1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → L1 = ⋆ →
+fact drop_inv_atom1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → L1 = ⋆ →
L2 = ⋆.
#d #e #L1 #L2 * -d e L1 L2
[ //
-| #L1 #L2 #I #V #_ #H destruct
+| #L #I #V #H destruct
| #L1 #L2 #I #V #e #_ #H destruct
| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
]
qed.
(* Basic-1: was: drop_gen_sort *)
-lemma drop_inv_sort1: ∀d,e,L2. ↓[d, e] ⋆ ≡ L2 → L2 = ⋆.
+lemma drop_inv_atom1: ∀d,e,L2. ↓[d, e] ⋆ ≡ L2 → L2 = ⋆.
/2 width=5/ qed.
fact drop_inv_O1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 →
(0 < e ∧ ↓[d, e - 1] K ≡ L2).
#d #e #L1 #L2 * -d e L1 L2
[ #d #e #_ #K #I #V #H destruct
-| #L1 #L2 #I #V #HL12 #H #K #J #W #HX destruct -L1 I V
- >(drop_inv_refl … HL12) -HL12 K /3/
+| #L #I #V #_ #K #J #W #HX destruct -L I V /3/
| #L1 #L2 #I #V #e #HL12 #_ #K #J #W #H destruct -L1 I V /3/
| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_false … H)
]
L2 = K2. 𝕓{I} V2.
#d #e #L1 #L2 * -d e L1 L2
[ #d #e #_ #I #K #V #H destruct
-| #L1 #L2 #I #V #_ #H elim (lt_refl_false … H)
+| #L #I #V #H elim (lt_refl_false … H)
| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
| #X #L2 #Y #Z #V2 #d #e #HL12 #HV12 #_ #I #L1 #V1 #H destruct -X Y Z
/2 width=5/
L1 = K1. 𝕓{I} V1.
#d #e #L1 #L2 * -d e L1 L2
[ #d #e #_ #I #K #V #H destruct
-| #L1 #L2 #I #V #_ #H elim (lt_refl_false … H)
+| #L #I #V #H elim (lt_refl_false … H)
| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
| #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #I #L2 #V2 #H destruct -X Y Z
/2 width=5/
(* Basic-1: was by definition: drop_refl *)
lemma drop_refl: ∀L. ↓[0, 0] L ≡ L.
-#L elim L -L /2/
+#L elim L -L //
qed.
lemma drop_drop_lt: ∀L1,L2,I,V,e.
↓[0, i] L2 ≡ K2. 𝕓{I} V.
#L1 #L2 #d #e #H elim H -H L1 L2 d e
[ #d #e #I #K1 #V #i #H
- lapply (drop_inv_sort1 … H) -H #H destruct
+ lapply (drop_inv_atom1 … H) -H #H destruct
| #L1 #L2 #I #K1 #V #i #_ #_ #H
elim (lt_zero_false … H)
| #L1 #L2 #I #V #e #HL12 #IHL12 #J #K1 #W #i #H #_ #Hie
lemma drop_fwd_drop2: ∀L1,I2,K2,V2,e. ↓[O, e] L1 ≡ K2. 𝕓{I2} V2 →
↓[O, e + 1] L1 ≡ K2.
#L1 elim L1 -L1
-[ #I2 #K2 #V2 #e #H lapply (drop_inv_sort1 … H) -H #H destruct
+[ #I2 #K2 #V2 #e #H lapply (drop_inv_atom1 … H) -H #H destruct
| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
elim (drop_inv_O1 … H) -H * #He #H
[ -IHL1; destruct -e K2 I2 V2 /2/
lemma drop_fwd_drop2_length: ∀L1,I2,K2,V2,e.
↓[0, e] L1 ≡ K2. 𝕓{I2} V2 → e < |L1|.
#L1 elim L1 -L1
-[ #I2 #K2 #V2 #e #H lapply (drop_inv_sort1 … H) -H #H destruct
+[ #I2 #K2 #V2 #e #H lapply (drop_inv_atom1 … H) -H #H destruct
| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
elim (drop_inv_O1 … H) -H * #He #H
[ -IHL1; destruct -e K2 I2 V2 //
lemma drop_fwd_O1_length: ∀L1,L2,e. ↓[0, e] L1 ≡ L2 → |L2| = |L1| - e.
#L1 elim L1 -L1
-[ #L2 #e #H >(drop_inv_sort1 … H) -H //
+[ #L2 #e #H >(drop_inv_atom1 … H) -H //
| #K1 #I1 #V1 #IHL1 #L2 #e #H
elim (drop_inv_O1 … H) -H * #He #H
[ -IHL1; destruct -e L2 //