(* *)
(**************************************************************************)
-include "Basic-2/substitution/leq.ma".
+include "Basic-2/grammar/leq.ma".
include "Basic-2/substitution/lift.ma".
(* 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)
(* Basic inversion lemmas ***************************************************)
-lemma drop_inv_refl_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
+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
[ //
| #L1 #L2 #I #V #_ #IHL12 #H1 #H2
]
qed.
+(* Basic-1: was: drop_gen_refl *)
lemma drop_inv_refl: ∀L1,L2. ↓[0, 0] L1 ≡ L2 → L1 = L2.
/2 width=5/ qed.
-lemma drop_inv_sort1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → L1 = ⋆ →
- L2 = ⋆.
+fact drop_inv_sort1_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
]
qed.
+(* Basic-1: was: drop_gen_sort *)
lemma drop_inv_sort1: ∀d,e,L2. ↓[d, e] ⋆ ≡ L2 → L2 = ⋆.
/2 width=5/ qed.
-lemma drop_inv_O1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 →
- ∀K,I,V. L1 = K. 𝕓{I} V →
- (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
- (0 < e ∧ ↓[d, e - 1] K ≡ L2).
+fact drop_inv_O1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 →
+ ∀K,I,V. L1 = K. 𝕓{I} V →
+ (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
+ (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
(0 < e ∧ ↓[0, e - 1] K ≡ L2).
/2/ qed.
+(* Basic-1: was: drop_gen_drop *)
lemma drop_inv_drop1: ∀e,K,I,V,L2.
↓[0, e] K. 𝕓{I} V ≡ L2 → 0 < e → ↓[0, e - 1] K ≡ L2.
#e #K #I #V #L2 #H #He
elim (lt_refl_false … He)
qed.
-lemma drop_inv_skip1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
- ∀I,K1,V1. L1 = K1. 𝕓{I} V1 →
- ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
- ↑[d - 1, e] V2 ≡ V1 &
- L2 = K2. 𝕓{I} V2.
+fact drop_inv_skip1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
+ ∀I,K1,V1. L1 = K1. 𝕓{I} V1 →
+ ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
+ ↑[d - 1, e] V2 ≡ V1 &
+ 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)
]
qed.
+(* Basic-1: was: drop_gen_skip_l *)
lemma drop_inv_skip1: ∀d,e,I,K1,V1,L2. ↓[d, e] K1. 𝕓{I} V1 ≡ L2 → 0 < d →
∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
↑[d - 1, e] V2 ≡ V1 &
L2 = K2. 𝕓{I} V2.
/2/ qed.
-lemma drop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
- ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
- ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 &
- ↑[d - 1, e] V2 ≡ V1 &
- L1 = K1. 𝕓{I} V1.
+fact drop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
+ ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
+ ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 &
+ ↑[d - 1, e] V2 ≡ V1 &
+ 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)
]
qed.
+(* Basic-1: was: drop_gen_skip_r *)
lemma drop_inv_skip2: ∀d,e,I,L1,K2,V2. ↓[d, e] L1 ≡ K2. 𝕓{I} V2 → 0 < d →
∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 & ↑[d - 1, e] V2 ≡ V1 &
L1 = K1. 𝕓{I} V1.
(* Basic properties *********************************************************)
+(* Basic-1: was by definition: drop_refl *)
lemma drop_refl: ∀L. ↓[0, 0] L ≡ L.
#L elim L -L /2/
qed.
#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
-| #L1 #L2 #I1 #I2 #V1 #V2 #_ #_ #I #K1 #V #i #_ #_ #H
+| #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
elim (drop_inv_O1 … H) -H * #Hi #HLK1
(* Basic forvard lemmas *****************************************************)
+(* Basic-1: was: drop_S *)
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
]
]
qed.
+
+(* Basic-1: removed theorems 18:
+ drop_skip_flat
+ cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
+ drop_clear drop_clear_O drop_clear_S
+ clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r
+ clear_gen_all clear_clear clear_mono clear_trans clear_ctail
+ clear_cle
+*)