(* Basic_1: was: ldrop_gen_refl *)
lemma ldrop_inv_refl: ∀L1,L2. ↓[0, 0] L1 ≡ L2 → L1 = L2.
-/2 width=5/ qed.
+/2 width=5/ qed-.
fact ldrop_inv_atom1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → L1 = ⋆ →
L2 = ⋆.
(* Basic_1: was: ldrop_gen_sort *)
lemma ldrop_inv_atom1: ∀d,e,L2. ↓[d, e] ⋆ ≡ L2 → L2 = ⋆.
-/2 width=5/ qed.
+/2 width=5/ qed-.
fact ldrop_inv_O1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 →
∀K,I,V. L1 = K. 𝕓{I} V →
lemma ldrop_inv_O1: ∀e,K,I,V,L2. ↓[0, e] K. 𝕓{I} V ≡ L2 →
(e = 0 ∧ L2 = K. 𝕓{I} V) ∨
(0 < e ∧ ↓[0, e - 1] K ≡ L2).
-/2/ qed.
+/2/ qed-.
(* Basic_1: was: ldrop_gen_ldrop *)
lemma ldrop_inv_ldrop1: ∀e,K,I,V,L2.
#e #K #I #V #L2 #H #He
elim (ldrop_inv_O1 … H) -H * // #H destruct -e;
elim (lt_refl_false … He)
-qed.
+qed-.
fact ldrop_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.
-/2/ qed.
+/2/ qed-.
fact ldrop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
lemma ldrop_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.
-/2/ qed.
+/2/ qed-.
(* Basic properties *********************************************************)
qed.
lemma ldrop_ldrop_lt: ∀L1,L2,I,V,e.
- ↓[0, e - 1] L1 ≡ L2 → 0 < e → ↓[0, e] L1. 𝕓{I} V ≡ L2.
+ ↓[0, e - 1] L1 ≡ L2 → 0 < e → ↓[0, e] L1. 𝕓{I} V ≡ L2.
#L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) /2/
qed.
| @ldrop_ldrop >(plus_minus_m_m e 1) /2/
]
]
-qed.
+qed-.
lemma ldrop_fwd_lw: ∀L1,L2,d,e. ↓[d, e] L1 ≡ L2 → #[L2] ≤ #[L1].
#L1 #L2 #d #e #H elim H -H L1 L2 d e // normalize
| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12
>(tw_lift … HV21) -HV21 /2/
]
-qed.
+qed-.
lemma ldrop_fwd_ldrop2_length: ∀L1,I2,K2,V2,e.
↓[0, e] L1 ≡ K2. 𝕓{I2} V2 → e < |L1|.
| lapply (IHL1 … H) -IHL1 H #HeK1 whd in ⊢ (? ? %) /2/
]
]
-qed.
+qed-.
lemma ldrop_fwd_O1_length: ∀L1,L2,e. ↓[0, e] L1 ≡ L2 → |L2| = |L1| - e.
#L1 elim L1 -L1
>minus_le_minus_minus_comm //
]
]
-qed.
+qed-.
(* Basic_1: removed theorems 49:
ldrop_skip_flat