+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "Basic_2/grammar/lenv_weight.ma".
-include "Basic_2/grammar/lsubs.ma".
-include "Basic_2/substitution/lift.ma".
-
-(* DROPPING *****************************************************************)
-
-(* Basic_1: includes: drop_skip_bind *)
-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 d e L1 L2 → ↑[d,e] V2 ≡ V1 →
- drop (d + 1) e (L1. 𝕓{I} V1) (L2. 𝕓{I} V2)
-.
-
-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 * -d e L1 L2
-[ //
-| //
-| #L1 #L2 #I #V #e #_ #_ #H
- elim (plus_S_eq_O_false … H)
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H
- elim (plus_S_eq_O_false … H)
-]
-qed.
-
-(* Basic_1: was: drop_gen_refl *)
-lemma drop_inv_refl: ∀L1,L2. ↓[0, 0] L1 ≡ L2 → L1 = L2.
-/2 width=5/ qed.
-
-fact drop_inv_atom1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → L1 = ⋆ →
- L2 = ⋆.
-#d #e #L1 #L2 * -d e L1 L2
-[ //
-| #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_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 →
- ∀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
-| #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)
-]
-qed.
-
-lemma drop_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.
-
-(* 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 (drop_inv_O1 … H) -H * // #H destruct -e;
-elim (lt_refl_false … He)
-qed.
-
-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
-| #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/
-]
-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.
-
-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
-| #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/
-]
-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.
-/2/ qed.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was by definition: drop_refl *)
-lemma drop_refl: ∀L. ↓[0, 0] L ≡ L.
-#L elim L -L //
-qed.
-
-lemma drop_drop_lt: ∀L1,L2,I,V,e.
- ↓[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.
-
-lemma drop_lsubs_drop1_abbr: ∀L1,L2,d,e. L1 [d, e] ≼ L2 →
- ∀K1,V,i. ↓[0, i] L1 ≡ K1. 𝕓{Abbr} V →
- d ≤ i → i < d + e →
- ∃∃K2. K1 [0, d + e - i - 1] ≼ K2 &
- ↓[0, i] L2 ≡ K2. 𝕓{Abbr} V.
-#L1 #L2 #d #e #H elim H -H L1 L2 d e
-[ #d #e #K1 #V #i #H
- lapply (drop_inv_atom1 … H) -H #H destruct
-| #L1 #L2 #K1 #V #i #_ #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #V #e #HL12 #IHL12 #K1 #W #i #H #_ #Hie
- elim (drop_inv_O1 … H) -H * #Hi #HLK1
- [ -IHL12 Hie; destruct -i K1 W;
- <minus_n_O <minus_plus_m_m /2/
- | -HL12;
- elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 // [2: /2/ ] -Hie >arith_g1 // /3/
- ]
-| #L1 #L2 #I #V1 #V2 #e #_ #IHL12 #K1 #W #i #H #_ #Hie
- elim (drop_inv_O1 … H) -H * #Hi #HLK1
- [ -IHL12 Hie Hi; destruct
- | elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 // [2: /2/ ] -Hie >arith_g1 // /3/
- ]
-| #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #IHL12 #K1 #V #i #H #Hdi >plus_plus_comm_23 #Hide
- lapply (plus_S_le_to_pos … Hdi) #Hi
- lapply (drop_inv_drop1 … H ?) -H // #HLK1
- elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 [2: /2/ |3: /2/ ] -Hdi Hide >arith_g1 // /3/
-]
-qed.
-
-(* 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
-[ #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/
- | @drop_drop >(plus_minus_m_m e 1) /2/
- ]
-]
-qed.
-
-lemma drop_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
-[ /2/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12
- >(tw_lift … HV21) -HV21 /2/
-]
-qed.
-
-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_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 //
- | lapply (IHL1 … H) -IHL1 H #HeK1 whd in ⊢ (? ? %) /2/
- ]
-]
-qed.
-
-lemma drop_fwd_O1_length: ∀L1,L2,e. ↓[0, e] L1 ≡ L2 → |L2| = |L1| - e.
-#L1 elim L1 -L1
-[ #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 //
- | lapply (IHL1 … H) -IHL1 H #H >H -H; normalize
- >minus_le_minus_minus_comm //
- ]
-]
-qed.
-
-(* Basic_1: removed theorems 49:
- 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
- getl_ctail_clen getl_gen_tail clear_getl_trans getl_clear_trans
- getl_clear_bind getl_clear_conf getl_dec getl_drop getl_drop_conf_lt
- getl_drop_conf_ge getl_conf_ge_drop getl_drop_conf_rev
- drop_getl_trans_lt drop_getl_trans_le drop_getl_trans_ge
- getl_drop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O
- getl_gen_S getl_gen_2 getl_gen_flat getl_gen_bind getl_conf_le
- getl_trans getl_refl getl_head getl_flat getl_ctail getl_mono
-*)
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