(* LOCAL ENVIRONMENT SLICING ************************************************)
-(* Basic_1: includes: ldrop_skip_bind *)
+(* Basic_1: includes: drop_skip_bind *)
inductive ldrop: nat → nat → relation lenv ≝
| ldrop_atom : ∀d,e. ldrop d e (⋆) (⋆)
-| ldrop_pair : ∀L,I,V. ldrop 0 0 (L. 𝕓{I} V) (L. 𝕓{I} V)
-| ldrop_ldrop: ∀L1,L2,I,V,e. ldrop 0 e L1 L2 → ldrop 0 (e + 1) (L1. 𝕓{I} V) L2
+| ldrop_pair : ∀L,I,V. ldrop 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
+| ldrop_ldrop: ∀L1,L2,I,V,e. ldrop 0 e L1 L2 → ldrop 0 (e + 1) (L1. ⓑ{I} V) L2
| ldrop_skip : ∀L1,L2,I,V1,V2,d,e.
ldrop d e L1 L2 → ⇧[d,e] V2 ≡ V1 →
- ldrop (d + 1) e (L1. 𝕓{I} V1) (L2. 𝕓{I} V2)
+ ldrop (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
.
interpretation "local slicing" 'RDrop d e L1 L2 = (ldrop d e L1 L2).
]
qed.
-(* Basic_1: was: ldrop_gen_refl *)
+(* Basic_1: was: drop_gen_refl *)
lemma ldrop_inv_refl: ∀L1,L2. ⇩[0, 0] L1 ≡ L2 → L1 = L2.
/2 width=5/ qed-.
]
qed.
-(* Basic_1: was: ldrop_gen_sort *)
+(* Basic_1: was: drop_gen_sort *)
lemma ldrop_inv_atom1: ∀d,e,L2. ⇩[d, e] ⋆ ≡ L2 → L2 = ⋆.
/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 →
- (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
+ ∀K,I,V. L1 = K. ⓑ{I} V →
+ (e = 0 ∧ L2 = K. ⓑ{I} V) ∨
(0 < e ∧ ⇩[d, e - 1] K ≡ L2).
#d #e #L1 #L2 * -d -e -L1 -L2
[ #d #e #_ #K #I #V #H destruct
]
qed.
-lemma ldrop_inv_O1: ∀e,K,I,V,L2. ⇩[0, e] K. 𝕓{I} V ≡ L2 →
- (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
+lemma ldrop_inv_O1: ∀e,K,I,V,L2. ⇩[0, e] K. ⓑ{I} V ≡ L2 →
+ (e = 0 ∧ L2 = K. ⓑ{I} V) ∨
(0 < e ∧ ⇩[0, e - 1] K ≡ L2).
/2 width=3/ qed-.
-(* Basic_1: was: ldrop_gen_ldrop *)
+(* Basic_1: was: drop_gen_drop *)
lemma ldrop_inv_ldrop1: ∀e,K,I,V,L2.
- ⇩[0, e] K. 𝕓{I} V ≡ L2 → 0 < e → ⇩[0, e - 1] K ≡ L2.
+ ⇩[0, e] K. ⓑ{I} V ≡ L2 → 0 < e → ⇩[0, e - 1] K ≡ L2.
#e #K #I #V #L2 #H #He
elim (ldrop_inv_O1 … H) -H * // #H destruct
elim (lt_refl_false … He)
qed-.
fact ldrop_inv_skip1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → 0 < d →
- ∀I,K1,V1. L1 = K1. 𝕓{I} V1 →
+ ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
∃∃K2,V2. ⇩[d - 1, e] K1 ≡ K2 &
⇧[d - 1, e] V2 ≡ V1 &
- L2 = K2. 𝕓{I} V2.
+ L2 = K2. ⓑ{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)
]
qed.
-(* Basic_1: was: ldrop_gen_skip_l *)
-lemma ldrop_inv_skip1: ∀d,e,I,K1,V1,L2. ⇩[d, e] K1. 𝕓{I} V1 ≡ L2 → 0 < d →
+(* Basic_1: was: drop_gen_skip_l *)
+lemma ldrop_inv_skip1: ∀d,e,I,K1,V1,L2. ⇩[d, e] K1. ⓑ{I} V1 ≡ L2 → 0 < d →
∃∃K2,V2. ⇩[d - 1, e] K1 ≡ K2 &
⇧[d - 1, e] V2 ≡ V1 &
- L2 = K2. 𝕓{I} V2.
+ L2 = K2. ⓑ{I} V2.
/2 width=3/ qed-.
fact ldrop_inv_skip2_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → 0 < d →
- ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
+ ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
∃∃K1,V1. ⇩[d - 1, e] K1 ≡ K2 &
⇧[d - 1, e] V2 ≡ V1 &
- L1 = K1. 𝕓{I} V1.
+ L1 = K1. ⓑ{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)
]
qed.
-(* Basic_1: was: ldrop_gen_skip_r *)
-lemma ldrop_inv_skip2: ∀d,e,I,L1,K2,V2. ⇩[d, e] L1 ≡ K2. 𝕓{I} V2 → 0 < d →
+(* Basic_1: was: drop_gen_skip_r *)
+lemma ldrop_inv_skip2: ∀d,e,I,L1,K2,V2. ⇩[d, e] L1 ≡ K2. ⓑ{I} V2 → 0 < d →
∃∃K1,V1. ⇩[d - 1, e] K1 ≡ K2 & ⇧[d - 1, e] V2 ≡ V1 &
- L1 = K1. 𝕓{I} V1.
+ L1 = K1. ⓑ{I} V1.
/2 width=3/ qed-.
(* Basic properties *********************************************************)
-(* Basic_1: was by definition: ldrop_refl *)
+(* Basic_1: was by definition: drop_refl *)
lemma ldrop_refl: ∀L. ⇩[0, 0] L ≡ L.
#L elim L -L //
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. ⓑ{I} V ≡ L2.
#L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/
qed.
lemma ldrop_lsubs_ldrop1_abbr: ∀L1,L2,d,e. L1 [d, e] ≼ L2 →
- ∀K1,V,i. ⇩[0, i] L1 ≡ K1. 𝕓{Abbr} V →
+ ∀K1,V,i. ⇩[0, i] L1 ≡ K1. ⓓV →
d ≤ i → i < d + e →
∃∃K2. K1 [0, d + e - i - 1] ≼ K2 &
- ⇩[0, i] L2 ≡ K2. 𝕓{Abbr} V.
+ ⇩[0, i] L2 ≡ K2. ⓓV.
#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
[ #d #e #K1 #V #i #H
lapply (ldrop_inv_atom1 … H) -H #H destruct
(* Basic forvard lemmas *****************************************************)
-(* Basic_1: was: ldrop_S *)
-lemma ldrop_fwd_ldrop2: ∀L1,I2,K2,V2,e. ⇩[O, e] L1 ≡ K2. 𝕓{I2} V2 →
+(* Basic_1: was: drop_S *)
+lemma ldrop_fwd_ldrop2: ∀L1,I2,K2,V2,e. ⇩[O, e] L1 ≡ K2. ⓑ{I2} V2 →
⇩[O, e + 1] L1 ≡ K2.
#L1 elim L1 -L1
[ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H #H destruct
qed-.
lemma ldrop_fwd_ldrop2_length: ∀L1,I2,K2,V2,e.
- ⇩[0, e] L1 ≡ K2. 𝕓{I2} V2 → e < |L1|.
+ ⇩[0, e] L1 ≡ K2. ⓑ{I2} V2 → e < |L1|.
#L1 elim L1 -L1
[ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H #H destruct
| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
qed-.
(* Basic_1: removed theorems 49:
- ldrop_skip_flat
+ drop_skip_flat
cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
- ldrop_clear ldrop_clear_O ldrop_clear_S
+ 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_ldrop getl_ldrop_conf_lt
- getl_ldrop_conf_ge getl_conf_ge_ldrop getl_ldrop_conf_rev
- ldrop_getl_trans_lt ldrop_getl_trans_le ldrop_getl_trans_ge
- getl_ldrop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O
+ 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
*)
projects / helm.git / blobdiff