(* *)
(**************************************************************************)
+include "basic_2/notation/relations/rdrop_4.ma".
include "basic_2/grammar/lenv_length.ma".
include "basic_2/grammar/lenv_weight.ma".
include "basic_2/relocation/lift.ma".
(* Basic_1: includes: drop_skip_bind *)
inductive ldrop: nat → nat → relation lenv ≝
-| ldrop_atom : ∀d,e. ldrop d e (⋆) (⋆)
+| ldrop_atom : ∀d. ldrop d 0 (⋆) (⋆)
| 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.
(* Basic inversion lemmas ***************************************************)
-fact ldrop_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 #_ #_ >commutative_plus normalize #H destruct
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-(* Basic_1: was: drop_gen_refl *)
-lemma ldrop_inv_refl: ∀L1,L2. ⇩[0, 0] L1 ≡ L2 → L1 = L2.
-/2 width=5/ qed-.
-
fact ldrop_inv_atom1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → L1 = ⋆ →
- L2 = ⋆.
+ L2 = ⋆ ∧ e = 0.
#d #e #L1 #L2 * -d -e -L1 -L2
-[ //
+[ /2 width=1/
| #L #I #V #H destruct
| #L1 #L2 #I #V #e #_ #H destruct
| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
]
-qed.
+qed-.
(* Basic_1: was: drop_gen_sort *)
-lemma ldrop_inv_atom1: ∀d,e,L2. ⇩[d, e] ⋆ ≡ L2 → L2 = ⋆.
-/2 width=5/ qed-.
+lemma ldrop_inv_atom1: ∀d,e,L2. ⇩[d, e] ⋆ ≡ L2 → L2 = ⋆ ∧ e = 0.
+/2 width=4 by ldrop_inv_atom1_aux/ qed-.
-fact ldrop_inv_O1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → d = 0 →
- ∀K,I,V. L1 = K. ⓑ{I} V →
- (e = 0 ∧ L2 = K. ⓑ{I} V) ∨
- (0 < e ∧ ⇩[d, e - 1] K ≡ L2).
+fact ldrop_inv_O1_pair1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → d = 0 →
+ ∀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
+[ #d #_ #K #I #V #H destruct
| #L #I #V #_ #K #J #W #HX destruct /3 width=1/
| #L1 #L2 #I #V #e #HL12 #_ #K #J #W #H destruct /3 width=1/
| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
]
-qed.
+qed-.
-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-.
+lemma ldrop_inv_O1_pair1: ∀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 by ldrop_inv_O1_pair1_aux/ qed-.
lemma ldrop_inv_pair1: ∀K,I,V,L2. ⇩[0, 0] K. ⓑ{I} V ≡ L2 → L2 = K. ⓑ{I} V.
#K #I #V #L2 #H
-elim (ldrop_inv_O1 … H) -H * // #H destruct
+elim (ldrop_inv_O1_pair1 … H) -H * // #H destruct
elim (lt_refl_false … H)
qed-.
lemma ldrop_inv_ldrop1: ∀e,K,I,V,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 (ldrop_inv_O1_pair1 … H) -H * // #H destruct
elim (lt_refl_false … He)
qed-.
⇧[d - 1, e] V2 ≡ V1 &
L2 = K2. ⓑ{I} V2.
#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #I #K #V #H destruct
+[ #d #_ #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 /2 width=5/
]
-qed.
+qed-.
(* 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. ⓑ{I} V2.
-/2 width=3/ qed-.
+/2 width=3 by ldrop_inv_skip1_aux/ qed-.
lemma ldrop_inv_O1_pair2: ∀I,K,V,e,L1. ⇩[0, e] L1 ≡ K. ⓑ{I} V →
(e = 0 ∧ L1 = K. ⓑ{I} V) ∨
∃∃I1,K1,V1. ⇩[0, e - 1] K1 ≡ K. ⓑ{I} V & L1 = K1.ⓑ{I1}V1 & 0 < e.
#I #K #V #e *
-[ #H lapply (ldrop_inv_atom1 … H) -H #H destruct
+[ #H elim (ldrop_inv_atom1 … H) -H #H destruct
| #L1 #I1 #V1 #H
- elim (ldrop_inv_O1 … H) -H *
+ elim (ldrop_inv_O1_pair1 … H) -H *
[ #H1 #H2 destruct /3 width=1/
| /3 width=5/
]
⇧[d - 1, e] V2 ≡ V1 &
L1 = K1. ⓑ{I} V1.
#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #I #K #V #H destruct
+[ #d #_ #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 /2 width=5/
]
-qed.
+qed-.
(* 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. ⓑ{I} V1.
-/2 width=3/ qed-.
+/2 width=3 by ldrop_inv_skip2_aux/ qed-.
(* Basic properties *********************************************************)
(* Basic_1: was by definition: drop_refl *)
-lemma ldrop_refl: ∀L. ⇩[0, 0] L ≡ L.
+lemma ldrop_refl: ∀L,d. ⇩[d, 0] L ≡ L.
#L elim L -L //
+#L #I #V #IHL #d @(nat_ind_plus … d) -d // /2 width=1/
qed.
lemma ldrop_ldrop_lt: ∀L1,L2,I,V,e.
#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV21 #Hd >(plus_minus_m_m d 1) // /2 width=1/
qed.
-lemma ldrop_O1_le: ∀i,L. i ≤ |L| → ∃K. ⇩[0, i] L ≡ K.
-#i @(nat_ind_plus … i) -i /2 width=2/
-#i #IHi *
+lemma ldrop_O1_le: ∀e,L. e ≤ |L| → ∃K. ⇩[0, e] L ≡ K.
+#e @(nat_ind_plus … e) -e /2 width=2/
+#e #IHe *
[ #H lapply (le_n_O_to_eq … H) -H >commutative_plus normalize #H destruct
| #L #I #V normalize #H
- elim (IHi L ?) -IHi /2 width=1/ -H /3 width=2/
+ elim (IHe L) -IHe /2 width=1/ -H /3 width=2/
]
qed.
-lemma ldrop_O1_lt: ∀L,i. i < |L| → ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V.
+lemma ldrop_O1_lt: ∀L,e. e < |L| → ∃∃I,K,V. ⇩[0, e] L ≡ K.ⓑ{I}V.
#L elim L -L
-[ #i #H elim (lt_zero_false … H)
-| #L #I #V #IHL #i @(nat_ind_plus … i) -i /2 width=4/
- #i #_ normalize #H
- elim (IHL i ? ) -IHL /2 width=1/ -H /3 width=4/
+[ #e #H elim (lt_zero_false … H)
+| #L #I #V #IHL #e @(nat_ind_plus … e) -e /2 width=4/
+ #e #_ normalize #H
+ elim (IHL e) -IHL /2 width=1/ -H /3 width=4/
]
qed.
(* Basic forvard lemmas *****************************************************)
-lemma ldrop_fwd_lw: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ♯{L2} ≤ ♯{L1}.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize
-[ /2 width=3/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12
- >(lift_fwd_tw … HV21) -HV21 /2 width=1/
-]
-qed-.
-
(* 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
+[ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H * #H destruct
| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #H
+ elim (ldrop_inv_O1_pair1 … H) -H * #He #H
[ -IHL1 destruct /2 width=1/
| @ldrop_ldrop >(plus_minus_m_m e 1) // /2 width=3/
]
]
qed-.
-lemma ldrop_fwd_length: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → |L2| ≤ |L1|.
+lemma ldrop_fwd_length: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → |L1| = |L2| + e.
#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize /2 width=1/
qed-.
-lemma ldrop_fwd_ldrop2_length: ∀L1,I2,K2,V2,e.
- ⇩[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
- elim (ldrop_inv_O1 … H) -H * #He #H
- [ -IHL1 destruct //
- | lapply (IHL1 … H) -IHL1 -H #HeK1 whd in ⊢ (? ? %); /2 width=1/
- ]
-]
+lemma ldrop_fwd_length_minus2: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → |L2| = |L1| - e.
+#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H /2 width=1/
qed-.
-lemma ldrop_fwd_O1_length: ∀L1,L2,e. ⇩[0, e] L1 ≡ L2 → |L2| = |L1| - e.
-#L1 elim L1 -L1
-[ #L2 #e #H >(ldrop_inv_atom1 … H) -H //
-| #K1 #I1 #V1 #IHL1 #L2 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #H
- [ -IHL1 destruct //
- | lapply (IHL1 … H) -IHL1 -H #H >H -H normalize
- >minus_le_minus_minus_comm //
- ]
-]
+lemma ldrop_fwd_length_minus4: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → e = |L1| - |L2|.
+#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H //
qed-.
-lemma ldrop_fwd_lw_eq: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 →
- |L1| = |L2| → ♯{L2} = ♯{L1}.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e //
-[ #L1 #L2 #I #V #e #HL12 #_
- lapply (ldrop_fwd_O1_length … HL12) -HL12 #HL21 >HL21 -HL21 normalize #H -I
- lapply (discr_plus_xy_minus_xz … H) -e #H destruct
-| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #HL12 normalize in ⊢ (??%%→??%%); #H -I
- >(lift_fwd_tw … HV21) -V2 /3 width=1 by eq_f2/ (**) (* auto is a bit slow without trace *)
+lemma ldrop_fwd_length_le2: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → e ≤ |L1|.
+#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H //
+qed-.
+
+lemma ldrop_fwd_length_le4: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → |L2| ≤ |L1|.
+#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H //
+qed-.
+
+lemma ldrop_fwd_length_lt2: ∀L1,I2,K2,V2,d,e.
+ ⇩[d, e] L1 ≡ K2. ⓑ{I2} V2 → e < |L1|.
+#L1 #I2 #K2 #V2 #d #e #H
+lapply (ldrop_fwd_length … H) normalize in ⊢ (%→?); -I2 -V2 //
+qed-.
+
+lemma ldrop_fwd_length_lt4: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → 0 < e → |L2| < |L1|.
+#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H /2 width=1/
+qed-.
+
+lemma ldrop_fwd_lw: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ♯{L2} ≤ ♯{L1}.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize
+[ /2 width=3/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12
+ >(lift_fwd_tw … HV21) -HV21 /2 width=1/
]
qed-.
-lemma ldrop_fwd_lw_lt: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 →
- |L2| < |L1| → ♯{L2} < ♯{L1}.
+lemma ldrop_fwd_lw_lt: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → 0 < e → ♯{L2} < ♯{L1}.
#L1 #L2 #d #e #H elim H -L1 -L2 -d -e //
[ #L #I #V #H elim (lt_refl_false … H)
| #L1 #L2 #I #V #e #HL12 #_ #_
lapply (ldrop_fwd_lw … HL12) -HL12 #HL12
@(le_to_lt_to_lt … HL12) -HL12 //
-| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12 normalize in ⊢ (?%%→?%%); #H -I
- >(lift_fwd_tw … HV21) -V2 /4 width=2 by lt_minus_to_plus, lt_plus_to_lt_l/ (**) (* auto too slow without trace *)
+| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12 #H normalize in ⊢ (?%%); -I
+ >(lift_fwd_tw … HV21) -V2 /3 by lt_minus_to_plus/ (**) (* auto too slow without trace *)
]
qed-.
+(* Advanced inversion lemmas ************************************************)
+
+fact ldrop_inv_O2_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → e = 0 → L1 = L2.
+#d #e #L1 #L2 #H elim H -d -e -L1 -L2
+[ //
+| //
+| #L1 #L2 #I #V #e #_ #_ >commutative_plus normalize #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12 #H
+ >(IHL12 H) -L1 >(lift_inv_O2_aux … HV21 … H) -V2 -d -e //
+]
+qed-.
+
+(* Basic_1: was: drop_gen_refl *)
+lemma ldrop_inv_O2: ∀L1,L2,d. ⇩[d, 0] L1 ≡ L2 → L1 = L2.
+/2 width=4 by ldrop_inv_O2_aux/ qed-.
+
+lemma ldrop_inv_length_eq: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → |L1| = |L2| → e = 0.
+#L1 #L2 #d #e #H #HL12 lapply (ldrop_fwd_length_minus4 … H) //
+qed-.
+
+lemma ldrop_inv_refl: ∀L,d,e. ⇩[d, e] L ≡ L → e = 0.
+/2 width=5 by ldrop_inv_length_eq/ qed-.
+
(* Basic_1: removed theorems 50:
drop_ctail drop_skip_flat
cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
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