(* *)
(**************************************************************************)
-include "basic_2/grammar/cl_weight.ma".
+include "ground_2/lib/bool.ma".
+include "ground_2/lib/lstar.ma".
+include "basic_2/notation/relations/rdrop_5.ma".
+include "basic_2/notation/relations/rdrop_4.ma".
+include "basic_2/notation/relations/rdrop_3.ma".
+include "basic_2/grammar/lenv_length.ma".
+include "basic_2/grammar/cl_restricted_weight.ma".
include "basic_2/substitution/lift.ma".
-include "basic_2/substitution/lsubr.ma".
-(* LOCAL ENVIRONMENT SLICING ************************************************)
+(* BASIC SLICING FOR LOCAL ENVIRONMENTS *************************************)
(* 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. ⓑ{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. ⓑ{I} V1) (L2. ⓑ{I} V2)
+inductive ldrop (s:bool): relation4 nat nat lenv lenv ≝
+| ldrop_atom: ∀d,e. (s = Ⓕ → e = 0) → ldrop s d e (⋆) (⋆)
+| ldrop_pair: ∀I,L,V. ldrop s 0 0 (L.ⓑ{I}V) (L.ⓑ{I}V)
+| ldrop_drop: ∀I,L1,L2,V,e. ldrop s 0 e L1 L2 → ldrop s 0 (e+1) (L1.ⓑ{I}V) L2
+| ldrop_skip: ∀I,L1,L2,V1,V2,d,e.
+ ldrop s d e L1 L2 → ⇧[d, e] V2 ≡ V1 →
+ ldrop s (d+1) e (L1.ⓑ{I}V1) (L2.ⓑ{I}V2)
.
-interpretation "local slicing" 'RDrop d e L1 L2 = (ldrop d e L1 L2).
+interpretation
+ "basic slicing (local environment) abstract"
+ 'RDrop s d e L1 L2 = (ldrop s d e L1 L2).
+(*
+interpretation
+ "basic slicing (local environment) general"
+ 'RDrop d e L1 L2 = (ldrop true d e L1 L2).
+*)
+interpretation
+ "basic slicing (local environment) lget"
+ 'RDrop e L1 L2 = (ldrop false O e L1 L2).
-definition l_liftable: (lenv → relation term) → Prop ≝
- λR. ∀K,T1,T2. R K T1 T2 → ∀L,d,e. ⇩[d, e] L ≡ K →
+definition l_liftable: predicate (lenv → relation term) ≝
+ λR. ∀K,T1,T2. R K T1 T2 → ∀L,s,d,e. ⇩[s, d, e] L ≡ K →
∀U1. ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 → R L U1 U2.
-definition l_deliftable_sn: (lenv → relation term) → Prop ≝
- λR. ∀L,U1,U2. R L U1 U2 → ∀K,d,e. ⇩[d, e] L ≡ K →
+definition l_deliftable_sn: predicate (lenv → relation term) ≝
+ λR. ∀L,U1,U2. R L U1 U2 → ∀K,s,d,e. ⇩[s, d, e] L ≡ K →
∀T1. ⇧[d, e] T1 ≡ U1 →
∃∃T2. ⇧[d, e] T2 ≡ U2 & R K T1 T2.
-definition dropable_sn: relation lenv → Prop ≝
- λR. ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀L2. R L1 L2 →
- ∃∃K2. R K1 K2 & ⇩[d, e] L2 ≡ K2.
-
-definition dedropable_sn: relation lenv → Prop ≝
- λR. ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀K2. R K1 K2 →
- ∃∃L2. R L1 L2 & ⇩[d, e] L2 ≡ K2.
+definition dropable_sn: predicate (relation lenv) ≝
+ λR. ∀L1,K1,s,d,e. ⇩[s, d, e] L1 ≡ K1 → ∀L2. R L1 L2 →
+ ∃∃K2. R K1 K2 & ⇩[s, d, e] L2 ≡ K2.
-definition dropable_dx: relation lenv → Prop ≝
- λR. ∀L1,L2. R L1 L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
- ∃∃K1. ⇩[0, e] L1 ≡ K1 & R K1 K2.
+definition dropable_dx: predicate (relation lenv) ≝
+ λR. ∀L1,L2. R L1 L2 → ∀K2,s,e. ⇩[s, 0, e] L2 ≡ K2 →
+ ∃∃K1. ⇩[s, 0, e] L1 ≡ K1 & R K1 K2.
(* 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 = ⋆.
-#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
+fact ldrop_inv_atom1_aux: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → L1 = ⋆ →
+ L2 = ⋆ ∧ (s = Ⓕ → e = 0).
+#L1 #L2 #s #d #e * -L1 -L2 -d -e
+[ /3 width=1 by conj/
+| #I #L #V #H destruct
+| #I #L1 #L2 #V #e #_ #H destruct
+| #I #L1 #L2 #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-.
-
-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).
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #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
+lemma ldrop_inv_atom1: ∀L2,s,d,e. ⇩[s, d, e] ⋆ ≡ L2 → L2 = ⋆ ∧ (s = Ⓕ → e = 0).
+/2 width=4 by ldrop_inv_atom1_aux/ qed-.
+
+fact ldrop_inv_O1_pair1_aux: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → d = 0 →
+ ∀K,I,V. L1 = K.ⓑ{I}V →
+ (e = 0 ∧ L2 = K.ⓑ{I}V) ∨
+ (0 < e ∧ ⇩[s, d, e-1] K ≡ L2).
+#L1 #L2 #s #d #e * -L1 -L2 -d -e
+[ #d #e #_ #_ #K #J #W #H destruct
+| #I #L #V #_ #K #J #W #HX destruct /3 width=1 by or_introl, conj/
+| #I #L1 #L2 #V #e #HL12 #_ #K #J #W #H destruct /3 width=1 by or_intror, conj/
+| #I #L1 #L2 #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: ∀I,K,L2,V,s,e. ⇩[s, 0, e] K. ⓑ{I} V ≡ L2 →
+ (e = 0 ∧ L2 = K.ⓑ{I}V) ∨
+ (0 < e ∧ ⇩[s, 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
+lemma ldrop_inv_pair1: ∀I,K,L2,V,s. ⇩[s, 0, 0] K.ⓑ{I}V ≡ L2 → L2 = K.ⓑ{I}V.
+#I #K #L2 #V #s #H
+elim (ldrop_inv_O1_pair1 … H) -H * // #H destruct
elim (lt_refl_false … H)
qed-.
(* Basic_1: was: drop_gen_drop *)
-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
+lemma ldrop_inv_drop1_lt: ∀I,K,L2,V,s,e.
+ ⇩[s, 0, e] K.ⓑ{I}V ≡ L2 → 0 < e → ⇩[s, 0, e-1] K ≡ L2.
+#I #K #L2 #V #s #e #H #He
+elim (ldrop_inv_O1_pair1 … 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. ⓑ{I} V1 →
- ∃∃K2,V2. ⇩[d - 1, e] K1 ≡ K2 &
- ⇧[d - 1, e] V2 ≡ V1 &
- 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)
-| #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/
+lemma ldrop_inv_drop1: ∀I,K,L2,V,s,e.
+ ⇩[s, 0, e+1] K.ⓑ{I}V ≡ L2 → ⇩[s, 0, e] K ≡ L2.
+#I #K #L2 #V #s #e #H lapply (ldrop_inv_drop1_lt … H ?) -H //
+qed-.
+
+fact ldrop_inv_skip1_aux: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → 0 < d →
+ ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⇩[s, d-1, e] K1 ≡ K2 &
+ ⇧[d-1, e] V2 ≡ V1 &
+ L2 = K2.ⓑ{I}V2.
+#L1 #L2 #s #d #e * -L1 -L2 -d -e
+[ #d #e #_ #_ #J #K1 #W1 #H destruct
+| #I #L #V #H elim (lt_refl_false … H)
+| #I #L1 #L2 #V #e #_ #H elim (lt_refl_false … H)
+| #I #L1 #L2 #V1 #V2 #d #e #HL12 #HV21 #_ #J #K1 #W1 #H destruct /2 width=5 by ex3_2_intro/
]
-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-.
-
-fact ldrop_inv_skip2_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → 0 < d →
- ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
- ∃∃K1,V1. ⇩[d - 1, e] K1 ≡ K2 &
- ⇧[d - 1, e] V2 ≡ 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)
-| #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/
+lemma ldrop_inv_skip1: ∀I,K1,V1,L2,s,d,e. ⇩[s, d, e] K1.ⓑ{I}V1 ≡ L2 → 0 < d →
+ ∃∃K2,V2. ⇩[s, d-1, e] K1 ≡ K2 &
+ ⇧[d-1, e] V2 ≡ V1 &
+ L2 = K2.ⓑ{I}V2.
+/2 width=3 by ldrop_inv_skip1_aux/ qed-.
+
+lemma ldrop_inv_O1_pair2: ∀I,K,V,s,e,L1. ⇩[s, 0, e] L1 ≡ K.ⓑ{I}V →
+ (e = 0 ∧ L1 = K.ⓑ{I}V) ∨
+ ∃∃I1,K1,V1. ⇩[s, 0, e-1] K1 ≡ K.ⓑ{I}V & L1 = K1.ⓑ{I1}V1 & 0 < e.
+#I #K #V #s #e *
+[ #H elim (ldrop_inv_atom1 … H) -H #H destruct
+| #L1 #I1 #V1 #H
+ elim (ldrop_inv_O1_pair1 … H) -H *
+ [ #H1 #H2 destruct /3 width=1 by or_introl, conj/
+ | /3 width=5 by ex3_3_intro, or_intror/
+ ]
]
-qed.
+qed-.
+
+fact ldrop_inv_skip2_aux: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → 0 < d →
+ ∀I,K2,V2. L2 = K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⇩[s, d-1, e] K1 ≡ K2 &
+ ⇧[d-1, e] V2 ≡ V1 &
+ L1 = K1.ⓑ{I}V1.
+#L1 #L2 #s #d #e * -L1 -L2 -d -e
+[ #d #e #_ #_ #J #K2 #W2 #H destruct
+| #I #L #V #H elim (lt_refl_false … H)
+| #I #L1 #L2 #V #e #_ #H elim (lt_refl_false … H)
+| #I #L1 #L2 #V1 #V2 #d #e #HL12 #HV21 #_ #J #K2 #W2 #H destruct /2 width=5 by ex3_2_intro/
+]
+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-.
+lemma ldrop_inv_skip2: ∀I,L1,K2,V2,s,d,e. ⇩[s, d, e] L1 ≡ K2.ⓑ{I}V2 → 0 < d →
+ ∃∃K1,V1. ⇩[s, d-1, e] K1 ≡ K2 & ⇧[d-1, e] V2 ≡ V1 &
+ L1 = K1.ⓑ{I}V1.
+/2 width=3 by ldrop_inv_skip2_aux/ qed-.
+
+lemma ldrop_inv_O1_gt: ∀L,K,e,s. ⇩[s, 0, e] L ≡ K → |L| < e →
+ s = Ⓣ ∧ K = ⋆.
+#L elim L -L [| #L #Z #X #IHL ] #K #e #s #H normalize in ⊢ (?%?→?); #H1e
+[ elim (ldrop_inv_atom1 … H) -H elim s -s /2 width=1 by conj/
+ #_ #Hs lapply (Hs ?) // -Hs #H destruct elim (lt_zero_false … H1e)
+| elim (ldrop_inv_O1_pair1 … H) -H * #H2e #HLK destruct
+ [ elim (lt_zero_false … H1e)
+ | elim (IHL … HLK) -IHL -HLK /2 width=1 by lt_plus_to_minus_r, conj/
+ ]
+]
+qed-.
(* Basic properties *********************************************************)
+lemma ldrop_refl_atom_O2: ∀s,d. ⇩[s, d, O] ⋆ ≡ ⋆.
+/2 width=1 by ldrop_atom/ qed.
+
(* Basic_1: was by definition: drop_refl *)
-lemma ldrop_refl: ∀L. ⇩[0, 0] L ≡ L.
+lemma ldrop_refl: ∀L,d,s. ⇩[s, d, 0] L ≡ L.
#L elim L -L //
+#L #I #V #IHL #d #s @(nat_ind_plus … d) -d /2 width=1 by ldrop_pair, ldrop_skip/
qed.
-lemma ldrop_ldrop_lt: ∀L1,L2,I,V,e.
- ⇩[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/
+lemma ldrop_drop_lt: ∀I,L1,L2,V,s,e.
+ ⇩[s, 0, e-1] L1 ≡ L2 → 0 < e → ⇩[s, 0, e] L1.ⓑ{I}V ≡ L2.
+#I #L1 #L2 #V #s #e #HL12 #He >(plus_minus_m_m e 1) /2 width=1 by ldrop_drop/
qed.
-lemma ldrop_skip_lt: ∀L1,L2,I,V1,V2,d,e.
- ⇩[d - 1, e] L1 ≡ L2 → ⇧[d - 1, e] V2 ≡ V1 → 0 < d →
- ⇩[d, e] L1. ⓑ{I} V1 ≡ L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV21 #Hd >(plus_minus_m_m d 1) // /2 width=1/
+lemma ldrop_skip_lt: ∀I,L1,L2,V1,V2,s,d,e.
+ ⇩[s, d-1, e] L1 ≡ L2 → ⇧[d-1, e] V2 ≡ V1 → 0 < d →
+ ⇩[s, d, e] L1. ⓑ{I} V1 ≡ L2.ⓑ{I}V2.
+#I #L1 #L2 #V1 #V2 #s #d #e #HL12 #HV21 #Hd >(plus_minus_m_m d 1) /2 width=1 by ldrop_skip/
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 *
-[ #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/
+lemma ldrop_O1_le: ∀s,e,L. e ≤ |L| → ∃K. ⇩[s, 0, e] L ≡ K.
+#s #e @(nat_ind_plus … e) -e /2 width=2 by ex_intro/
+#e #IHe *
+[ #H elim (le_plus_xSy_O_false … H)
+| #L #I #V normalize #H elim (IHe L) -IHe /3 width=2 by ldrop_drop, monotonic_pred, ex_intro/
]
-qed.
+qed-.
-lemma ldrop_O1_lt: ∀L,i. i < |L| → ∃∃I,K,V. ⇩[0, i] 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/
+lemma ldrop_O1_lt: ∀s,L,e. e < |L| → ∃∃I,K,V. ⇩[s, 0, e] L ≡ K.ⓑ{I}V.
+#s #L elim L -L
+[ #e #H elim (lt_zero_false … H)
+| #L #I #V #IHL #e @(nat_ind_plus … e) -e /2 width=4 by ldrop_pair, ex1_3_intro/
+ #e #_ normalize #H elim (IHL e) -IHL /3 width=4 by ldrop_drop, lt_plus_to_minus_r, lt_plus_to_lt_l, ex1_3_intro/
]
+qed-.
+
+lemma ldrop_O1_pair: ∀L,K,e,s. ⇩[s, 0, e] L ≡ K → e ≤ |L| → ∀I,V.
+ ∃∃J,W. ⇩[s, 0, e] L.ⓑ{I}V ≡ K.ⓑ{J}W.
+#L elim L -L [| #L #Z #X #IHL ] #K #e #s #H normalize #He #I #V
+[ elim (ldrop_inv_atom1 … H) -H #H <(le_n_O_to_eq … He) -e
+ #Hs destruct /2 width=3 by ex1_2_intro/
+| elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK destruct /2 width=3 by ex1_2_intro/
+ elim (IHL … HLK … Z X) -IHL -HLK
+ /3 width=3 by ldrop_drop_lt, le_plus_to_minus, ex1_2_intro/
+]
+qed-.
+
+lemma ldrop_O1_ge: ∀L,e. |L| ≤ e → ⇩[Ⓣ, 0, e] L ≡ ⋆.
+#L elim L -L [ #e #_ @ldrop_atom #H destruct ]
+#L #I #V #IHL #e @(nat_ind_plus … e) -e [ #H elim (le_plus_xSy_O_false … H) ]
+normalize /4 width=1 by ldrop_drop, monotonic_pred/
qed.
-lemma ldrop_lsubr_ldrop2_abbr: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
- ∀K2,V,i. ⇩[0, i] L2 ≡ K2. ⓓV →
- d ≤ i → i < d + e →
- ∃∃K1. K1 ⊑ [0, d + e - i - 1] K2 &
- ⇩[0, i] L1 ≡ K1. ⓓ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
-| #L1 #L2 #K1 #V #i #_ #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #V #e #HL12 #IHL12 #K1 #W #i #H #_ #Hie
- elim (ldrop_inv_O1 … H) -H * #Hi #HLK1
- [ -IHL12 -Hie destruct
- <minus_n_O <minus_plus_m_m // /2 width=3/
- | -HL12
- elim (IHL12 … HLK1 ? ?) -IHL12 -HLK1 // /2 width=1/ -Hie >minus_minus_comm >arith_b1 // /4 width=3/
- ]
-| #L1 #L2 #I #V1 #V2 #e #_ #IHL12 #K1 #W #i #H #_ #Hie
- elim (ldrop_inv_O1 … H) -H * #Hi #HLK1
- [ -IHL12 -Hie -Hi destruct
- | elim (IHL12 … HLK1 ? ?) -IHL12 -HLK1 // /2 width=1/ -Hie >minus_minus_comm >arith_b1 // /3 width=3/
+lemma ldrop_O1_eq: ∀L,s. ⇩[s, 0, |L|] L ≡ ⋆.
+#L elim L -L /2 width=1 by ldrop_drop, ldrop_atom/
+qed.
+
+lemma ldrop_split: ∀L1,L2,d,e2,s. ⇩[s, d, e2] L1 ≡ L2 → ∀e1. e1 ≤ e2 →
+ ∃∃L. ⇩[s, d, e2 - e1] L1 ≡ L & ⇩[s, d, e1] L ≡ L2.
+#L1 #L2 #d #e2 #s #H elim H -L1 -L2 -d -e2
+[ #d #e2 #Hs #e1 #He12 @(ex2_intro … (⋆))
+ @ldrop_atom #H lapply (Hs H) -s #H destruct /2 width=1 by le_n_O_to_eq/
+| #I #L1 #V #e1 #He1 lapply (le_n_O_to_eq … He1) -He1
+ #H destruct /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #e2 #HL12 #IHL12 #e1 @(nat_ind_plus … e1) -e1
+ [ /3 width=3 by ldrop_drop, ex2_intro/
+ | -HL12 #e1 #_ #He12 lapply (le_plus_to_le_r … He12) -He12
+ #He12 elim (IHL12 … He12) -IHL12 >minus_plus_plus_l
+ #L #HL1 #HL2 elim (lt_or_ge (|L1|) (e2-e1)) #H0
+ [ elim (ldrop_inv_O1_gt … HL1 H0) -HL1 #H1 #H2 destruct
+ elim (ldrop_inv_atom1 … HL2) -HL2 #H #_ destruct
+ @(ex2_intro … (⋆)) [ @ldrop_O1_ge normalize // ]
+ @ldrop_atom #H destruct
+ | elim (ldrop_O1_pair … HL1 H0 I V) -HL1 -H0 /3 width=5 by ldrop_drop, ex2_intro/
+ ]
]
-| #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #IHL12 #K1 #V #i #H #Hdi >plus_plus_comm_23 #Hide
- elim (le_inv_plus_l … Hdi) #Hdim #Hi
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HLK1
- elim (IHL12 … HLK1 ? ?) -IHL12 -HLK1 // /2 width=1/ -Hdi -Hide >minus_minus_comm >arith_b1 // /3 width=3/
+| #I #L1 #L2 #V1 #V2 #d #e2 #_ #HV21 #IHL12 #e1 #He12 elim (IHL12 … He12) -IHL12
+ #L #HL1 #HL2 elim (lift_split … HV21 d e1) -HV21 /3 width=5 by ldrop_skip, ex2_intro/
]
+qed-.
+
+lemma ldrop_FT: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → ⇩[Ⓣ, d, e] L1 ≡ L2.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+/3 width=1 by ldrop_atom, ldrop_drop, ldrop_skip/
qed.
-lemma dropable_sn_TC: ∀R. dropable_sn R → dropable_sn (TC … R).
-#R #HR #L1 #K1 #d #e #HLK1 #L2 #H elim H -L2
-[ #L2 #HL12
- elim (HR … HLK1 … HL12) -HR -L1 /3 width=3/
-| #L #L2 #_ #HL2 * #K #HK1 #HLK
- elim (HR … HLK … HL2) -HR -L /3 width=3/
+lemma ldrop_gen: ∀L1,L2,s,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → ⇩[s, d, e] L1 ≡ L2.
+#L1 #L2 * /2 width=1 by ldrop_FT/
+qed-.
+
+lemma ldrop_T: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → ⇩[Ⓣ, d, e] L1 ≡ L2.
+#L1 #L2 * /2 width=1 by ldrop_FT/
+qed-.
+
+lemma l_liftable_LTC: ∀R. l_liftable R → l_liftable (LTC … R).
+#R #HR #K #T1 #T2 #H elim H -T2
+[ /3 width=10 by inj/
+| #T #T2 #_ #HT2 #IHT1 #L #s #d #e #HLK #U1 #HTU1 #U2 #HTU2
+ elim (lift_total T d e) /4 width=12 by step/
]
-qed.
+qed-.
-lemma dedropable_sn_TC: ∀R. dedropable_sn R → dedropable_sn (TC … R).
-#R #HR #L1 #K1 #d #e #HLK1 #K2 #H elim H -K2
-[ #K2 #HK12
- elim (HR … HLK1 … HK12) -HR -K1 /3 width=3/
-| #K #K2 #_ #HK2 * #L #HL1 #HLK
- elim (HR … HLK … HK2) -HR -K /3 width=3/
+lemma l_deliftable_sn_LTC: ∀R. l_deliftable_sn R → l_deliftable_sn (LTC … R).
+#R #HR #L #U1 #U2 #H elim H -U2
+[ #U2 #HU12 #K #s #d #e #HLK #T1 #HTU1
+ elim (HR … HU12 … HLK … HTU1) -HR -L -U1 /3 width=3 by inj, ex2_intro/
+| #U #U2 #_ #HU2 #IHU1 #K #s #d #e #HLK #T1 #HTU1
+ elim (IHU1 … HLK … HTU1) -IHU1 -U1 #T #HTU #HT1
+ elim (HR … HU2 … HLK … HTU) -HR -L -U /3 width=5 by step, ex2_intro/
]
-qed.
+qed-.
+
+lemma dropable_sn_TC: ∀R. dropable_sn R → dropable_sn (TC … R).
+#R #HR #L1 #K1 #s #d #e #HLK1 #L2 #H elim H -L2
+[ #L2 #HL12 elim (HR … HLK1 … HL12) -HR -L1
+ /3 width=3 by inj, ex2_intro/
+| #L #L2 #_ #HL2 * #K #HK1 #HLK elim (HR … HLK … HL2) -HR -L
+ /3 width=3 by step, ex2_intro/
+]
+qed-.
lemma dropable_dx_TC: ∀R. dropable_dx R → dropable_dx (TC … R).
#R #HR #L1 #L2 #H elim H -L2
-[ #L2 #HL12 #K2 #e #HLK2
- elim (HR … HL12 … HLK2) -HR -L2 /3 width=3/
-| #L #L2 #_ #HL2 #IHL1 #K2 #e #HLK2
- elim (HR … HL2 … HLK2) -HR -L2 #K #HLK #HK2
- elim (IHL1 … HLK) -L /3 width=5/
+[ #L2 #HL12 #K2 #s #e #HLK2 elim (HR … HL12 … HLK2) -HR -L2
+ /3 width=3 by inj, ex2_intro/
+| #L #L2 #_ #HL2 #IHL1 #K2 #s #e #HLK2 elim (HR … HL2 … HLK2) -HR -L2
+ #K #HLK #HK2 elim (IHL1 … HLK) -L
+ /3 width=5 by step, ex2_intro/
]
-qed.
+qed-.
+
+lemma l_deliftable_sn_llstar: ∀R. l_deliftable_sn R →
+ ∀l. l_deliftable_sn (llstar … R l).
+#R #HR #l #L #U1 #U2 #H @(lstar_ind_r … l U2 H) -l -U2
+[ /2 width=3 by lstar_O, ex2_intro/
+| #l #U #U2 #_ #HU2 #IHU1 #K #s #d #e #HLK #T1 #HTU1
+ elim (IHU1 … HLK … HTU1) -IHU1 -U1 #T #HTU #HT1
+ elim (HR … HU2 … HLK … HTU) -HR -L -U /3 width=5 by lstar_dx, ex2_intro/
+]
+qed-.
(* Basic forvard lemmas *****************************************************)
(* 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.
+lemma ldrop_fwd_drop2: ∀L1,I2,K2,V2,s,e. ⇩[s, O, e] L1 ≡ K2. ⓑ{I2} V2 →
+ ⇩[s, O, e + 1] L1 ≡ K2.
#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 /2 width=1/
- | @ldrop_ldrop >(plus_minus_m_m e 1) // /2 width=3/
+[ #I2 #K2 #V2 #s #e #H lapply (ldrop_inv_atom1 … H) -H * #H destruct
+| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #s #e #H
+ elim (ldrop_inv_O1_pair1 … H) -H * #He #H
+ [ -IHL1 destruct /2 width=1 by ldrop_drop/
+ | @ldrop_drop >(plus_minus_m_m e 1) /2 width=3 by/
]
]
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
- >(tw_lift … HV21) -HV21 /2 width=1/
+lemma ldrop_fwd_length_ge: ∀L1,L2,d,e,s. ⇩[s, d, e] L1 ≡ L2 → |L1| ≤ d → |L2| = |L1|.
+#L1 #L2 #d #e #s #H elim H -L1 -L2 -d -e // normalize
+[ #I #L1 #L2 #V #e #_ #_ #H elim (le_plus_xSy_O_false … H)
+| /4 width=2 by le_plus_to_le_r, eq_f/
]
qed-.
-lemma ldrop_pair2_fwd_fw: ∀I,L,K,V,d,e. ⇩[d, e] L ≡ K. ⓑ{I} V →
- ∀T. ♯{K, V} < ♯{L, T}.
-#I #L #K #V #d #e #H #T
-lapply (ldrop_fwd_lw … H) -H #H
-@(le_to_lt_to_lt … H) -H /3 width=1/
+lemma ldrop_fwd_length_le_le: ∀L1,L2,d,e,s. ⇩[s, d, e] L1 ≡ L2 → d ≤ |L1| → e ≤ |L1| - d → |L2| = |L1| - e.
+#L1 #L2 #d #e #s #H elim H -L1 -L2 -d -e // normalize
+[ /3 width=2 by le_plus_to_le_r/
+| #I #L1 #L2 #V1 #V2 #d #e #_ #_ #IHL12 >minus_plus_plus_l
+ #Hd #He lapply (le_plus_to_le_r … Hd) -Hd
+ #Hd >IHL12 // -L2 >plus_minus /2 width=3 by transitive_le/
+]
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_le_ge: ∀L1,L2,d,e,s. ⇩[s, d, e] L1 ≡ L2 → d ≤ |L1| → |L1| - d ≤ e → |L2| = d.
+#L1 #L2 #d #e #s #H elim H -L1 -L2 -d -e normalize
+[ /2 width=1 by le_n_O_to_eq/
+| #I #L #V #_ <minus_n_O #H elim (le_plus_xSy_O_false … H)
+| /3 width=2 by le_plus_to_le_r/
+| /4 width=2 by le_plus_to_le_r, eq_f/
]
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: ∀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 by/
+qed-.
+
+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 by plus_minus, le_n/
+qed-.
+
+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_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 by lt_minus_to_plus_r/
+qed-.
+
+lemma ldrop_fwd_length_eq1: ∀L1,L2,K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ |L1| = |L2| → |K1| = |K2|.
+#L1 #L2 #K1 #K2 #d #e #HLK1 #HLK2 #HL12
+lapply (ldrop_fwd_length … HLK1) -HLK1
+lapply (ldrop_fwd_length … HLK2) -HLK2
+/2 width=2 by injective_plus_r/
+qed-.
+
+lemma ldrop_fwd_length_eq2: ∀L1,L2,K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ |K1| = |K2| → |L1| = |L2|.
+#L1 #L2 #K1 #K2 #d #e #HLK1 #HLK2 #HL12
+lapply (ldrop_fwd_length … HLK1) -HLK1
+lapply (ldrop_fwd_length … HLK2) -HLK2 //
+qed-.
+
+lemma ldrop_fwd_lw: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → ♯{L2} ≤ ♯{L1}.
+#L1 #L2 #s #d #e #H elim H -L1 -L2 -d -e // normalize
+[ /2 width=3 by transitive_le/
+| #I #L1 #L2 #V1 #V2 #d #e #_ #HV21 #IHL12
+ >(lift_fwd_tw … HV21) -HV21 /2 width=1 by monotonic_le_plus_l/
+]
+qed-.
+
+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
+[ #d #e #H >H -H //
+| #I #L #V #H elim (lt_refl_false … H)
+| #I #L1 #L2 #V #e #HL12 #_ #_
+ lapply (ldrop_fwd_lw … HL12) -HL12 #HL12
+ @(le_to_lt_to_lt … HL12) -HL12 //
+| #I #L1 #L2 #V1 #V2 #d #e #_ #HV21 #IHL12 #H normalize in ⊢ (?%%); -I
+ >(lift_fwd_tw … HV21) -V2 /3 by lt_minus_to_plus/
+]
+qed-.
+
+lemma ldrop_fwd_rfw: ∀I,L,K,V,i. ⇩[i] L ≡ K.ⓑ{I}V → ∀T. ♯{K, V} < ♯{L, T}.
+#I #L #K #V #i #HLK lapply (ldrop_fwd_lw … HLK) -HLK
+normalize in ⊢ (%→?→?%%); /3 width=3 by le_to_lt_to_lt/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+fact ldrop_inv_O2_aux: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → e = 0 → L1 = L2.
+#L1 #L2 #s #d #e #H elim H -L1 -L2 -d -e
+[ //
+| //
+| #I #L1 #L2 #V #e #_ #_ >commutative_plus normalize #H destruct
+| #I #L1 #L2 #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,s,d. ⇩[s, d, 0] L1 ≡ L2 → L1 = L2.
+/2 width=5 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-.
+
+fact ldrop_inv_FT_aux: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 →
+ ∀I,K,V. L2 = K.ⓑ{I}V → s = Ⓣ → d = 0 →
+ ⇩[Ⓕ, d, e] L1 ≡ K.ⓑ{I}V.
+#L1 #L2 #s #d #e #H elim H -L1 -L2 -d -e
+[ #d #e #_ #J #K #W #H destruct
+| #I #L #V #J #K #W #H destruct //
+| #I #L1 #L2 #V #e #_ #IHL12 #J #K #W #H1 #H2 destruct
+ /3 width=1 by ldrop_drop/
+| #I #L1 #L2 #V1 #V2 #d #e #_ #_ #_ #J #K #W #_ #_
+ <plus_n_Sm #H destruct
+]
+qed-.
+
+lemma ldrop_inv_FT: ∀I,L,K,V,e. ⇩[Ⓣ, 0, e] L ≡ K.ⓑ{I}V → ⇩[e] L ≡ K.ⓑ{I}V.
+/2 width=5 by ldrop_inv_FT_aux/ qed.
+
+lemma ldrop_inv_gen: ∀I,L,K,V,s,e. ⇩[s, 0, e] L ≡ K.ⓑ{I}V → ⇩[e] L ≡ K.ⓑ{I}V.
+#I #L #K #V * /2 width=1 by ldrop_inv_FT/
+qed-.
+
+lemma ldrop_inv_T: ∀I,L,K,V,s,e. ⇩[Ⓣ, 0, e] L ≡ K.ⓑ{I}V → ⇩[s, 0, e] L ≡ K.ⓑ{I}V.
+#I #L #K #V * /2 width=1 by ldrop_inv_FT/
+qed-.
+
(* Basic_1: removed theorems 50:
drop_ctail drop_skip_flat
cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
projects / helm.git / blobdiff