(* *)
(**************************************************************************)
-include "basic_2/grammar/lenv_length.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR SUBSTITUTION ****************************)
-
-inductive lsubr: nat → nat → relation lenv ≝
-| lsubr_sort: ∀d,e. lsubr d e (⋆) (⋆)
-| lsubr_OO: ∀L1,L2. lsubr 0 0 L1 L2
-| lsubr_abbr: ∀L1,L2,V,e. lsubr 0 e L1 L2 →
- lsubr 0 (e + 1) (L1. ⓓV) (L2.ⓓV)
-| lsubr_abst: ∀L1,L2,I,V1,V2,e. lsubr 0 e L1 L2 →
- lsubr 0 (e + 1) (L1. ⓑ{I}V1) (L2. ⓛV2)
-| lsubr_skip: ∀L1,L2,I1,I2,V1,V2,d,e.
- lsubr d e L1 L2 → lsubr (d + 1) e (L1. ⓑ{I1} V1) (L2. ⓑ{I2} V2)
+include "basic_2/notation/relations/lrsubeq_2.ma".
+include "basic_2/relocation/ldrop.ma".
+
+(* RESTRICTED LOCAL ENVIRONMENT REFINEMENT **********************************)
+
+inductive lsubr: relation lenv ≝
+| lsubr_sort: ∀L. lsubr L (⋆)
+| lsubr_bind: ∀I,L1,L2,V. lsubr L1 L2 → lsubr (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+| lsubr_abst: ∀L1,L2,V,W. lsubr L1 L2 → lsubr (L1.ⓓⓝW.V) (L2.ⓛW)
.
interpretation
- "local environment refinement (substitution)"
- 'SubEq L1 d e L2 = (lsubr d e L1 L2).
-
-definition lsubr_trans: ∀S. (lenv → relation S) → Prop ≝ λS,R.
- ∀L2,s1,s2. R L2 s1 s2 →
- ∀L1,d,e. L1 ⊑ [d, e] L2 → R L1 s1 s2.
+ "local environment refinement (restricted)"
+ 'LRSubEq L1 L2 = (lsubr L1 L2).
(* Basic properties *********************************************************)
-lemma lsubr_bind_eq: ∀L1,L2,e. L1 ⊑ [0, e] L2 → ∀I,V.
- L1. ⓑ{I} V ⊑ [0, e + 1] L2.ⓑ{I} V.
-#L1 #L2 #e #HL12 #I #V elim I -I /2 width=1/
-qed.
-
-lemma lsubr_abbr_lt: ∀L1,L2,V,e. L1 ⊑ [0, e - 1] L2 → 0 < e →
- L1. ⓓV ⊑ [0, e] L2.ⓓV.
-#L1 #L2 #V #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/
-qed.
-
-lemma lsubr_abst_lt: ∀L1,L2,I,V1,V2,e. L1 ⊑ [0, e - 1] L2 → 0 < e →
- L1. ⓑ{I}V1 ⊑ [0, e] L2. ⓛV2.
-#L1 #L2 #I #V1 #V2 #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/
-qed.
-
-lemma lsubr_skip_lt: ∀L1,L2,d,e. L1 ⊑ [d - 1, e] L2 → 0 < d →
- ∀I1,I2,V1,V2. L1. ⓑ{I1} V1 ⊑ [d, e] L2. ⓑ{I2} V2.
-#L1 #L2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) // /2 width=1/
-qed.
-
-lemma lsubr_bind_lt: ∀I,L1,L2,V,e. L1 ⊑ [0, e - 1] L2 → 0 < e →
- L1. ⓓV ⊑ [0, e] L2. ⓑ{I}V.
-* /2 width=1/ qed.
-
-lemma lsubr_refl: ∀d,e,L. L ⊑ [d, e] L.
-#d elim d -d
-[ #e elim e -e // #e #IHe #L elim L -L // /2 width=1/
-| #d #IHd #e #L elim L -L // /2 width=1/
-]
-qed.
-
-lemma TC_lsubr_trans: ∀S,R. lsubr_trans S R → lsubr_trans S (λL. (TC … (R L))).
-#S #R #HR #L1 #s1 #s2 #H elim H -s2
-[ /3 width=5/
-| #s #s2 #_ #Hs2 #IHs1 #L2 #d #e #HL12
- lapply (HR … Hs2 … HL12) -HR -Hs2 -HL12 /3 width=3/
-]
+lemma lsubr_refl: ∀L. L ⊑ L.
+#L elim L -L /2 width=1 by lsubr_sort, lsubr_bind/
qed.
(* Basic inversion lemmas ***************************************************)
-fact lsubr_inv_atom1_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 → L1 = ⋆ →
- L2 = ⋆ ∨ (d = 0 ∧ e = 0).
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ /2 width=1/
-| /3 width=1/
-| #L1 #L2 #W #e #_ #H destruct
-| #L1 #L2 #I #W1 #W2 #e #_ #H destruct
-| #L1 #L2 #I1 #I2 #W1 #W2 #d #e #_ #H destruct
+fact lsubr_inv_atom1_aux: ∀L1,L2. L1 ⊑ L2 → L1 = ⋆ → L2 = ⋆.
+#L1 #L2 * -L1 -L2 //
+[ #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #_ #H destruct
]
-qed.
+qed-.
-lemma lsubr_inv_atom1: ∀L2,d,e. ⋆ ⊑ [d, e] L2 →
- L2 = ⋆ ∨ (d = 0 ∧ e = 0).
-/2 width=3/ qed-.
-
-fact lsubr_inv_skip1_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
- ∀I1,K1,V1. L1 = K1.ⓑ{I1}V1 → 0 < d →
- ∃∃I2,K2,V2. K1 ⊑ [d - 1, e] K2 & L2 = K2.ⓑ{I2}V2.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #d #e #I1 #K1 #V1 #H destruct
-| #L1 #L2 #I1 #K1 #V1 #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #W #e #_ #I1 #K1 #V1 #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #I #W1 #W2 #e #_ #I1 #K1 #V1 #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #J1 #J2 #W1 #W2 #d #e #HL12 #I1 #K1 #V1 #H #_ destruct /2 width=5/
-]
-qed.
+lemma lsubr_inv_atom1: ∀L2. ⋆ ⊑ L2 → L2 = ⋆.
+/2 width=3 by lsubr_inv_atom1_aux/ qed-.
-lemma lsubr_inv_skip1: ∀I1,K1,L2,V1,d,e. K1.ⓑ{I1}V1 ⊑ [d, e] L2 → 0 < d →
- ∃∃I2,K2,V2. K1 ⊑ [d - 1, e] K2 & L2 = K2.ⓑ{I2}V2.
-/2 width=5/ qed-.
-
-fact lsubr_inv_atom2_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 → L2 = ⋆ →
- L1 = ⋆ ∨ (d = 0 ∧ e = 0).
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ /2 width=1/
-| /3 width=1/
-| #L1 #L2 #W #e #_ #H destruct
-| #L1 #L2 #I #W1 #W2 #e #_ #H destruct
-| #L1 #L2 #I1 #I2 #W1 #W2 #d #e #_ #H destruct
+fact lsubr_inv_abst1_aux: ∀L1,L2. L1 ⊑ L2 → ∀K1,W. L1 = K1.ⓛW →
+ L2 = ⋆ ∨ ∃∃K2. K1 ⊑ K2 & L2 = K2.ⓛW.
+#L1 #L2 * -L1 -L2
+[ #L #K1 #W #H destruct /2 width=1 by or_introl/
+| #I #L1 #L2 #V #HL12 #K1 #W #H destruct /3 width=3 by ex2_intro, or_intror/
+| #L1 #L2 #V1 #V2 #_ #K1 #W #H destruct
]
-qed.
-
-lemma lsubr_inv_atom2: ∀L1,d,e. L1 ⊑ [d, e] ⋆ →
- L1 = ⋆ ∨ (d = 0 ∧ e = 0).
-/2 width=3/ qed-.
-
-fact lsubr_inv_abbr2_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
- ∀K2,V. L2 = K2.ⓓV → d = 0 → 0 < e →
- ∃∃K1. K1 ⊑ [0, e - 1] K2 & L1 = K1.ⓓV.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #d #e #K1 #V #H destruct
-| #L1 #L2 #K1 #V #_ #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #W #e #HL12 #K1 #V #H #_ #_ destruct /2 width=3/
-| #L1 #L2 #I #W1 #W2 #e #_ #K1 #V #H destruct
-| #L1 #L2 #I1 #I2 #W1 #W2 #d #e #_ #K1 #V #_ >commutative_plus normalize #H destruct
+qed-.
+
+lemma lsubr_inv_abst1: ∀K1,L2,W. K1.ⓛW ⊑ L2 →
+ L2 = ⋆ ∨ ∃∃K2. K1 ⊑ K2 & L2 = K2.ⓛW.
+/2 width=3 by lsubr_inv_abst1_aux/ qed-.
+
+fact lsubr_inv_abbr2_aux: ∀L1,L2. L1 ⊑ L2 → ∀K2,W. L2 = K2.ⓓW →
+ ∃∃K1. K1 ⊑ K2 & L1 = K1.ⓓW.
+#L1 #L2 * -L1 -L2
+[ #L #K2 #W #H destruct
+| #I #L1 #L2 #V #HL12 #K2 #W #H destruct /2 width=3 by ex2_intro/
+| #L1 #L2 #V1 #V2 #_ #K2 #W #H destruct
]
-qed.
+qed-.
-lemma lsubr_inv_abbr2: ∀L1,K2,V,e. L1 ⊑ [0, e] K2.ⓓV → 0 < e →
- ∃∃K1. K1 ⊑ [0, e - 1] K2 & L1 = K1.ⓓV.
-/2 width=5/ qed-.
-
-fact lsubr_inv_skip2_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
- ∀I2,K2,V2. L2 = K2.ⓑ{I2}V2 → 0 < d →
- ∃∃I1,K1,V1. K1 ⊑ [d - 1, e] K2 & L1 = K1.ⓑ{I1}V1.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #d #e #I1 #K1 #V1 #H destruct
-| #L1 #L2 #I1 #K1 #V1 #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #W #e #_ #I1 #K1 #V1 #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #I #W1 #W2 #e #_ #I1 #K1 #V1 #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #J1 #J2 #W1 #W2 #d #e #HL12 #I1 #K1 #V1 #H #_ destruct /2 width=5/
-]
-qed.
-
-lemma lsubr_inv_skip2: ∀I2,L1,K2,V2,d,e. L1 ⊑ [d, e] K2.ⓑ{I2}V2 → 0 < d →
- ∃∃I1,K1,V1. K1 ⊑ [d - 1, e] K2 & L1 = K1.ⓑ{I1}V1.
-/2 width=5/ qed-.
+lemma lsubr_inv_abbr2: ∀L1,K2,W. L1 ⊑ K2.ⓓW →
+ ∃∃K1. K1 ⊑ K2 & L1 = K1.ⓓW.
+/2 width=3 by lsubr_inv_abbr2_aux/ qed-.
(* Basic forward lemmas *****************************************************)
-fact lsubr_fwd_length_full1_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
- d = 0 → e = |L1| → |L1| ≤ |L2|.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize
-[ //
-| /2 width=1/
-| /3 width=1/
-| /3 width=1/
-| #L1 #L2 #_ #_ #_ #_ #d #e #_ #_ >commutative_plus normalize #H destruct
+lemma lsubr_fwd_length: ∀L1,L2. L1 ⊑ L2 → |L2| ≤ |L1|.
+#L1 #L2 #H elim H -L1 -L2 /2 width=1 by monotonic_le_plus_l/
+qed-.
+
+lemma lsubr_fwd_ldrop2_bind: ∀L1,L2. L1 ⊑ L2 →
+ ∀I,K2,W,s,i. ⇩[s, 0, i] L2 ≡ K2.ⓑ{I}W →
+ (∃∃K1. K1 ⊑ K2 & ⇩[s, 0, i] L1 ≡ K1.ⓑ{I}W) ∨
+ ∃∃K1,V. K1 ⊑ K2 & ⇩[s, 0, i] L1 ≡ K1.ⓓⓝW.V & I = Abst.
+#L1 #L2 #H elim H -L1 -L2
+[ #L #I #K2 #W #s #i #H
+ elim (ldrop_inv_atom1 … H) -H #H destruct
+| #J #L1 #L2 #V #HL12 #IHL12 #I #K2 #W #s #i #H
+ elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
+ [ /3 width=3 by ldrop_pair, ex2_intro, or_introl/
+ | elim (IHL12 … HLK2) -IHL12 -HLK2 *
+ /4 width=4 by ldrop_drop_lt, ex3_2_intro, ex2_intro, or_introl, or_intror/
+ ]
+| #L1 #L2 #V1 #V2 #HL12 #IHL12 #I #K2 #W #s #i #H
+ elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
+ [ /3 width=4 by ldrop_pair, ex3_2_intro, or_intror/
+ | elim (IHL12 … HLK2) -IHL12 -HLK2 *
+ /4 width=4 by ldrop_drop_lt, ex3_2_intro, ex2_intro, or_introl, or_intror/
+ ]
]
-qed.
-
-lemma lsubr_fwd_length_full1: ∀L1,L2. L1 ⊑ [0, |L1|] L2 → |L1| ≤ |L2|.
-/2 width=5/ qed-.
-
-fact lsubr_fwd_length_full2_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
- d = 0 → e = |L2| → |L2| ≤ |L1|.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize
-[ //
-| /2 width=1/
-| /3 width=1/
-| /3 width=1/
-| #L1 #L2 #_ #_ #_ #_ #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma lsubr_fwd_length_full2: ∀L1,L2. L1 ⊑ [0, |L2|] L2 → |L2| ≤ |L1|.
-/2 width=5/ qed-.
+qed-.
+
+lemma lsubr_fwd_ldrop2_abbr: ∀L1,L2. L1 ⊑ L2 →
+ ∀K2,V,s,i. ⇩[s, 0, i] L2 ≡ K2.ⓓV →
+ ∃∃K1. K1 ⊑ K2 & ⇩[s, 0, i] L1 ≡ K1.ⓓV.
+#L1 #L2 #HL12 #K2 #V #s #i #HLK2 elim (lsubr_fwd_ldrop2_bind … HL12 … HLK2) -L2 // *
+#K1 #W #_ #_ #H destruct
+qed-.
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