(**************************************************************************)
include "basic_2/notation/relations/lrsubeqc_2.ma".
-include "basic_2/substitution/drop.ma".
+include "basic_2/syntax/lenv.ma".
-(* RESTRICTED LOCAL ENVIRONMENT REFINEMENT **********************************)
+(* RESTRICTED REFINEMENT FOR LOCAL ENVIRONMENTS *****************************)
+(* Basic_2A1: just tpr_cpr and tprs_cprs require the extended lsubr_atom *)
+(* Basic_2A1: includes: lsubr_pair *)
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)
+| lsubr_atom: lsubr (⋆) (⋆)
+| lsubr_bind: ∀I,L1,L2. lsubr L1 L2 → lsubr (L1.ⓘ{I}) (L2.ⓘ{I})
+| lsubr_beta: ∀L1,L2,V,W. lsubr L1 L2 → lsubr (L1.ⓓⓝW.V) (L2.ⓛW)
+| lsubr_unit: ∀I1,I2,L1,L2,V. lsubr L1 L2 → lsubr (L1.ⓑ{I1}V) (L2.ⓤ{I2})
.
interpretation
- "local environment refinement (restricted)"
+ "restricted refinement (local environment)"
'LRSubEqC L1 L2 = (lsubr L1 L2).
(* Basic properties *********************************************************)
lemma lsubr_refl: ∀L. L ⫃ L.
-#L elim L -L /2 width=1 by lsubr_sort, lsubr_bind/
+#L elim L -L /2 width=1 by lsubr_atom, lsubr_bind/
qed.
(* Basic inversion lemmas ***************************************************)
fact lsubr_inv_atom1_aux: ∀L1,L2. L1 ⫃ L2 → L1 = ⋆ → L2 = ⋆.
#L1 #L2 * -L1 -L2 //
-[ #I #L1 #L2 #V #_ #H destruct
+[ #I #L1 #L2 #_ #H destruct
| #L1 #L2 #V #W #_ #H destruct
+| #I1 #I2 #L1 #L2 #V #_ #H destruct
]
qed-.
lemma lsubr_inv_atom1: ∀L2. ⋆ ⫃ L2 → L2 = ⋆.
/2 width=3 by lsubr_inv_atom1_aux/ qed-.
-fact lsubr_inv_abst1_aux: ∀L1,L2. L1 ⫃ L2 → ∀K1,W. L1 = K1.ⓛW →
- L2 = ⋆ ∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓛW.
+fact lsubr_inv_bind1_aux: ∀L1,L2. L1 ⫃ L2 → ∀I,K1. L1 = K1.ⓘ{I} →
+ ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓘ{I}
+ | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW &
+ I = BPair Abbr (ⓝW.V)
+ | ∃∃J1,J2,K2,V. K1 ⫃ K2 & L2 = K2.ⓤ{J2} &
+ I = BPair J1 V.
#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
+[ #J #K1 #H destruct
+| #I #L1 #L2 #HL12 #J #K1 #H destruct /3 width=3 by or3_intro0, ex2_intro/
+| #L1 #L2 #V #W #HL12 #J #K1 #H destruct /3 width=6 by or3_intro1, ex3_3_intro/
+| #I1 #I2 #L1 #L2 #V #HL12 #J #K1 #H destruct /3 width=4 by or3_intro2, ex3_4_intro/
]
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-.
+(* Basic_2A1: uses: lsubr_inv_pair1 *)
+lemma lsubr_inv_bind1: ∀I,K1,L2. K1.ⓘ{I} ⫃ L2 →
+ ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓘ{I}
+ | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW &
+ I = BPair Abbr (ⓝW.V)
+ | ∃∃J1,J2,K2,V. K1 ⫃ K2 & L2 = K2.ⓤ{J2} &
+ I = BPair J1 V.
+/2 width=3 by lsubr_inv_bind1_aux/ qed-.
+
+fact lsubr_inv_atom2_aux: ∀L1,L2. L1 ⫃ L2 → L2 = ⋆ → L1 = ⋆.
+#L1 #L2 * -L1 -L2 //
+[ #I #L1 #L2 #_ #H destruct
+| #L1 #L2 #V #W #_ #H destruct
+| #I1 #I2 #L1 #L2 #V #_ #H destruct
+]
+qed-.
+
+lemma lsubr_inv_atom2: ∀L1. L1 ⫃ ⋆ → L1 = ⋆.
+/2 width=3 by lsubr_inv_atom2_aux/ qed-.
-fact lsubr_inv_abbr2_aux: ∀L1,L2. L1 ⫃ L2 → ∀K2,W. L2 = K2.ⓓW →
- ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓓW.
+fact lsubr_inv_bind2_aux: ∀L1,L2. L1 ⫃ L2 → ∀I,K2. L2 = K2.ⓘ{I} →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓘ{I}
+ | ∃∃K1,W,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = BPair Abst W
+ | ∃∃J1,J2,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J1}V & I = BUnit J2.
#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
+[ #J #K2 #H destruct
+| #I #L1 #L2 #HL12 #J #K2 #H destruct /3 width=3 by ex2_intro, or3_intro0/
+| #L1 #L2 #V1 #V2 #HL12 #J #K2 #H destruct /3 width=6 by ex3_3_intro, or3_intro1/
+| #I1 #I2 #L1 #L2 #V #HL12 #J #K2 #H destruct /3 width=5 by ex3_4_intro, or3_intro2/
]
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-.
+lemma lsubr_inv_bind2: ∀I,L1,K2. L1 ⫃ K2.ⓘ{I} →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓘ{I}
+ | ∃∃K1,W,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = BPair Abst W
+ | ∃∃J1,J2,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J1}V & I = BUnit J2.
+/2 width=3 by lsubr_inv_bind2_aux/ qed-.
-(* Basic forward lemmas *****************************************************)
+(* Advanced inversion lemmas ************************************************)
+
+lemma lsubr_inv_abst1: ∀K1,L2,W. K1.ⓛW ⫃ L2 →
+ ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓛW
+ | ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓤ{I2}.
+#K1 #L2 #W #H elim (lsubr_inv_bind1 … H) -H *
+/3 width=4 by ex2_2_intro, ex2_intro, or_introl, or_intror/
+#K2 #V2 #W2 #_ #_ #H destruct
+qed-.
+
+lemma lsubr_inv_unit1: ∀I,K1,L2. K1.ⓤ{I} ⫃ L2 →
+ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓤ{I}.
+#I #K1 #L2 #H elim (lsubr_inv_bind1 … H) -H *
+[ #K2 #HK12 #H destruct /2 width=3 by ex2_intro/
+| #K2 #V #W #_ #_ #H destruct
+| #I1 #I2 #K2 #V #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubr_inv_pair2: ∀I,L1,K2,W. L1 ⫃ K2.ⓑ{I}W →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓑ{I}W
+ | ∃∃K1,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = Abst.
+#I #L1 #K2 #W #H elim (lsubr_inv_bind2 … H) -H *
+[ /3 width=3 by ex2_intro, or_introl/
+| #K2 #X #V #HK12 #H1 #H2 destruct /3 width=4 by ex3_2_intro, or_intror/
+| #I1 #I1 #K2 #V #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubr_inv_abbr2: ∀L1,K2,V. L1 ⫃ K2.ⓓV →
+ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓓV.
+#L1 #K2 #V #H elim (lsubr_inv_pair2 … H) -H *
+[ /2 width=3 by ex2_intro/
+| #K1 #X #_ #_ #H destruct
+]
+qed-.
-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/
+lemma lsubr_inv_abst2: ∀L1,K2,W. L1 ⫃ K2.ⓛW →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓛW
+ | ∃∃K1,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V.
+#L1 #K2 #W #H elim (lsubr_inv_pair2 … H) -H *
+/3 width=4 by ex2_2_intro, ex2_intro, or_introl, or_intror/
qed-.
-lemma lsubr_fwd_drop2_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 (drop_inv_atom1 … H) -H #H destruct
-| #J #L1 #L2 #V #HL12 #IHL12 #I #K2 #W #s #i #H
- elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
- [ /3 width=3 by drop_pair, ex2_intro, or_introl/
- | elim (IHL12 … HLK2) -IHL12 -HLK2 *
- /4 width=4 by drop_drop_lt, ex3_2_intro, ex2_intro, or_introl, or_intror/
- ]
-| #L1 #L2 #V1 #V2 #HL12 #IHL12 #I #K2 #W #s #i #H
- elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
- [ /3 width=4 by drop_pair, ex3_2_intro, or_intror/
- | elim (IHL12 … HLK2) -IHL12 -HLK2 *
- /4 width=4 by drop_drop_lt, ex3_2_intro, ex2_intro, or_introl, or_intror/
- ]
+lemma lsubr_inv_unit2: ∀I,L1,K2. L1 ⫃ K2.ⓤ{I} →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓤ{I}
+ | ∃∃J,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J}V.
+#I #L1 #K2 #H elim (lsubr_inv_bind2 … H) -H *
+[ /3 width=3 by ex2_intro, or_introl/
+| #K1 #W #V #_ #_ #H destruct
+| #I1 #I2 #K1 #V #HK12 #H1 #H2 destruct /3 width=5 by ex2_3_intro, or_intror/
]
qed-.
-lemma lsubr_fwd_drop2_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_drop2_bind … HL12 … HLK2) -L2 // *
-#K1 #W #_ #_ #H destruct
+(* Basic forward lemmas *****************************************************)
+
+lemma lsubr_fwd_bind1: ∀I1,K1,L2. K1.ⓘ{I1} ⫃ L2 →
+ ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓘ{I2}.
+#I1 #K1 #L2 #H elim (lsubr_inv_bind1 … H) -H *
+[ #K2 #HK12 #H destruct /3 width=4 by ex2_2_intro/
+| #K2 #W1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
+| #I1 #I2 #K2 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lsubr_fwd_bind2: ∀I2,L1,K2. L1 ⫃ K2.ⓘ{I2} →
+ ∃∃I1,K1. K1 ⫃ K2 & L1 = K1.ⓘ{I1}.
+#I2 #L1 #K2 #H elim (lsubr_inv_bind2 … H) -H *
+[ #K1 #HK12 #H destruct /3 width=4 by ex2_2_intro/
+| #K1 #W1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
+| #I1 #I2 #K1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
+]
qed-.
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