(* *)
(**************************************************************************)
-include "basic_2/equivalence/lsubse.ma".
-include "basic_2/dynamic/snv.ma".
+include "basic_2/notation/relations/lrsubeqv_5.ma".
+include "basic_2/dynamic/shnv.ma".
(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
(* Note: this is not transitive *)
-inductive lsubsv (h:sh) (g:sd h): relation lenv ≝
-| lsubsv_atom: lsubsv h g (⋆) (⋆)
-| lsubsv_pair: ∀I,L1,L2,V. lsubsv h g L1 L2 →
- lsubsv h g (L1. ⓑ{I} V) (L2. ⓑ{I} V)
-| lsubsv_abbr: ∀L1,L2,V1,V2,W1,W2,l. ⦃h, L1⦄ ⊩ V1 :[g] → ⦃h, L1⦄ ⊢ V1 •[g, l+1] W1 →
- L1 ⊢ W1 ⬌* W2 → ⦃h, L2⦄ ⊩ W2 :[g] → ⦃h, L2⦄ ⊢ W2 •[g, l] V2 →
- lsubsv h g L1 L2 → lsubsv h g (L1. ⓓV1) (L2. ⓛW2)
+inductive lsubsv (h) (o) (G): relation lenv ≝
+| lsubsv_atom: lsubsv h o G (⋆) (⋆)
+| lsubsv_pair: ∀I,L1,L2,V. lsubsv h o G L1 L2 →
+ lsubsv h o G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+| lsubsv_beta: ∀L1,L2,W,V,d1. ⦃G, L1⦄ ⊢ ⓝW.V ¡[h, o, d1] → ⦃G, L2⦄ ⊢ W ¡[h, o] →
+ ⦃G, L1⦄ ⊢ V ▪[h, o] d1+1 → ⦃G, L2⦄ ⊢ W ▪[h, o] d1 →
+ lsubsv h o G L1 L2 → lsubsv h o G (L1.ⓓⓝW.V) (L2.ⓛW)
.
interpretation
"local environment refinement (stratified native validity)"
- 'CrSubEqV h g L1 L2 = (lsubsv h g L1 L2).
+ 'LRSubEqV h o G L1 L2 = (lsubsv h o G L1 L2).
(* Basic inversion lemmas ***************************************************)
-fact lsubsv_inv_atom1_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 = ⋆ → L2 = ⋆.
-#h #g #L1 #L2 * -L1 -L2
+fact lsubsv_inv_atom1_aux: ∀h,o,G,L1,L2. G ⊢ L1 ⫃¡[h, o] L2 → L1 = ⋆ → L2 = ⋆.
+#h #o #G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #_ #_ #H destruct
+| #L1 #L2 #W #V #d1 #_ #_ #_ #_ #_ #H destruct
]
qed-.
-lemma lsubsv_inv_atom1: ∀h,g,L2. h ⊢ ⋆ ⊩:⊑[g] L2 → L2 = ⋆.
-/2 width=5 by lsubsv_inv_atom1_aux/ qed-.
-
-fact lsubsv_inv_pair1_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 →
- ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
- (∃∃K2. h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨
- ∃∃K2,V2,W1,W2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 &
- K1 ⊢ W1 ⬌* W2 & ⦃h, K2⦄ ⊩ W2 :[g] & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
- h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr.
-#h #g #L1 #L2 * -L1 -L2
-[ #J #K1 #U1 #H destruct
-| #I #L1 #L2 #V #HL12 #J #K1 #U1 #H destruct /3 width=3/
-| #L1 #L2 #V1 #V2 #W1 #W2 #l #HV1 #HVW1 #HW12 #HW2 #HWV2 #HL12 #J #K1 #U1 #H destruct /3 width=11/
+lemma lsubsv_inv_atom1: ∀h,o,G,L2. G ⊢ ⋆ ⫃¡[h, o] L2 → L2 = ⋆.
+/2 width=6 by lsubsv_inv_atom1_aux/ qed-.
+
+fact lsubsv_inv_pair1_aux: ∀h,o,G,L1,L2. G ⊢ L1 ⫃¡[h, o] L2 →
+ ∀I,K1,X. L1 = K1.ⓑ{I}X →
+ (∃∃K2. G ⊢ K1 ⫃¡[h, o] K2 & L2 = K2.ⓑ{I}X) ∨
+ ∃∃K2,W,V,d1. ⦃G, K1⦄ ⊢ ⓝW.V ¡[h, o, d1] & ⦃G, K2⦄ ⊢ W ¡[h, o] &
+ ⦃G, K1⦄ ⊢ V ▪[h, o] d1+1 & ⦃G, K2⦄ ⊢ W ▪[h, o] d1 &
+ G ⊢ K1 ⫃¡[h, o] K2 &
+ I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+#h #o #G #L1 #L2 * -L1 -L2
+[ #J #K1 #X #H destruct
+| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #d1 #HWV #HW #HVd1 #HWd1 #HL12 #J #K1 #X #H destruct /3 width=11 by or_intror, ex8_4_intro/
]
qed-.
-lemma lsubsv_inv_pair1: ∀h,g,I,K1,L2,V1. h ⊢ K1. ⓑ{I} V1 ⊩:⊑[g] L2 →
- (∃∃K2. h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨
- ∃∃K2,V2,W1,W2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 &
- K1 ⊢ W1 ⬌* W2 & ⦃h, K2⦄ ⊩ W2 :[g] & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
- h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr.
+lemma lsubsv_inv_pair1: ∀h,o,I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⫃¡[h, o] L2 →
+ (∃∃K2. G ⊢ K1 ⫃¡[h, o] K2 & L2 = K2.ⓑ{I}X) ∨
+ ∃∃K2,W,V,d1. ⦃G, K1⦄ ⊢ ⓝW.V ¡[h, o, d1] & ⦃G, K2⦄ ⊢ W ¡[h, o] &
+ ⦃G, K1⦄ ⊢ V ▪[h, o] d1+1 & ⦃G, K2⦄ ⊢ W ▪[h, o] d1 &
+ G ⊢ K1 ⫃¡[h, o] K2 &
+ I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
/2 width=3 by lsubsv_inv_pair1_aux/ qed-.
-fact lsubsv_inv_atom2_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L2 = ⋆ → L1 = ⋆.
-#h #g #L1 #L2 * -L1 -L2
+fact lsubsv_inv_atom2_aux: ∀h,o,G,L1,L2. G ⊢ L1 ⫃¡[h, o] L2 → L2 = ⋆ → L1 = ⋆.
+#h #o #G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #_ #_ #H destruct
+| #L1 #L2 #W #V #d1 #_ #_ #_ #_ #_ #H destruct
]
qed-.
-lemma lsubsv_inv_atom2: ∀h,g,L1. h ⊢ L1 ⊩:⊑[g] ⋆ → L1 = ⋆.
-/2 width=5 by lsubsv_inv_atom2_aux/ qed-.
-
-fact lsubsv_inv_pair2_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 →
- ∀I,K2,W2. L2 = K2. ⓑ{I} W2 →
- (∃∃K1. h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨
- ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 &
- K1 ⊢ W1 ⬌* W2 & ⦃h, K2⦄ ⊩ W2 :[g] & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
- h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst.
-#h #g #L1 #L2 * -L1 -L2
-[ #J #K2 #U2 #H destruct
-| #I #L1 #L2 #V #HL12 #J #K2 #U2 #H destruct /3 width=3/
-| #L1 #L2 #V1 #V2 #W1 #W2 #l #HV #HVW1 #HW12 #HW2 #HWV2 #HL12 #J #K2 #U2 #H destruct /3 width=11/
+lemma lsubsv_inv_atom2: ∀h,o,G,L1. G ⊢ L1 ⫃¡[h, o] ⋆ → L1 = ⋆.
+/2 width=6 by lsubsv_inv_atom2_aux/ qed-.
+
+fact lsubsv_inv_pair2_aux: ∀h,o,G,L1,L2. G ⊢ L1 ⫃¡[h, o] L2 →
+ ∀I,K2,W. L2 = K2.ⓑ{I}W →
+ (∃∃K1. G ⊢ K1 ⫃¡[h, o] K2 & L1 = K1.ⓑ{I}W) ∨
+ ∃∃K1,V,d1. ⦃G, K1⦄ ⊢ ⓝW.V ¡[h, o, d1] & ⦃G, K2⦄ ⊢ W ¡[h, o] &
+ ⦃G, K1⦄ ⊢ V ▪[h, o] d1+1 & ⦃G, K2⦄ ⊢ W ▪[h, o] d1 &
+ G ⊢ K1 ⫃¡[h, o] K2 & I = Abst & L1 = K1.ⓓⓝW.V.
+#h #o #G #L1 #L2 * -L1 -L2
+[ #J #K2 #U #H destruct
+| #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #d1 #HWV #HW #HVd1 #HWd1 #HL12 #J #K2 #U #H destruct /3 width=8 by or_intror, ex7_3_intro/
]
qed-.
-lemma lsubsv_inv_pair2: ∀h,g,I,L1,K2,W2. h ⊢ L1 ⊩:⊑[g] K2. ⓑ{I} W2 →
- (∃∃K1. h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨
- ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 &
- K1 ⊢ W1 ⬌* W2 & ⦃h, K2⦄ ⊩ W2 :[g] & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
- h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst.
+lemma lsubsv_inv_pair2: ∀h,o,I,G,L1,K2,W. G ⊢ L1 ⫃¡[h, o] K2.ⓑ{I}W →
+ (∃∃K1. G ⊢ K1 ⫃¡[h, o] K2 & L1 = K1.ⓑ{I}W) ∨
+ ∃∃K1,V,d1. ⦃G, K1⦄ ⊢ ⓝW.V ¡[h, o, d1] & ⦃G, K2⦄ ⊢ W ¡[h, o] &
+ ⦃G, K1⦄ ⊢ V ▪[h, o] d1+1 & ⦃G, K2⦄ ⊢ W ▪[h, o] d1 &
+ G ⊢ K1 ⫃¡[h, o] K2 & I = Abst & L1 = K1.ⓓⓝW.V.
/2 width=3 by lsubsv_inv_pair2_aux/ qed-.
-(* Basic_forward lemmas *****************************************************)
+(* Basic forward lemmas *****************************************************)
-lemma lsubsv_fwd_lsubse: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → h ⊢ L1 ⊢•⊑[g] L2.
-#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ /2 width=6/
-qed-.
-
-lemma lsubsv_fwd_lsubs1: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 ≼[0, |L1|] L2.
-/3 width=3 by lsubsv_fwd_lsubse, lsubse_fwd_lsubs1/
-qed-.
-
-lemma lsubsv_fwd_lsubs2: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 ≼[0, |L2|] L2.
-/3 width=3 by lsubsv_fwd_lsubse, lsubse_fwd_lsubs2/
+lemma lsubsv_fwd_lsubr: ∀h,o,G,L1,L2. G ⊢ L1 ⫃¡[h, o] L2 → L1 ⫃ L2.
+#h #o #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_pair, lsubr_beta/
qed-.
(* Basic properties *********************************************************)
-lemma lsubsv_refl: ∀h,g,L. h ⊢ L ⊩:⊑[g] L.
-#h #g #L elim L -L // /2 width=1/
+lemma lsubsv_refl: ∀h,o,G,L. G ⊢ L ⫃¡[h, o] L.
+#h #o #G #L elim L -L /2 width=1 by lsubsv_pair/
qed.
-lemma lsubsv_cprs_trans: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 →
- ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
-/3 width=5 by lsubsv_fwd_lsubse, lsubse_cprs_trans/
+lemma lsubsv_cprs_trans: ∀h,o,G,L1,L2. G ⊢ L1 ⫃¡[h, o] L2 →
+ ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ➡* T2 → ⦃G, L1⦄ ⊢ T1 ➡* T2.
+/3 width=6 by lsubsv_fwd_lsubr, lsubr_cprs_trans/
+qed-.
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsubsv_drop_O1_conf: ∀h,o,G,L1,L2. G ⊢ L1 ⫃¡[h, o] L2 →
+ ∀K1,b,k. ⬇[b, 0, k] L1 ≘ K1 →
+ ∃∃K2. G ⊢ K1 ⫃¡[h, o] K2 & ⬇[b, 0, k] L2 ≘ K2.
+#h #o #G #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #_ #IHL12 #K1 #b #k #H
+ elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
+ [ destruct
+ elim (IHL12 L1 b 0) -IHL12 // #X #HL12 #H
+ <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubsv_pair, drop_pair, ex2_intro/
+ | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
+ ]
+| #L1 #L2 #W #V #d1 #HWV #HW #HVd1 #HWd1 #_ #IHL12 #K1 #b #k #H
+ elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
+ [ destruct
+ elim (IHL12 L1 b 0) -IHL12 // #X #HL12 #H
+ <(drop_inv_O2 … H) in HL12; -H /3 width=4 by lsubsv_beta, drop_pair, ex2_intro/
+ | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
+ ]
+]
+qed-.
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsubsv_drop_O1_trans: ∀h,o,G,L1,L2. G ⊢ L1 ⫃¡[h, o] L2 →
+ ∀K2,b, k. ⬇[b, 0, k] L2 ≘ K2 →
+ ∃∃K1. G ⊢ K1 ⫃¡[h, o] K2 & ⬇[b, 0, k] L1 ≘ K1.
+#h #o #G #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #_ #IHL12 #K2 #b #k #H
+ elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
+ [ destruct
+ elim (IHL12 L2 b 0) -IHL12 // #X #HL12 #H
+ <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubsv_pair, drop_pair, ex2_intro/
+ | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
+ ]
+| #L1 #L2 #W #V #d1 #HWV #HW #HVd1 #HWd1 #_ #IHL12 #K2 #b #k #H
+ elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
+ [ destruct
+ elim (IHL12 L2 b 0) -IHL12 // #X #HL12 #H
+ <(drop_inv_O2 … H) in HL12; -H /3 width=4 by lsubsv_beta, drop_pair, ex2_intro/
+ | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
+ ]
+]
qed-.
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