(**************************************************************************)
include "basic_2/notation/relations/lrsubeqv_5.ma".
-include "basic_2/dynamic/snv.ma".
+include "basic_2/dynamic/shnv.ma".
(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
| lsubsv_atom: lsubsv h g G (⋆) (⋆)
| lsubsv_pair: ∀I,L1,L2,V. lsubsv h g G L1 L2 →
lsubsv h g G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
-| lsubsv_abbr: ∀L1,L2,W,V,l. ⦃G, L1⦄ ⊢ W ¡[h, g] → ⦃G, L1⦄ ⊢ V ¡[h, g] →
- scast h g l G L1 V W → ⦃G, L2⦄ ⊢ W ¡[h, g] →
- ⦃G, L1⦄ ⊢ V ▪[h, g] l+1 → ⦃G, L2⦄ ⊢ W ▪[h, g] l →
+| lsubsv_beta: ∀L1,L2,W,V,d1. ⦃G, L1⦄ ⊢ ⓝW.V ¡[h, g, d1] → ⦃G, L2⦄ ⊢ W ¡[h, g] →
+ ⦃G, L1⦄ ⊢ V ▪[h, g] d1+1 → ⦃G, L2⦄ ⊢ W ▪[h, g] d1 →
lsubsv h g G L1 L2 → lsubsv h g G (L1.ⓓⓝW.V) (L2.ⓛW)
.
(* Basic inversion lemmas ***************************************************)
-fact lsubsv_inv_atom1_aux: ∀h,g,G,L1,L2. G ⊢ L1 ¡⫃[h, g] L2 → L1 = ⋆ → L2 = ⋆.
+fact lsubsv_inv_atom1_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 → L1 = ⋆ → L2 = ⋆.
#h #g #G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #W #V #l #_ #_ #_ #_ #_ #_ #_ #H destruct
+| #L1 #L2 #W #V #d1 #_ #_ #_ #_ #_ #H destruct
]
qed-.
-lemma lsubsv_inv_atom1: ∀h,g,G,L2. G ⊢ ⋆ ¡⫃[h, g] L2 → L2 = ⋆.
+lemma lsubsv_inv_atom1: ∀h,g,G,L2. G ⊢ ⋆ ⫃¡[h, g] L2 → L2 = ⋆.
/2 width=6 by lsubsv_inv_atom1_aux/ qed-.
-fact lsubsv_inv_pair1_aux: ∀h,g,G,L1,L2. G ⊢ L1 ¡⫃[h, g] L2 →
+fact lsubsv_inv_pair1_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
∀I,K1,X. L1 = K1.ⓑ{I}X →
- (∃∃K2. G ⊢ K1 ¡⫃[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
- ∃∃K2,W,V,l. ⦃G, K1⦄ ⊢ W ¡[h, g] & ⦃G, K1⦄ ⊢ V ¡[h, g] &
- scast h g l G K1 V W & ⦃G, K2⦄ ⊢ W ¡[h, g] &
- ⦃G, K1⦄ ⊢ V ▪[h, g] l+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] l &
- G ⊢ K1 ¡⫃[h, g] K2 &
- I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+ (∃∃K2. G ⊢ K1 ⫃¡[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
+ ∃∃K2,W,V,d1. ⦃G, K1⦄ ⊢ ⓝW.V ¡[h, g, d1] & ⦃G, K2⦄ ⊢ W ¡[h, g] &
+ ⦃G, K1⦄ ⊢ V ▪[h, g] d1+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d1 &
+ G ⊢ K1 ⫃¡[h, g] K2 &
+ I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
#h #g #G #L1 #L2 * -L1 -L2
[ #J #K1 #X #H destruct
-| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3/
-| #L1 #L2 #W #V #l #H1W #HV #HVW #H2W #H1l #H2l #HL12 #J #K1 #X #H destruct /3 width=13/
+| #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,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ¡⫃[h, g] L2 →
- (∃∃K2. G ⊢ K1 ¡⫃[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
- ∃∃K2,W,V,l. ⦃G, K1⦄ ⊢ W ¡[h, g] & ⦃G, K1⦄ ⊢ V ¡[h, g] &
- scast h g l G K1 V W & ⦃G, K2⦄ ⊢ W ¡[h, g] &
- ⦃G, K1⦄ ⊢ V ▪[h, g] l+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] l &
- G ⊢ K1 ¡⫃[h, g] K2 &
- I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+lemma lsubsv_inv_pair1: ∀h,g,I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⫃¡[h, g] L2 →
+ (∃∃K2. G ⊢ K1 ⫃¡[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
+ ∃∃K2,W,V,d1. ⦃G, K1⦄ ⊢ ⓝW.V ¡[h, g, d1] & ⦃G, K2⦄ ⊢ W ¡[h, g] &
+ ⦃G, K1⦄ ⊢ V ▪[h, g] d1+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d1 &
+ G ⊢ K1 ⫃¡[h, g] 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,G,L1,L2. G ⊢ L1 ¡⫃[h, g] L2 → L2 = ⋆ → L1 = ⋆.
+fact lsubsv_inv_atom2_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 → L2 = ⋆ → L1 = ⋆.
#h #g #G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #W #V #l #_ #_ #_ #_ #_ #_ #_ #H destruct
+| #L1 #L2 #W #V #d1 #_ #_ #_ #_ #_ #H destruct
]
qed-.
-lemma lsubsv_inv_atom2: ∀h,g,G,L1. G ⊢ L1 ¡⫃[h, g] ⋆ → L1 = ⋆.
+lemma lsubsv_inv_atom2: ∀h,g,G,L1. G ⊢ L1 ⫃¡[h, g] ⋆ → L1 = ⋆.
/2 width=6 by lsubsv_inv_atom2_aux/ qed-.
-fact lsubsv_inv_pair2_aux: ∀h,g,G,L1,L2. G ⊢ L1 ¡⫃[h, g] L2 →
+fact lsubsv_inv_pair2_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
∀I,K2,W. L2 = K2.ⓑ{I}W →
- (∃∃K1. G ⊢ K1 ¡⫃[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
- ∃∃K1,V,l. ⦃G, K1⦄ ⊢ W ¡[h, g] & ⦃G, K1⦄ ⊢ V ¡[h, g] &
- scast h g l G K1 V W & ⦃G, K2⦄ ⊢ W ¡[h, g] &
- ⦃G, K1⦄ ⊢ V ▪[h, g] l+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] l &
- G ⊢ K1 ¡⫃[h, g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
+ (∃∃K1. G ⊢ K1 ⫃¡[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
+ ∃∃K1,V,d1. ⦃G, K1⦄ ⊢ ⓝW.V ¡[h, g, d1] & ⦃G, K2⦄ ⊢ W ¡[h, g] &
+ ⦃G, K1⦄ ⊢ V ▪[h, g] d1+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d1 &
+ G ⊢ K1 ⫃¡[h, g] K2 & I = Abst & L1 = K1.ⓓⓝW.V.
#h #g #G #L1 #L2 * -L1 -L2
[ #J #K2 #U #H destruct
-| #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3/
-| #L1 #L2 #W #V #l #H1W #HV #HVW #H2W #H1l #H2l #HL12 #J #K2 #U #H destruct /3 width=10/
+| #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,G,L1,K2,W. G ⊢ L1 ¡⫃[h, g] K2.ⓑ{I}W →
- (∃∃K1. G ⊢ K1 ¡⫃[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
- ∃∃K1,V,l. ⦃G, K1⦄ ⊢ W ¡[h, g] & ⦃G, K1⦄ ⊢ V ¡[h, g] &
- scast h g l G K1 V W & ⦃G, K2⦄ ⊢ W ¡[h, g] &
- ⦃G, K1⦄ ⊢ V ▪[h, g] l+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] l &
- G ⊢ K1 ¡⫃[h, g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
+lemma lsubsv_inv_pair2: ∀h,g,I,G,L1,K2,W. G ⊢ L1 ⫃¡[h, g] K2.ⓑ{I}W →
+ (∃∃K1. G ⊢ K1 ⫃¡[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
+ ∃∃K1,V,d1. ⦃G, K1⦄ ⊢ ⓝW.V ¡[h, g, d1] & ⦃G, K2⦄ ⊢ W ¡[h, g] &
+ ⦃G, K1⦄ ⊢ V ▪[h, g] d1+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d1 &
+ G ⊢ K1 ⫃¡[h, g] 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_lsubr: ∀h,g,G,L1,L2. G ⊢ L1 ¡⫃[h, g] L2 → L1 ⫃ L2.
-#h #g #G #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
+lemma lsubsv_fwd_lsubr: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 → L1 ⫃ L2.
+#h #g #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_pair, lsubr_beta/
qed-.
(* Basic properties *********************************************************)
-lemma lsubsv_refl: ∀h,g,G,L. G ⊢ L ¡⫃[h, g] L.
-#h #g #G #L elim L -L // /2 width=1/
+lemma lsubsv_refl: ∀h,g,G,L. G ⊢ L ⫃¡[h, g] L.
+#h #g #G #L elim L -L /2 width=1 by lsubsv_pair/
qed.
-lemma lsubsv_cprs_trans: ∀h,g,G,L1,L2. G ⊢ L1 ¡⫃[h, g] L2 →
+lemma lsubsv_cprs_trans: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] 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,g,G,L1,L2. G ⊢ L1 ¡⫃[h, g] L2 →
- ∀K1,s,e. ⇩[s, 0, e] L1 ≡ K1 →
- ∃∃K2. G ⊢ K1 ¡⫃[h, g] K2 & ⇩[s, 0, e] L2 ≡ K2.
+lemma lsubsv_drop_O1_conf: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
+ ∀K1,s,m. ⬇[s, 0, m] L1 ≡ K1 →
+ ∃∃K2. G ⊢ K1 ⫃¡[h, g] K2 & ⬇[s, 0, m] L2 ≡ K2.
#h #g #G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
-| #I #L1 #L2 #V #_ #IHL12 #K1 #s #e #H
- elim (drop_inv_O1_pair1 … H) -H * #He #HLK1
+| #I #L1 #L2 #V #_ #IHL12 #K1 #s #m #H
+ elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
[ destruct
elim (IHL12 L1 s 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 #l #H1W #HV #HVW #H2W #H1l #H2l #_ #IHL12 #K1 #s #e #H
- elim (drop_inv_O1_pair1 … H) -H * #He #HLK1
+| #L1 #L2 #W #V #d1 #HWV #HW #HVd1 #HWd1 #_ #IHL12 #K1 #s #m #H
+ elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
[ destruct
elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
- <(drop_inv_O2 … H) in HL12; -H /3 width=4 by lsubsv_abbr, drop_pair, ex2_intro/
+ <(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,g,G,L1,L2. G ⊢ L1 ¡⫃[h, g] L2 →
- ∀K2,s, e. ⇩[s, 0, e] L2 ≡ K2 →
- ∃∃K1. G ⊢ K1 ¡⫃[h, g] K2 & ⇩[s, 0, e] L1 ≡ K1.
+lemma lsubsv_drop_O1_trans: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
+ ∀K2,s, m. ⬇[s, 0, m] L2 ≡ K2 →
+ ∃∃K1. G ⊢ K1 ⫃¡[h, g] K2 & ⬇[s, 0, m] L1 ≡ K1.
#h #g #G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
-| #I #L1 #L2 #V #_ #IHL12 #K2 #s #e #H
- elim (drop_inv_O1_pair1 … H) -H * #He #HLK2
+| #I #L1 #L2 #V #_ #IHL12 #K2 #s #m #H
+ elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
[ destruct
elim (IHL12 L2 s 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 #l #H1W #HV #HVW #H2W #H1l #H2l #_ #IHL12 #K2 #s #e #H
- elim (drop_inv_O1_pair1 … H) -H * #He #HLK2
+| #L1 #L2 #W #V #d1 #HWV #HW #HVd1 #HWd1 #_ #IHL12 #K2 #s #m #H
+ elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
[ destruct
elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
- <(drop_inv_O2 … H) in HL12; -H /3 width=4 by lsubsv_abbr, drop_pair, ex2_intro/
+ <(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/
]
]