(* *)
(**************************************************************************)
-include "basic_2/notation/relations/crsubeqv_4.ma".
+include "basic_2/notation/relations/lrsubeqv_5.ma".
include "basic_2/dynamic/snv.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,W,V,W1,V2,l. ⦃h, L1⦄ ⊢ ⓝW.V ¡[g] → ⦃h, L2⦄ ⊢ W ¡[g] →
- ⦃h, L1⦄ ⊢ V •[g] ⦃l+1, W1⦄ → ⦃h, L2⦄ ⊢ W •[g] ⦃l, V2⦄ →
- lsubsv h g L1 L2 → lsubsv h g (L1.ⓓⓝW.V) (L2.ⓛW)
+inductive lsubsv (h) (g) (G): relation lenv ≝
+| 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 h g G L1 L2 → lsubsv h g 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 g G L1 L2 = (lsubsv h g 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,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 #V1 #V2 #l #_ #_ #_ #_ #_ #H destruct
+| #L1 #L2 #W #V #l #_ #_ #_ #_ #_ #_ #_ #H destruct
]
qed-.
-lemma lsubsv_inv_atom1: ∀h,g,L2. h ⊢ ⋆ ¡⊑[g] L2 → L2 = ⋆.
-/2 width=5 by lsubsv_inv_atom1_aux/ qed-.
+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,L1,L2. h ⊢ L1 ¡⊑[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. h ⊢ K1 ¡⊑[g] K2 & L2 = K2.ⓑ{I}X) ∨
- ∃∃K2,W,V,W1,V2,l. ⦃h, K1⦄ ⊢ X ¡[g] & ⦃h, K2⦄ ⊢ W ¡[g] &
- ⦃h, K1⦄ ⊢ V •[g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W •[g] ⦃l, V2⦄ &
- h ⊢ K1 ¡⊑[g] K2 &
- I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
-#h #g #L1 #L2 * -L1 -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.
+#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 #W1 #V2 #l #HV #HW #HW1 #HV2 #HL12 #J #K1 #X #H destruct /3 width=12/
+| #L1 #L2 #W #V #l #H1W #HV #HVW #H2W #H1l #H2l #HL12 #J #K1 #X #H destruct /3 width=13/
]
qed-.
-lemma lsubsv_inv_pair1: ∀h,g,I,K1,L2,X. h ⊢ K1.ⓑ{I}X ¡⊑[g] L2 →
- (∃∃K2. h ⊢ K1 ¡⊑[g] K2 & L2 = K2.ⓑ{I}X) ∨
- ∃∃K2,W,V,W1,V2,l. ⦃h, K1⦄ ⊢ X ¡[g] & ⦃h, K2⦄ ⊢ W ¡[g] &
- ⦃h, K1⦄ ⊢ V •[g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W •[g] ⦃l, V2⦄ &
- h ⊢ K1 ¡⊑[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,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.
/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,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 #V1 #V2 #l #_ #_ #_ #_ #_ #H destruct
+| #L1 #L2 #W #V #l #_ #_ #_ #_ #_ #_ #_ #H destruct
]
qed-.
-lemma lsubsv_inv_atom2: ∀h,g,L1. h ⊢ L1 ¡⊑[g] ⋆ → L1 = ⋆.
-/2 width=5 by lsubsv_inv_atom2_aux/ qed-.
+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,L1,L2. h ⊢ L1 ¡⊑[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. h ⊢ K1 ¡⊑[g] K2 & L1 = K1.ⓑ{I}W) ∨
- ∃∃K1,V,W1,V2,l. ⦃h, K1⦄ ⊢ ⓝW.V ¡[g] & ⦃h, K2⦄ ⊢ W ¡[g] &
- ⦃h, K1⦄ ⊢ V •[g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W •[g] ⦃l, V2⦄ &
- h ⊢ K1 ¡⊑[g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
-#h #g #L1 #L2 * -L1 -L2
+ (∃∃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.
+#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 #W1 #V2 #l #HV #HW #HW1 #HV2 #HL12 #J #K2 #U #H destruct /3 width=10/
+| #L1 #L2 #W #V #l #H1W #HV #HVW #H2W #H1l #H2l #HL12 #J #K2 #U #H destruct /3 width=10/
]
qed-.
-lemma lsubsv_inv_pair2: ∀h,g,I,L1,K2,W. h ⊢ L1 ¡⊑[g] K2.ⓑ{I}W →
- (∃∃K1. h ⊢ K1 ¡⊑[g] K2 & L1 = K1.ⓑ{I}W) ∨
- ∃∃K1,V,W1,V2,l. ⦃h, K1⦄ ⊢ ⓝW.V ¡[g] & ⦃h, K2⦄ ⊢ W ¡[g] &
- ⦃h, K1⦄ ⊢ V •[g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W •[g] ⦃l, V2⦄ &
- h ⊢ K1 ¡⊑[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,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.
/2 width=3 by lsubsv_inv_pair2_aux/ qed-.
(* Basic_forward lemmas *****************************************************)
-lemma lsubsv_fwd_lsubx: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 → L1 ⓝ⊑ L2.
-#h #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/
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,g,G,L. G ⊢ L ¡⫃[h, g] L.
+#h #g #G #L elim L -L // /2 width=1/
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_lsubx, lsubx_cprs_trans/
+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_ldrop_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.
+#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 (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
+ [ destruct
+ elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsubsv_pair, ldrop_pair, ex2_intro/
+ | elim (IHL12 … HLK1) -L1 /3 width=3 by ldrop_drop_lt, ex2_intro/
+ ]
+| #L1 #L2 #W #V #l #H1W #HV #HVW #H2W #H1l #H2l #_ #IHL12 #K1 #s #e #H
+ elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
+ [ destruct
+ elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_O2 … H) in HL12; -H /3 width=4 by lsubsv_abbr, ldrop_pair, ex2_intro/
+ | elim (IHL12 … HLK1) -L1 /3 width=3 by ldrop_drop_lt, ex2_intro/
+ ]
+]
+qed-.
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsubsv_ldrop_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.
+#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 (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
+ [ destruct
+ elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsubsv_pair, ldrop_pair, ex2_intro/
+ | elim (IHL12 … HLK2) -L2 /3 width=3 by ldrop_drop_lt, ex2_intro/
+ ]
+| #L1 #L2 #W #V #l #H1W #HV #HVW #H2W #H1l #H2l #_ #IHL12 #K2 #s #e #H
+ elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
+ [ destruct
+ elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_O2 … H) in HL12; -H /3 width=4 by lsubsv_abbr, ldrop_pair, ex2_intro/
+ | elim (IHL12 … HLK2) -L2 /3 width=3 by ldrop_drop_lt, ex2_intro/
+ ]
+]
qed-.
projects / helm.git / blobdiff