(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
(* Note: this is not transitive *)
-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_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)
+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)"
- 'LRSubEqV h g G L1 L2 = (lsubsv h g 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,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 → L1 = ⋆ → L2 = ⋆.
-#h #g #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 #W #V #d1 #_ #_ #_ #_ #_ #H destruct
]
qed-.
-lemma lsubsv_inv_atom1: ∀h,g,G,L2. G ⊢ ⋆ ⫃¡[h, g] L2 → L2 = ⋆.
+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,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
+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, 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 &
+ (∃∃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 #g #G #L1 #L2 * -L1 -L2
+#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,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 &
+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,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 → L2 = ⋆ → L1 = ⋆.
-#h #g #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 #W #V #d1 #_ #_ #_ #_ #_ #H destruct
]
qed-.
-lemma lsubsv_inv_atom2: ∀h,g,G,L1. G ⊢ L1 ⫃¡[h, g] ⋆ → L1 = ⋆.
+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,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
+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, 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
+ (∃∃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,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.
+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 *****************************************************)
-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/
+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,G,L. G ⊢ L ⫃¡[h, g] L.
-#h #g #G #L elim L -L /2 width=1 by lsubsv_pair/
+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,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
+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,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
+lemma lsubsv_drop_O1_conf: ∀h,o,G,L1,L2. G ⊢ L1 ⫃¡[h, o] L2 →
+ ∀K1,c,k. ⬇[c, 0, k] L1 ≡ K1 →
+ ∃∃K2. G ⊢ K1 ⫃¡[h, o] K2 & ⬇[c, 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 #s #m #H
+| #I #L1 #L2 #V #_ #IHL12 #K1 #c #k #H
elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
[ destruct
- elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+ elim (IHL12 L1 c 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 #s #m #H
+| #L1 #L2 #W #V #d1 #HWV #HW #HVd1 #HWd1 #_ #IHL12 #K1 #c #k #H
elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
[ destruct
- elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+ elim (IHL12 L1 c 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,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
+lemma lsubsv_drop_O1_trans: ∀h,o,G,L1,L2. G ⊢ L1 ⫃¡[h, o] L2 →
+ ∀K2,c, k. ⬇[c, 0, k] L2 ≡ K2 →
+ ∃∃K1. G ⊢ K1 ⫃¡[h, o] K2 & ⬇[c, 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 #s #m #H
+| #I #L1 #L2 #V #_ #IHL12 #K2 #c #k #H
elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
[ destruct
- elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+ elim (IHL12 L2 c 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 #s #m #H
+| #L1 #L2 #W #V #d1 #HWV #HW #HVd1 #HWd1 #_ #IHL12 #K2 #c #k #H
elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
[ destruct
- elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+ elim (IHL12 L2 c 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/
]