(* *)
(**************************************************************************)
-include "basic_2/computation/cpds_lift.ma".
-include "basic_2/equivalence/cpcs_cpcs.ma".
+include "basic_2/multiple/fqus_alt.ma".
+include "basic_2/computation/scpds_lift.ma".
include "basic_2/dynamic/snv.ma".
(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
(* Relocation properties ****************************************************)
-lemma snv_lift: ∀h,g,G,K,T. ⦃G, K⦄ ⊢ T ¡[h, g] → ∀L,s,d,e. ⇩[s, d, e] L ≡ K →
- ∀U. ⇧[d, e] T ≡ U → ⦃G, L⦄ ⊢ U ¡[h, g].
-#h #g #G #K #T #H elim H -G -K -T
-[ #G #K #k #L #s #d #e #_ #X #H
- >(lift_inv_sort1 … H) -X -K -d -e //
-| #I #G #K #K0 #V #i #HK0 #_ #IHV #L #s #d #e #HLK #X #H
- elim (lift_inv_lref1 … H) * #Hid #H destruct
- [ elim (ldrop_trans_le … HLK … HK0) -K /2 width=2 by lt_to_le/ #X #HL0 #H
- elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #L0 #W #HLK0 #HVW #H destruct
+| nv_lref: ∀I,G,L,K,V,i. ⬇[i] L ≘ K.ⓑ{I}V → nv a h G K V → nv a h G L (#i)
+
+lemma nv_inv_lref: ∀a,h,G,L,i. ⦃G, L⦄ ⊢ #i ![a, h] →
+ ∃∃I,K,V. ⬇[i] L ≘ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ![a, h].
+/2 width=3 by nv_inv_lref_aux/ qed-.
+
+
+lemma snv_lift: ∀h,o,G,K,T. ⦃G, K⦄ ⊢ T ¡[h, o] → ∀L,b,l,k. ⬇[b, l, k] L ≘ K →
+ ∀U. ⬆[l, k] T ≘ U → ⦃G, L⦄ ⊢ U ¡[h, o].
+#h #o #G #K #T #H elim H -G -K -T
+[ #G #K #s #L #b #l #k #_ #X #H
+ >(lift_inv_sort1 … H) -X -K -l -k //
+| #I #G #K #K0 #V #i #HK0 #_ #IHV #L #b #l #k #HLK #X #H
+ elim (lift_inv_lref1 … H) * #Hil #H destruct
+ [ elim (drop_trans_le … HLK … HK0) -K /2 width=2 by ylt_fwd_le/ #X #HL0 #H
+ elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #L0 #W #HLK0 #HVW #H destruct
/3 width=9 by snv_lref/
- | lapply (ldrop_trans_ge … HLK … HK0 ?) -K
- /3 width=9 by snv_lref, ldrop_inv_gen/
+ | lapply (drop_trans_ge … HLK … HK0 ?) -K
+ /3 width=9 by snv_lref, drop_inv_gen/
]
-| #a #I #G #K #V #T #_ #_ #IHV #IHT #L #s #d #e #HLK #X #H
+| #a #I #G #K #V #T #_ #_ #IHV #IHT #L #b #l #k #HLK #X #H
elim (lift_inv_bind1 … H) -H #W #U #HVW #HTU #H destruct
- /4 width=5 by snv_bind, ldrop_skip/
-| #a #G #K #V #V0 #V1 #T #T1 #l #_ #_ #Hl #HV0 #HV01 #HT1 #IHV #IHT #L #s #d #e #HLK #X #H
+ /4 width=5 by snv_bind, drop_skip/
+| #a #G #K #V #W0 #T #U0 #d #_ #_ #HVW0 #HTU0 #IHV #IHT #L #b #l #k #HLK #X #H
elim (lift_inv_flat1 … H) -H #W #U #HVW #HTU #H destruct
- elim (lift_total V0 d e) #W0 #HVW0
- elim (lift_total V1 d e) #W1 #HVW1
- elim (lift_total T1 (d+1) e) #U1 #HTU1
- @(snv_appl … a … W0 … W1 … U1 l)
- [1,2,3,4,5: /2 width=10 by cprs_lift, ssta_lift, da_lift/ ]
- @(cpds_lift … HT1 … HLK … HTU) /2 width=1 by lift_bind/ (**) (* full auto raises typecjhecker failure *)
-| #G #K #V0 #T #V #l #_ #_ #Hl #HTV #HV0 #IHV0 #IHT #L #s #d #e #HLK #X #H
- elim (lift_inv_flat1 … H) -H #W0 #U #HVW0 #HTU #H destruct
- elim (lift_total V d e)
- /3 width=12 by snv_cast, cpcs_lift, ssta_lift, da_lift/
+ elim (lift_total W0 l k)
+ elim (lift_total U0 (l+1) k)
+ /4 width=17 by snv_appl, scpds_lift, lift_bind/
+| #G #K #V #T #U0 #_ #_ #HVU0 #HTU0 #IHV #IHT #L #b #l #k #HLK #X #H
+ elim (lift_inv_flat1 … H) -H #W #U #HVW #HTU #H destruct
+ elim (lift_total U0 l k)
+ /3 width=12 by snv_cast, scpds_lift/
]
qed.
-lemma snv_inv_lift: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ U ¡[h, g] → ∀K,s,d,e. ⇩[s, d, e] L ≡ K →
- â\88\80T. â\87§[d, e] T â\89¡ U â\86\92 â¦\83G, Kâ¦\84 â\8a¢ T ¡[h, g].
-#h #g #G #L #U #H elim H -G -L -U
-[ #G #L #k #K #s #d #e #_ #X #H
- >(lift_inv_sort2 … H) -X -L -d -e //
-| #I #G #L #L0 #W #i #HL0 #_ #IHW #K #s #d #e #HLK #X #H
- elim (lift_inv_lref2 … H) * #Hid #H destruct
- [ elim (ldrop_conf_le … HLK … HL0) -L /2 width=2 by lt_to_le/ #X #HK0 #H
- elim (ldrop_inv_skip1 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K0 #V #HLK0 #HVW #H destruct
+lemma snv_inv_lift: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ U ¡[h, o] → ∀K,b,l,k. ⬇[b, l, k] L ≘ K →
+ â\88\80T. â¬\86[l, k] T â\89\98 U â\86\92 â¦\83G, Kâ¦\84 â\8a¢ T ¡[h, o].
+#h #o #G #L #U #H elim H -G -L -U
+[ #G #L #s #K #b #l #k #_ #X #H
+ >(lift_inv_sort2 … H) -X -L -l -k //
+| #I #G #L #L0 #W #i #HL0 #_ #IHW #K #b #l #k #HLK #X #H
+ elim (lift_inv_lref2 … H) * #Hil #H destruct
+ [ elim (drop_conf_le … HLK … HL0) -L /2 width=2 by ylt_fwd_le/ #X #HK0 #H
+ elim (drop_inv_skip1 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K0 #V #HLK0 #HVW #H destruct
/3 width=12 by snv_lref/
- | lapply (ldrop_conf_ge … HLK … HL0 ?) -L /3 width=9 by snv_lref/
+ | lapply (drop_conf_ge … HLK … HL0 ?) -L /3 width=9 by snv_lref/
]
-| #a #I #G #L #W #U #_ #_ #IHW #IHU #K #s #d #e #HLK #X #H
+| #a #I #G #L #W #U #_ #_ #IHW #IHU #K #b #l #k #HLK #X #H
elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct
- /4 width=5 by snv_bind, ldrop_skip/
-| #a #G #L #W #W0 #W1 #U #U1 #l #_ #_ #Hl #HW0 #HW01 #HU1 #IHW #IHU #K #s #d #e #HLK #X #H
+ /4 width=5 by snv_bind, drop_skip/
+| #a #G #L #W #W1 #U #U1 #d #_ #_ #HW1 #HU1 #IHW #IHU #K #b #l #k #HLK #X #H
+ elim (lift_inv_flat2 … H) -H #V #T #HVW #HTU #H destruct
+ elim (scpds_inv_lift1 … HW1 … HLK … HVW) -HW1 #W0 #HW01 #HVW0
+ elim (scpds_inv_lift1 … HU1 … HLK … HTU) -HU1 #X #H #HTU0
+ elim (lift_inv_bind2 … H) -H #Y #U0 #HY #HU01 #H destruct
+ lapply (lift_inj … HY … HW01) -HY #H destruct
+ /3 width=6 by snv_appl/
+| #G #L #W #U #U1 #_ #_ #HWU1 #HU1 #IHW #IHU #K #b #l #k #HLK #X #H
elim (lift_inv_flat2 … H) -H #V #T #HVW #HTU #H destruct
- lapply (da_inv_lift … Hl … HLK … HVW) -Hl #Hl
- elim (ssta_inv_lift1 … HW0 … HLK … HVW) -HW0 #V0 #HVW0 #HV0
- elim (cprs_inv_lift1 … HW01 … HLK … HVW0) -W0 #V1 #HVW1 #HV01
- elim (cpds_inv_lift1 … HU1 … HLK … HTU) -HU1 #X #H #HTU
- elim (lift_inv_bind2 … H) -H #Y #T1 #HY #HTU1 #H destruct
- lapply (lift_inj … HY … HVW1) -HY #H destruct
- /3 width=8 by snv_appl/
-| #G #L #W0 #U #W #l #_ #_ #Hl #HUW #HW0 #IHW0 #IHU #K #s #d #e #HLK #X #H
- elim (lift_inv_flat2 … H) -H #V0 #T #HVW0 #HTU #H destruct
- lapply (da_inv_lift … Hl … HLK … HTU) -Hl #Hl
- elim (ssta_inv_lift1 … HUW … HLK … HTU) -HUW #V #HVW #HTV
- lapply (cpcs_inv_lift G … HLK … HVW … HVW0 ?) // -W
- /3 width=8 by snv_cast/
+ elim (scpds_inv_lift1 … HWU1 … HLK … HVW) -HWU1 #U0 #HU01 #HVU0
+ elim (scpds_inv_lift1 … HU1 … HLK … HTU) -HU1 #X #HX #HTU0
+ lapply (lift_inj … HX … HU01) -HX #H destruct
+ /3 width=5 by snv_cast/
]
qed-.
-(* Advanced properties ******************************************************)
+(* Properties on subclosure *************************************************)
-lemma snv_fqu_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ T1 ¡[h, g] → ⦃G2, L2⦄ ⊢ T2 ¡[h, g].
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+lemma snv_fqu_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ⦃G1, L1⦄ ⊢ T1 ¡[h, o] → ⦃G2, L2⦄ ⊢ T2 ¡[h, o].
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ #I1 #G1 #L1 #V1 #H
elim (snv_inv_lref … H) -H #I2 #L2 #V2 #H #HV2
- lapply (ldrop_inv_O2 … H) -H #H destruct //
+ lapply (drop_inv_O2 … H) -H #H destruct //
|2: *
|5,6: /3 width=8 by snv_inv_lift/
]
]
]
qed-.
+
+lemma snv_fquq_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ⦃G1, L1⦄ ⊢ T1 ¡[h, o] → ⦃G2, L2⦄ ⊢ T2 ¡[h, o].
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fquq_inv_gen … H) -H [|*]
+/2 width=5 by snv_fqu_conf/
+qed-.
+
+lemma snv_fqup_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+ ⦃G1, L1⦄ ⊢ T1 ¡[h, o] → ⦃G2, L2⦄ ⊢ T2 ¡[h, o].
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+/3 width=5 by fqup_strap1, snv_fqu_conf/
+qed-.
+
+lemma snv_fqus_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ⦃G1, L1⦄ ⊢ T1 ¡[h, o] → ⦃G2, L2⦄ ⊢ T2 ¡[h, o].
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqus_inv_gen … H) -H [|*]
+/2 width=5 by snv_fqup_conf/
+qed-.
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