(* Advanced properties ******************************************************)
(* Basic_2A1: was: aaa_lref *)
-lemma aaa_lref_gen: ∀I,G,K,V,B,i,L. ⬇*[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ⁝ B → ⦃G, L⦄ ⊢ #i ⁝ B.
+lemma aaa_lref_drops: ∀I,G,K,V,B,i,L. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ⁝ B → ⦃G, L⦄ ⊢ #i ⁝ B.
#I #G #K #V #B #i elim i -i
[ #L #H lapply (drops_fwd_isid … H ?) -H //
#H destruct /2 width=1 by aaa_zero/
-| #i #IH #L <uni_succ #H #HB lapply (drops_inv_pair2_isuni_next … H) -H // *
- #Z #Y #X #HY #H destruct /3 width=1 by aaa_lref/
+| #i #IH #L <uni_succ #H #HB lapply (drops_inv_bind2_isuni_next … H) -H // *
+ #Z #Y #HY #H destruct /3 width=1 by aaa_lref/
]
qed.
(* Advanced inversion lemmas ************************************************)
(* Basic_2A1: was: aaa_inv_lref *)
-lemma aaa_inv_lref_gen: ∀G,A,i,L. ⦃G, L⦄ ⊢ #i ⁝ A →
- ∃∃I,K,V. ⬇*[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
+lemma aaa_inv_lref_drops: ∀G,A,i,L. ⦃G, L⦄ ⊢ #i ⁝ A →
+ ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
#G #A #i elim i -i
[ #L #H elim (aaa_inv_zero … H) -H /3 width=5 by drops_refl, ex2_3_intro/
| #i #IH #L #H elim (aaa_inv_lref … H) -H
- #I #K #V #H #HA destruct elim (IH … HA) -IH -HA /3 width=5 by drops_drop, ex2_3_intro/
+ #I #K #H #HA destruct elim (IH … HA) -IH -HA /3 width=5 by drops_drop, ex2_3_intro/
]
qed-.
(* Properties with generic slicing for local environments *******************)
(* Basic_2A1: includes: aaa_lift *)
-lemma aaa_lifts: ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀b,f,L2. ⬇*[b, f] L2 ≡ L1 →
- ∀T2. ⬆*[f] T1 ≡ T2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
-@fqup_wf_ind_eq #G0 #L0 #T0 #IH #G #L1 * *
+(* Note: it should use drops_split_trans_pair2 *)
+lemma aaa_lifts: ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀b,f,L2. ⬇*[b, f] L2 ≘ L1 →
+ ∀T2. ⬆*[f] T1 ≘ T2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
+@(fqup_wf_ind_eq (Ⓣ)) #G0 #L0 #T0 #IH #G #L1 * *
[ #s #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX -b -IH
lapply (aaa_inv_sort … H) -H #H destruct
>(lifts_inv_sort1 … HX) -HX //
| #i1 #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX
- elim (aaa_inv_lref_gen … H) -H #J #K1 #V1 #HLK1 #HA
+ elim (aaa_inv_lref_drops … H) -H #J #K1 #V1 #HLK1 #HA
elim (lifts_inv_lref1 … HX) -HX #i2 #Hf #H destruct
lapply (drops_trans … HL21 … HLK1 ??) -HL21 [1,2: // ] #H
elim (drops_split_trans … H) -H [ |*: /2 width=6 by after_uni_dx/ ] #Y #HLK2 #HY
- lapply (drops_inv_tls_at … Hf … HY) -HY #HY -Hf
- elim (drops_inv_skip2 … HY) -HY #K2 #V2 #HK21 #HV12 #H destruct
- /4 width=12 by aaa_lref_gen, fqup_lref, drops_inv_gen/
+ lapply (drops_tls_at … Hf … HY) -HY #HY -Hf
+ elim (drops_inv_skip2 … HY) -HY #Z #K2 #HK21 #HZ #H destruct
+ elim (liftsb_inv_pair_sn … HZ) -HZ #V2 #HV12 #H destruct
+ /4 width=12 by aaa_lref_drops, fqup_lref, drops_inv_gen/
| #l #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX -b -f -IH
elim (aaa_inv_gref … H)
| #p * #V1 #T1 #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX
[ elim (aaa_inv_abbr … H) -H #B #HB #HA
elim (lifts_inv_bind1 … HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
- /4 width=9 by aaa_abbr, drops_skip/
+ /5 width=9 by aaa_abbr, drops_skip, ext2_pair/
| elim (aaa_inv_abst … H) -H #B #A0 #HB #HA #H0
- elim (lifts_inv_bind1 … HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
- /4 width=8 by aaa_abst, drops_skip/
+ elim (lifts_inv_bind1 … HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
+ /5 width=8 by aaa_abst, drops_skip, ext2_pair/
]
| * #V1 #T1 #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX
[ elim (aaa_inv_appl … H) -H #B #HB #HA
(* Inversion lemmas with generic slicing for local environments *************)
(* Basic_2A1: includes: aaa_inv_lift *)
-lemma aaa_inv_lifts: â\88\80G,L2,T2,A. â¦\83G, L2â¦\84 â\8a¢ T2 â\81\9d A â\86\92 â\88\80b,f,L1. â¬\87*[b, f] L2 â\89¡ L1 →
- â\88\80T1. â¬\86*[f] T1 â\89¡ T2 → ⦃G, L1⦄ ⊢ T1 ⁝ A.
-@fqup_wf_ind_eq #G0 #L0 #T0 #IH #G #L2 * *
+lemma aaa_inv_lifts: â\88\80G,L2,T2,A. â¦\83G, L2â¦\84 â\8a¢ T2 â\81\9d A â\86\92 â\88\80b,f,L1. â¬\87*[b, f] L2 â\89\98 L1 →
+ â\88\80T1. â¬\86*[f] T1 â\89\98 T2 → ⦃G, L1⦄ ⊢ T1 ⁝ A.
+@(fqup_wf_ind_eq (Ⓣ)) #G0 #L0 #T0 #IH #G #L2 * *
[ #s #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX -b -IH
lapply (aaa_inv_sort … H) -H #H destruct
>(lifts_inv_sort2 … HX) -HX //
| #i2 #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX
- elim (aaa_inv_lref_gen … H) -H #J #K2 #V2 #HLK2 #HA
+ elim (aaa_inv_lref_drops … H) -H #J #K2 #V2 #HLK2 #HA
elim (lifts_inv_lref2 … HX) -HX #i1 #Hf #H destruct
lapply (drops_split_div … HL21 (𝐔❴i1❵) ???) -HL21 [4: * |*: // ] #Y #HLK1 #HY
lapply (drops_conf … HLK2 … HY ??) -HY [1,2: /2 width=6 by after_uni_dx/ ] #HY
- lapply (drops_inv_tls_at … Hf … HY) -HY #HY -Hf
- elim (drops_inv_skip1 … HY) -HY #K1 #V1 #HK21 #HV12 #H destruct
- /4 width=12 by aaa_lref_gen, fqup_lref, drops_inv_F/
+ lapply (drops_tls_at … Hf … HY) -HY #HY -Hf
+ elim (drops_inv_skip1 … HY) -HY #Z #K1 #HK21 #HZ #H destruct
+ elim (liftsb_inv_pair_dx … HZ) -HZ #V1 #HV12 #H destruct
+ /4 width=12 by aaa_lref_drops, fqup_lref, drops_inv_F/
| #l #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX -IH -b -f
elim (aaa_inv_gref … H)
| #p * #V2 #T2 #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX
[ elim (aaa_inv_abbr … H) -H #B #HB #HA
elim (lifts_inv_bind2 … HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
- /4 width=9 by aaa_abbr, drops_skip/
+ /5 width=9 by aaa_abbr, drops_skip, ext2_pair/
| elim (aaa_inv_abst … H) -H #B #A0 #HB #HA #H0
elim (lifts_inv_bind2 … HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
- /4 width=8 by aaa_abst, drops_skip/
+ /5 width=8 by aaa_abst, drops_skip, ext2_pair/
]
| * #V2 #T2 #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX
[ elim (aaa_inv_appl … H) -H #B #HB #HA
projects / helm.git / blobdiff