(* activate genv *)
inductive aaa: relation4 genv lenv term aarity ≝
| aaa_sort: ∀G,L,k. aaa G L (⋆k) (⓪)
-| aaa_lref: ∀I,G,L,K,V,B,i. ⇩[0, i] L ≡ K. ⓑ{I}V → aaa G K V B → aaa G L (#i) B
+| aaa_lref: ∀I,G,L,K,V,B,i. ⇩[i] L ≡ K. ⓑ{I}V → aaa G K V B → aaa G L (#i) B
| aaa_abbr: ∀a,G,L,V,T,B,A.
aaa G L V B → aaa G (L.ⓓV) T A → aaa G L (ⓓ{a}V.T) A
| aaa_abst: ∀a,G,L,V,T,B,A.
/2 width=6 by aaa_inv_sort_aux/ qed-.
fact aaa_inv_lref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀i. T = #i →
- ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I} V & ⦃G, K⦄ ⊢ V ⁝ A.
+ ∃∃I,K,V. ⇩[i] L ≡ K.ⓑ{I} V & ⦃G, K⦄ ⊢ V ⁝ A.
#G #L #T #A * -G -L -T -A
[ #G #L #k #i #H destruct
-| #I #G #L #K #V #B #j #HLK #HB #i #H destruct /2 width=5/
+| #I #G #L #K #V #B #j #HLK #HB #i #H destruct /2 width=5 by ex2_3_intro/
| #a #G #L #V #T #B #A #_ #_ #i #H destruct
| #a #G #L #V #T #B #A #_ #_ #i #H destruct
| #G #L #V #T #B #A #_ #_ #i #H destruct
qed-.
lemma aaa_inv_lref: ∀G,L,A,i. ⦃G, L⦄ ⊢ #i ⁝ A →
- ∃∃I,K,V. ⇩[0, i] L ≡ K. ⓑ{I} V & ⦃G, K⦄ ⊢ V ⁝ A.
+ ∃∃I,K,V. ⇩[i] L ≡ K. ⓑ{I} V & ⦃G, K⦄ ⊢ V ⁝ A.
/2 width=3 by aaa_inv_lref_aux/ qed-.
fact aaa_inv_gref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p. T = §p → ⊥.
#G #L #T #A * -G -L -T -A
[ #G #L #k #a #W #U #H destruct
| #I #G #L #K #V #B #i #_ #_ #a #W #U #H destruct
-| #b #G #L #V #T #B #A #HV #HT #a #W #U #H destruct /2 width=2/
+| #b #G #L #V #T #B #A #HV #HT #a #W #U #H destruct /2 width=2 by ex2_intro/
| #b #G #L #V #T #B #A #_ #_ #a #W #U #H destruct
| #G #L #V #T #B #A #_ #_ #a #W #U #H destruct
| #G #L #V #T #A #_ #_ #a #W #U #H destruct
[ #G #L #k #a #W #U #H destruct
| #I #G #L #K #V #B #i #_ #_ #a #W #U #H destruct
| #b #G #L #V #T #B #A #_ #_ #a #W #U #H destruct
-| #b #G #L #V #T #B #A #HV #HT #a #W #U #H destruct /2 width=5/
+| #b #G #L #V #T #B #A #HV #HT #a #W #U #H destruct /2 width=5 by ex3_2_intro/
| #G #L #V #T #B #A #_ #_ #a #W #U #H destruct
| #G #L #V #T #A #_ #_ #a #W #U #H destruct
]
| #I #G #L #K #V #B #i #_ #_ #W #U #H destruct
| #a #G #L #V #T #B #A #_ #_ #W #U #H destruct
| #a #G #L #V #T #B #A #_ #_ #W #U #H destruct
-| #G #L #V #T #B #A #HV #HT #W #U #H destruct /2 width=3/
+| #G #L #V #T #B #A #HV #HT #W #U #H destruct /2 width=3 by ex2_intro/
| #G #L #V #T #A #_ #_ #W #U #H destruct
]
qed-.
| #a #G #L #V #T #B #A #_ #_ #W #U #H destruct
| #a #G #L #V #T #B #A #_ #_ #W #U #H destruct
| #G #L #V #T #B #A #_ #_ #W #U #H destruct
-| #G #L #V #T #A #HV #HT #W #U #H destruct /2 width=1/
+| #G #L #V #T #A #HV #HT #W #U #H destruct /2 width=1 by conj/
]
qed-.
[ elim (aaa_inv_appl … H)
| elim (aaa_inv_cast … H)
] -H /2 width=2 by ex_intro/
-| /3 width=8 by aaa_inv_lift, ex_intro/
+| /3 width=9 by aaa_inv_lift, ex_intro/
]
qed-.
(* Properties on basic relocation *******************************************)
-lemma aaa_lift: ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀L2,d,e. ⇩[d, e] L2 ≡ L1 →
+lemma aaa_lift: ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀L2,s,d,e. ⇩[s, d, e] L2 ≡ L1 →
∀T2. ⇧[d, e] T1 ≡ T2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
#G #L1 #T1 #A #H elim H -G -L1 -T1 -A
-[ #G #L1 #k #L2 #d #e #_ #T2 #H
+[ #G #L1 #k #L2 #s #d #e #_ #T2 #H
>(lift_inv_sort1 … H) -H //
-| #I #G #L1 #K1 #V1 #B #i #HLK1 #_ #IHB #L2 #d #e #HL21 #T2 #H
+| #I #G #L1 #K1 #V1 #B #i #HLK1 #_ #IHB #L2 #s #d #e #HL21 #T2 #H
elim (lift_inv_lref1 … H) -H * #Hid #H destruct
[ elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
- /3 width=8/
- | lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 /3 width=8 by aaa_lref/
+ /3 width=9 by aaa_lref/
+ | lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1
+ /3 width=9 by aaa_lref, ldrop_inv_gen/
]
-| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #d #e #HL21 #X #H
+| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #s #d #e #HL21 #X #H
elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- /4 width=4 by aaa_abbr, ldrop_skip/
-| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #d #e #HL21 #X #H
+ /4 width=5 by aaa_abbr, ldrop_skip/
+| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #s #d #e #HL21 #X #H
elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- /4 width=4 by aaa_abst, ldrop_skip/
-| #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #d #e #HL21 #X #H
+ /4 width=5 by aaa_abst, ldrop_skip/
+| #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #s #d #e #HL21 #X #H
elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- /3 width=4 by aaa_appl/
-| #G #L1 #V1 #T1 #A #_ #_ #IH1 #IH2 #L2 #d #e #HL21 #X #H
+ /3 width=5 by aaa_appl/
+| #G #L1 #V1 #T1 #A #_ #_ #IH1 #IH2 #L2 #s #d #e #HL21 #X #H
elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- /3 width=4 by aaa_cast/
+ /3 width=5 by aaa_cast/
]
qed.
-lemma aaa_inv_lift: ∀G,L2,T2,A. ⦃G, L2⦄ ⊢ T2 ⁝ A → ∀L1,d,e. ⇩[d, e] L2 ≡ L1 →
+lemma aaa_inv_lift: ∀G,L2,T2,A. ⦃G, L2⦄ ⊢ T2 ⁝ A → ∀L1,s,d,e. ⇩[s, d, e] L2 ≡ L1 →
∀T1. ⇧[d, e] T1 ≡ T2 → ⦃G, L1⦄ ⊢ T1 ⁝ A.
#G #L2 #T2 #A #H elim H -G -L2 -T2 -A
-[ #G #L2 #k #L1 #d #e #_ #T1 #H
+[ #G #L2 #k #L1 #s #d #e #_ #T1 #H
>(lift_inv_sort2 … H) -H //
-| #I #G #L2 #K2 #V2 #B #i #HLK2 #_ #IHB #L1 #d #e #HL21 #T1 #H
+| #I #G #L2 #K2 #V2 #B #i #HLK2 #_ #IHB #L1 #s #d #e #HL21 #T1 #H
elim (lift_inv_lref2 … H) -H * #Hid #H destruct
- [ elim (ldrop_conf_lt … HL21 … HLK2) -L2 /3 width=8 by aaa_lref/
- | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 /3 width=8 by aaa_lref/
+ [ elim (ldrop_conf_lt … HL21 … HLK2) -L2 /3 width=9 by aaa_lref/
+ | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 /3 width=9 by aaa_lref/
]
-| #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #d #e #HL21 #X #H
+| #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #s #d #e #HL21 #X #H
elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
- /4 width=4 by aaa_abbr, ldrop_skip/
-| #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #d #e #HL21 #X #H
+ /4 width=5 by aaa_abbr, ldrop_skip/
+| #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #s #d #e #HL21 #X #H
elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
- /4 width=4 by aaa_abst, ldrop_skip/
-| #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #d #e #HL21 #X #H
+ /4 width=5 by aaa_abst, ldrop_skip/
+| #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #s #d #e #HL21 #X #H
elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
- /3 width=4 by aaa_appl/
-| #G #L2 #V2 #T2 #A #_ #_ #IH1 #IH2 #L1 #d #e #HL21 #X #H
+ /3 width=5 by aaa_appl/
+| #G #L2 #V2 #T2 #A #_ #_ #IH1 #IH2 #L1 #s #d #e #HL21 #X #H
elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
- /3 width=4 by aaa_cast/
+ /3 width=5 by aaa_cast/
]
qed-.
(* Properties concerning generic relocation *********************************)
-lemma aaa_lifts: ∀G,L1,L2,T2,A,des. ⇩*[des] L2 ≡ L1 → ∀T1. ⇧*[des] T1 ≡ T2 →
+lemma aaa_lifts: ∀G,L1,L2,T2,A,s,des. ⇩*[s, des] L2 ≡ L1 → ∀T1. ⇧*[des] T1 ≡ T2 →
⦃G, L1⦄ ⊢ T1 ⁝ A → ⦃G, L2⦄ ⊢ T2 ⁝ A.
-#G #L1 #L2 #T2 #A #des #H elim H -L1 -L2 -des
+#G #L1 #L2 #T2 #A #s #des #H elim H -L1 -L2 -des
[ #L #T1 #H #HT1
<(lifts_inv_nil … H) -H //
| #L1 #L #L2 #des #d #e #_ #HL2 #IHL1 #T1 #H #HT1
- elim (lifts_inv_cons … H) -H /3 width=9/
+ elim (lifts_inv_cons … H) -H /3 width=10 by aaa_lift/
]
qed.
(* activate genv *)
inductive da (h:sh) (g:sd h): relation4 genv lenv term nat ≝
| da_sort: ∀G,L,k,l. deg h g k l → da h g G L (⋆k) l
-| da_ldef: ∀G,L,K,V,i,l. ⇩[0, i] L ≡ K.ⓓV → da h g G K V l → da h g G L (#i) l
-| da_ldec: ∀G,L,K,W,i,l. ⇩[0, i] L ≡ K.ⓛW → da h g G K W l → da h g G L (#i) (l+1)
+| da_ldef: ∀G,L,K,V,i,l. ⇩[i] L ≡ K.ⓓV → da h g G K V l → da h g G L (#i) l
+| da_ldec: ∀G,L,K,W,i,l. ⇩[i] L ≡ K.ⓛW → da h g G K W l → da h g G L (#i) (l+1)
| da_bind: ∀a,I,G,L,V,T,l. da h g G (L.ⓑ{I}V) T l → da h g G L (ⓑ{a,I}V.T) l
| da_flat: ∀I,G,L,V,T,l. da h g G L T l → da h g G L (ⓕ{I}V.T) l
.
/2 width=5 by da_inv_sort_aux/ qed-.
fact da_inv_lref_aux: ∀h,g,G,L,T,l. ⦃G, L⦄ ⊢ T ▪[h, g] l → ∀j. T = #j →
- (∃∃K,V. ⇩[0, j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ▪[h, g] l) ∨
- (∃∃K,W,l0. ⇩[0, j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] l0 &
+ (∃∃K,V. ⇩[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ▪[h, g] l) ∨
+ (∃∃K,W,l0. ⇩[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] l0 &
l = l0 + 1
).
#h #g #G #L #T #l * -G -L -T -l
[ #G #L #k #l #_ #j #H destruct
-| #G #L #K #V #i #l #HLK #HV #j #H destruct /3 width=4/
-| #G #L #K #W #i #l #HLK #HW #j #H destruct /3 width=6/
+| #G #L #K #V #i #l #HLK #HV #j #H destruct /3 width=4 by ex2_2_intro, or_introl/
+| #G #L #K #W #i #l #HLK #HW #j #H destruct /3 width=6 by ex3_3_intro, or_intror/
| #a #I #G #L #V #T #l #_ #j #H destruct
| #I #G #L #V #T #l #_ #j #H destruct
]
qed-.
lemma da_inv_lref: ∀h,g,G,L,j,l. ⦃G, L⦄ ⊢ #j ▪[h, g] l →
- (∃∃K,V. ⇩[0, j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ▪[h, g] l) ∨
- (∃∃K,W,l0. ⇩[0, j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] l0 & l = l0+1).
+ (∃∃K,V. ⇩[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ▪[h, g] l) ∨
+ (∃∃K,W,l0. ⇩[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] l0 & l = l0+1).
/2 width=3 by da_inv_lref_aux/ qed-.
fact da_inv_gref_aux: ∀h,g,G,L,T,l. ⦃G, L⦄ ⊢ T ▪[h, g] l → ∀p0. T = §p0 → ⊥.
(* Properties on relocation *************************************************)
lemma da_lift: ∀h,g,G,L1,T1,l. ⦃G, L1⦄ ⊢ T1 ▪[h, g] l →
- ∀L2,d,e. ⇩[d, e] L2 ≡ L1 → ∀T2. ⇧[d, e] T1 ≡ T2 →
+ ∀L2,s,d,e. ⇩[s, d, e] L2 ≡ L1 → ∀T2. ⇧[d, e] T1 ≡ T2 →
⦃G, L2⦄ ⊢ T2 ▪[h, g] l.
#h #g #G #L1 #T1 #l #H elim H -G -L1 -T1 -l
-[ #G #L1 #k #l #Hkl #L2 #d #e #_ #X #H
- >(lift_inv_sort1 … H) -X /2 width=1/
-| #G #L1 #K1 #V1 #i #l #HLK1 #_ #IHV1 #L2 #d #e #HL21 #X #H
+[ #G #L1 #k #l #Hkl #L2 #s #d #e #_ #X #H
+ >(lift_inv_sort1 … H) -X /2 width=1 by da_sort/
+| #G #L1 #K1 #V1 #i #l #HLK1 #_ #IHV1 #L2 #s #d #e #HL21 #X #H
elim (lift_inv_lref1 … H) * #Hid #H destruct
- [ elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2/ #X #HLK2 #H
- elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
- /3 width=7/
- | lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=7/
+ [ elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+ elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
+ /3 width=9 by da_ldef/
+ | lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1
+ /3 width=8 by da_ldef, ldrop_inv_gen/
]
-| #G #L1 #K1 #W1 #i #l #HLK1 #_ #IHW1 #L2 #d #e #HL21 #X #H
+| #G #L1 #K1 #W1 #i #l #HLK1 #_ #IHW1 #L2 #s #d #e #HL21 #X #H
elim (lift_inv_lref1 … H) * #Hid #H destruct
- [ elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2/ #X #HLK2 #H
- elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #K2 #W2 #HK21 #HW12 #H destruct
- /3 width=7/
- | lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=7/
+ [ elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+ elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #W2 #HK21 #HW12 #H destruct
+ /3 width=8 by da_ldec/
+ | lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1
+ /3 width=8 by da_ldec, ldrop_inv_gen/
]
-| #a #I #G #L1 #V1 #T1 #l #_ #IHT1 #L2 #d #e #HL21 #X #H
- elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct /4 width=4/
-| #I #G #L1 #V1 #T1 #l #_ #IHT1 #L2 #d #e #HL21 #X #H
- elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct /3 width=4/
+| #a #I #G #L1 #V1 #T1 #l #_ #IHT1 #L2 #s #d #e #HL21 #X #H
+ elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct
+ /4 width=5 by da_bind, ldrop_skip/
+| #I #G #L1 #V1 #T1 #l #_ #IHT1 #L2 #s #d #e #HL21 #X #H
+ elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct
+ /3 width=5 by da_flat/
]
qed.
(* Inversion lemmas on relocation *******************************************)
lemma da_inv_lift: ∀h,g,G,L2,T2,l. ⦃G, L2⦄ ⊢ T2 ▪[h, g] l →
- ∀L1,d,e. ⇩[d, e] L2 ≡ L1 → ∀T1. ⇧[d, e] T1 ≡ T2 →
+ ∀L1,s,d,e. ⇩[s, d, e] L2 ≡ L1 → ∀T1. ⇧[d, e] T1 ≡ T2 →
⦃G, L1⦄ ⊢ T1 ▪[h, g] l.
#h #g #G #L2 #T2 #l #H elim H -G -L2 -T2 -l
-[ #G #L2 #k #l #Hkl #L1 #d #e #_ #X #H
- >(lift_inv_sort2 … H) -X /2 width=1/
-| #G #L2 #K2 #V2 #i #l #HLK2 #HV2 #IHV2 #L1 #d #e #HL21 #X #H
+[ #G #L2 #k #l #Hkl #L1 #s #d #e #_ #X #H
+ >(lift_inv_sort2 … H) -X /2 width=1 by da_sort/
+| #G #L2 #K2 #V2 #i #l #HLK2 #HV2 #IHV2 #L1 #s #d #e #HL21 #X #H
elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HV2 | -IHV2 ]
- [ elim (ldrop_conf_lt … HL21 … HLK2) -L2 // /3 width=7/
- | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 // /2 width=4/
+ [ elim (ldrop_conf_lt … HL21 … HLK2) -L2 /3 width=8 by da_ldef/
+ | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 /2 width=4 by da_ldef/
]
-| #G #L2 #K2 #W2 #i #l #HLK2 #HW2 #IHW2 #L1 #d #e #HL21 #X #H
+| #G #L2 #K2 #W2 #i #l #HLK2 #HW2 #IHW2 #L1 #s #d #e #HL21 #X #H
elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HW2 | -IHW2 ]
- [ elim (ldrop_conf_lt … HL21 … HLK2) -L2 // /3 width=7/
- | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 // /2 width=4/
+ [ elim (ldrop_conf_lt … HL21 … HLK2) -L2 /3 width=8 by da_ldec/
+ | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 /2 width=4 by da_ldec/
]
-| #a #I #G #L2 #V2 #T2 #l #_ #IHT2 #L1 #d #e #HL21 #X #H
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct /4 width=4/
-| #I #G #L2 #V2 #T2 #l #_ #IHT2 #L1 #d #e #HL21 #X #H
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct /3 width=4/
+| #a #I #G #L2 #V2 #T2 #l #_ #IHT2 #L1 #s #d #e #HL21 #X #H
+ elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+ /4 width=5 by da_bind, ldrop_skip/
+| #I #G #L2 #V2 #T2 #l #_ #IHT2 #L1 #s #d #e #HL21 #X #H
+ elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+ /3 width=5 by da_flat/
]
qed-.
(* Properties concerning basic local environment slicing ********************)
(* Note: the constant 0 cannot be generalized *)
-lemma lsuba_ldrop_O1_conf: ∀G,L1,L2. G ⊢ L1 ⁝⊑ L2 → ∀K1,e. ⇩[0, e] L1 ≡ K1 →
- ∃∃K2. G ⊢ K1 ⁝⊑ K2 & ⇩[0, e] L2 ≡ K2.
+lemma lsuba_ldrop_O1_conf: ∀G,L1,L2. G ⊢ L1 ⁝⊑ L2 → ∀K1,s,e. ⇩[s, 0, e] L1 ≡ K1 →
+ ∃∃K2. G ⊢ K1 ⁝⊑ K2 & ⇩[s, 0, e] L2 ≡ K2.
#G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3/
-| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H
+| #I #L1 #L2 #V #_ #IHL12 #K1 #s #e #H
elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
[ destruct
- elim (IHL12 L1 0) -IHL12 // #X #HL12 #H
- <(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
- | elim (IHL12 … HLK1) -L1 /3 width=3/
+ elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_pair, ldrop_pair, ex2_intro/
+ | elim (IHL12 … HLK1) -L1 /3 width=3 by ldrop_drop_lt, ex2_intro/
]
-| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K1 #e #H
+| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K1 #s #e #H
elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
[ destruct
- elim (IHL12 L1 0) -IHL12 // #X #HL12 #H
- <(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
- | elim (IHL12 … HLK1) -L1 /3 width=3/
+ elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_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 lsuba_ldrop_O1_trans: ∀G,L1,L2. G ⊢ L1 ⁝⊑ L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
- ∃∃K1. G ⊢ K1 ⁝⊑ K2 & ⇩[0, e] L1 ≡ K1.
+lemma lsuba_ldrop_O1_trans: ∀G,L1,L2. G ⊢ L1 ⁝⊑ L2 → ∀K2,s,e. ⇩[s, 0, e] L2 ≡ K2 →
+ ∃∃K1. G ⊢ K1 ⁝⊑ K2 & ⇩[s, 0, e] L1 ≡ K1.
#G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3/
-| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H
+| #I #L1 #L2 #V #_ #IHL12 #K2 #s #e #H
elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
[ destruct
- elim (IHL12 L2 0) -IHL12 // #X #HL12 #H
- <(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
- | elim (IHL12 … HLK2) -L2 /3 width=3/
+ elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_pair, ldrop_pair, ex2_intro/
+ | elim (IHL12 … HLK2) -L2 /3 width=3 by ldrop_drop_lt, ex2_intro/
]
-| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K2 #e #H
+| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K2 #s #e #H
elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
[ destruct
- elim (IHL12 L2 0) -IHL12 // #X #HL12 #H
- <(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
- | elim (IHL12 … HLK2) -L2 /3 width=3/
+ elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_abbr, ldrop_pair, ex2_intro/
+ | elim (IHL12 … HLK2) -L2 /3 width=3 by ldrop_drop_lt, ex2_intro/
]
]
qed-.
(* Basic_forward lemmas *****************************************************)
lemma lsubd_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/
+#h #g #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_bind, lsubr_abst/
qed-.
(* Basic inversion lemmas ***************************************************)
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 #HV #HW #HL12 #J #K1 #X #H destruct /3 width=9/
+| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #l #HV #HW #HL12 #J #K1 #X #H destruct /3 width=9 by ex6_4_intro, or_intror/
]
qed-.
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 #HV #HW #HL12 #J #K2 #U #H destruct /3 width=7/
+| #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #l #HV #HW #HL12 #J #K2 #U #H destruct /3 width=7 by ex5_3_intro, or_intror/
]
qed-.
(* Basic properties *********************************************************)
lemma lsubd_refl: ∀h,g,G,L. G ⊢ L ▪⊑[h, g] L.
-#h #g #G #L elim L -L // /2 width=1/
+#h #g #G #L elim L -L /2 width=1 by lsubd_pair/
qed.
(* Note: the constant 0 cannot be generalized *)
lemma lsubd_ldrop_O1_conf: ∀h,g,G,L1,L2. G ⊢ L1 ▪⊑[h, g] L2 →
- ∀K1,e. ⇩[0, e] L1 ≡ K1 →
- ∃∃K2. G ⊢ K1 ▪⊑[h, g] K2 & ⇩[0, e] L2 ≡ K2.
+ ∀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/
-| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H
+[ /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 0) -IHL12 // #X #HL12 #H
- <(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
- | elim (IHL12 … HLK1) -L1 /3 width=3/
+ elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_pair, ldrop_pair, ex2_intro/
+ | elim (IHL12 … HLK1) -L1 /3 width=3 by ldrop_drop_lt, ex2_intro/
]
-| #L1 #L2 #W #V #l #HV #HW #_ #IHL12 #K1 #e #H
+| #L1 #L2 #W #V #l #HV #HW #_ #IHL12 #K1 #s #e #H
elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
[ destruct
- elim (IHL12 L1 0) -IHL12 // #X #HL12 #H
- <(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
- | elim (IHL12 … HLK1) -L1 /3 width=3/
+ elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_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 lsubd_ldrop_O1_trans: ∀h,g,G,L1,L2. G ⊢ L1 ▪⊑[h, g] L2 →
- ∀K2,e. ⇩[0, e] L2 ≡ K2 →
- ∃∃K1. G ⊢ K1 ▪⊑[h, g] K2 & ⇩[0, e] L1 ≡ K1.
+ ∀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/
-| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H
+[ /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 0) -IHL12 // #X #HL12 #H
- <(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
- | elim (IHL12 … HLK2) -L2 /3 width=3/
+ elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_pair, ldrop_pair, ex2_intro/
+ | elim (IHL12 … HLK2) -L2 /3 width=3 by ldrop_drop_lt, ex2_intro/
]
-| #L1 #L2 #W #V #l #HV #HW #_ #IHL12 #K2 #e #H
+| #L1 #L2 #W #V #l #HV #HW #_ #IHL12 #K2 #s #e #H
elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
[ destruct
- elim (IHL12 L2 0) -IHL12 // #X #HL12 #H
- <(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
- | elim (IHL12 … HLK2) -L2 /3 width=3/
+ elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_abbr, ldrop_pair, ex2_intro/
+ | elim (IHL12 … HLK2) -L2 /3 width=3 by ldrop_drop_lt, ex2_intro/
]
]
qed-.
(* activate genv *)
inductive ssta (h) (g): relation4 genv lenv term term ≝
| ssta_sort: ∀G,L,k. ssta h g G L (⋆k) (⋆(next h k))
-| ssta_ldef: ∀G,L,K,V,U,W,i. ⇩[0, i] L ≡ K.ⓓV → ssta h g G K V W →
+| ssta_ldef: ∀G,L,K,V,U,W,i. ⇩[i] L ≡ K.ⓓV → ssta h g G K V W →
⇧[0, i + 1] W ≡ U → ssta h g G L (#i) U
-| ssta_ldec: ∀G,L,K,W,U,l,i. ⇩[0, i] L ≡ K.ⓛW → ⦃G, K⦄ ⊢ W ▪[h, g] l →
+| ssta_ldec: ∀G,L,K,W,U,l,i. ⇩[i] L ≡ K.ⓛW → ⦃G, K⦄ ⊢ W ▪[h, g] l →
⇧[0, i + 1] W ≡ U → ssta h g G L (#i) U
| ssta_bind: ∀a,I,G,L,V,T,U. ssta h g G (L.ⓑ{I}V) T U →
ssta h g G L (ⓑ{a,I}V.T) (ⓑ{a,I}V.U)
/2 width=6 by ssta_inv_sort1_aux/ qed-.
fact ssta_inv_lref1_aux: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h, g] U → ∀j. T = #j →
- (∃∃K,V,W. ⇩[0, j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •[h, g] W &
+ (∃∃K,V,W. ⇩[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •[h, g] W &
⇧[0, j + 1] W ≡ U
) ∨
- (∃∃K,W,l. ⇩[0, j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] l &
+ (∃∃K,W,l. ⇩[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] l &
⇧[0, j + 1] W ≡ U
).
#h #g #G #L #T #U * -G -L -T -U
[ #G #L #k #j #H destruct
-| #G #L #K #V #U #W #i #HLK #HVW #HWU #j #H destruct /3 width=6/
-| #G #L #K #W #U #l #i #HLK #HWl #HWU #j #H destruct /3 width=6/
+| #G #L #K #V #U #W #i #HLK #HVW #HWU #j #H destruct /3 width=6 by ex3_3_intro, or_introl/
+| #G #L #K #W #U #l #i #HLK #HWl #HWU #j #H destruct /3 width=6 by ex3_3_intro, or_intror/
| #a #I #G #L #V #T #U #_ #j #H destruct
| #G #L #V #T #U #_ #j #H destruct
| #G #L #W #T #U #_ #j #H destruct
qed-.
lemma ssta_inv_lref1: ∀h,g,G,L,U,i. ⦃G, L⦄ ⊢ #i •[h, g] U →
- (∃∃K,V,W. ⇩[0, i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •[h, g] W &
+ (∃∃K,V,W. ⇩[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •[h, g] W &
⇧[0, i + 1] W ≡ U
) ∨
- (∃∃K,W,l. ⇩[0, i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] l &
+ (∃∃K,W,l. ⇩[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] l &
⇧[0, i + 1] W ≡ U
).
/2 width=3 by ssta_inv_lref1_aux/ qed-.
[ #G #L #k #b #J #X #Y #H destruct
| #G #L #K #V #U #W #i #_ #_ #_ #b #J #X #Y #H destruct
| #G #L #K #W #U #l #i #_ #_ #_ #b #J #X #Y #H destruct
-| #a #I #G #L #V #T #U #HTU #b #J #X #Y #H destruct /2 width=3/
+| #a #I #G #L #V #T #U #HTU #b #J #X #Y #H destruct /2 width=3 by ex2_intro/
| #G #L #V #T #U #_ #b #J #X #Y #H destruct
| #G #L #W #T #U #_ #b #J #X #Y #H destruct
]
| #G #L #K #V #U #W #i #_ #_ #_ #X #Y #H destruct
| #G #L #K #W #U #l #i #_ #_ #_ #X #Y #H destruct
| #a #I #G #L #V #T #U #_ #X #Y #H destruct
-| #G #L #V #T #U #HTU #X #Y #H destruct /2 width=3/
+| #G #L #V #T #U #HTU #X #Y #H destruct /2 width=3 by ex2_intro/
| #G #L #W #T #U #_ #X #Y #H destruct
]
qed-.
lemma ssta_inv_da: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h, g] U →
∃l. ⦃G, L⦄ ⊢ T ▪[h, g] l.
#h #g #G #L #T #U #H elim H -G -L -T -U
-[ #G #L #k elim (deg_total h g k) /3 width=2/
-| #G #L #K #V #U #W #i #HLK #_ #_ * /3 width=5/
-| #G #L #K #W #U #l #i #HLK #HWl #_ /3 width=5/
-| #a #I #G #L #V #T #U #_ * /3 width=2/
-| #G #L #V #T #U #_ * /3 width=2/
-| #G #L #W #T #U #_ * /3 width=2/
+[ #G #L #k elim (deg_total h g k) /3 width=2 by da_sort, ex_intro/
+| #G #L #K #V #U #W #i #HLK #_ #_ * /3 width=5 by da_ldef, ex_intro/
+| #G #L #K #W #U #l #i #HLK #HWl #_ /3 width=5 by da_ldec, ex_intro/
+| #a #I #G #L #V #T #U #_ * /3 width=2 by da_bind, ex_intro/
+| #G #L #V #T #U #_ * /3 width=2 by da_flat, ex_intro/
+| #G #L #W #T #U #_ * /3 width=2 by da_flat, ex_intro/
]
qed-.
#h #g #G #L #T #l #H elim H -G -L -T -l
[ /2 width=2/
| #G #L #K #V #i #l #HLK #_ * #W #HVW
- elim (lift_total W 0 (i+1)) /3 width=7/
+ elim (lift_total W 0 (i+1)) /3 width=7 by ssta_ldef, ex_intro/
| #G #L #K #W #i #l #HLK #HW #_
- elim (lift_total W 0 (i+1)) /3 width=7/
-| #a #I #G #L #V #T #l #_ * /3 width=2/
-| * #G #L #V #T #l #_ * /3 width=2/
+ elim (lift_total W 0 (i+1)) /3 width=7 by ssta_ldec, ex_intro/
+| #a #I #G #L #V #T #l #_ * /3 width=2 by ssta_bind, ex_intro/
+| * #G #L #V #T #l #_ * /3 width=2 by ssta_appl, ssta_cast, ex_intro/
]
qed-.
lemma ssta_lift: ∀h,g,G. l_liftable (ssta h g G).
#h #g #G #L1 #T1 #U1 #H elim H -G -L1 -T1 -U1
-[ #G #L1 #k #L2 #d #e #HL21 #X1 #H1 #X2 #H2
+[ #G #L1 #k #L2 #s #d #e #HL21 #X1 #H1 #X2 #H2
>(lift_inv_sort1 … H1) -X1
>(lift_inv_sort1 … H2) -X2 //
-| #G #L1 #K1 #V1 #U1 #W1 #i #HLK1 #_ #HWU1 #IHVW1 #L2 #d #e #HL21 #X #H #U2 #HU12
+| #G #L1 #K1 #V1 #U1 #W1 #i #HLK1 #_ #HWU1 #IHVW1 #L2 #s #d #e #HL21 #X #H #U2 #HU12
elim (lift_inv_lref1 … H) * #Hid #H destruct
[ elim (lift_trans_ge … HWU1 … HU12) -U1 // #W2 #HW12 #HWU2
- elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2/ #X #HLK2 #H
- elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
- /3 width=8/
- | lapply (lift_trans_be … HWU1 … HU12 ? ?) -U1 // /2 width=1/ #HW1U2
- lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/
+ elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+ elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
+ /3 width=9 by ssta_ldef/
+ | lapply (lift_trans_be … HWU1 … HU12 ? ?) -U1 /2 width=1 by le_S/ #HW1U2
+ lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid
+ /3 width=9 by ssta_ldef, ldrop_inv_gen/
]
-| #G #L1 #K1 #W1 #U1 #l #i #HLK1 #HW1l #HWU1 #L2 #d #e #HL21 #X #H #U2 #HU12
+| #G #L1 #K1 #W1 #U1 #l #i #HLK1 #HW1l #HWU1 #L2 #s #d #e #HL21 #X #H #U2 #HU12
elim (lift_inv_lref1 … H) * #Hid #H destruct
[ elim (lift_trans_ge … HWU1 … HU12) -U1 // <minus_plus #W2 #HW12 #HWU2
- elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2/ #X #HLK2 #H
- elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #K2 #W #HK21 #HW1 #H destruct
+ elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+ elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #W #HK21 #HW1 #H destruct
lapply (lift_mono … HW1 … HW12) -HW1 #H destruct
- /3 width=10 by da_lift, ssta_ldec/
- | lapply (lift_trans_be … HWU1 … HU12 ? ?) -U1 // /2 width=1/ #HW1U2
- lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/
+ /3 width=11 by da_lift, ssta_ldec/
+ | lapply (lift_trans_be … HWU1 … HU12 ? ?) -U1 /2 width=1 by le_S/ #HW1U2
+ lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid
+ /3 width=8 by ssta_ldec, ldrop_inv_gen/
]
-| #a #I #G #L1 #V1 #T1 #U1 #_ #IHTU1 #L2 #d #e #HL21 #X1 #H1 #X2 #H2
+| #a #I #G #L1 #V1 #T1 #U1 #_ #IHTU1 #L2 #s #d #e #HL21 #X1 #H1 #X2 #H2
elim (lift_inv_bind1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
elim (lift_inv_bind1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
- lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=5/
-| #G #L1 #V1 #T1 #U1 #_ #IHTU1 #L2 #d #e #HL21 #X1 #H1 #X2 #H2
+ lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=6 by ssta_bind, ldrop_skip/
+| #G #L1 #V1 #T1 #U1 #_ #IHTU1 #L2 #s #d #e #HL21 #X1 #H1 #X2 #H2
elim (lift_inv_flat1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
elim (lift_inv_flat1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
- lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=5/
-| #G #L1 #W1 #T1 #U1 #_ #IHTU1 #L2 #d #e #HL21 #X #H #U2 #HU12
- elim (lift_inv_flat1 … H) -H #W2 #T2 #_ #HT12 #H destruct /3 width=5/
+ lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=6 by ssta_appl/
+| #G #L1 #W1 #T1 #U1 #_ #IHTU1 #L2 #s #d #e #HL21 #X #H #U2 #HU12
+ elim (lift_inv_flat1 … H) -H #W2 #T2 #_ #HT12 #H destruct /3 width=6 by ssta_cast/
]
qed.
lemma ssta_inv_lift1: ∀h,g,G. l_deliftable_sn (ssta h g G).
#h #g #G #L2 #T2 #U2 #H elim H -G -L2 -T2 -U2
-[ #G #L2 #k #L1 #d #e #_ #X #H
- >(lift_inv_sort2 … H) -X /2 width=3/
-| #G #L2 #K2 #V2 #U2 #W2 #i #HLK2 #HVW2 #HWU2 #IHVW2 #L1 #d #e #HL21 #X #H
+[ #G #L2 #k #L1 #s #d #e #_ #X #H
+ >(lift_inv_sort2 … H) -X /2 width=3 by ssta_sort, lift_sort, ex2_intro/
+| #G #L2 #K2 #V2 #U2 #W2 #i #HLK2 #HVW2 #HWU2 #IHVW2 #L1 #s #d #e #HL21 #X #H
elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HVW2 | -IHVW2 ]
[ elim (ldrop_conf_lt … HL21 … HLK2) -L2 // #K1 #V1 #HLK1 #HK21 #HV12
elim (IHVW2 … HK21 … HV12) -K2 -V2 #W1 #HW12 #HVW1
- elim (lift_trans_le … HW12 … HWU2) -W2 // >minus_plus <plus_minus_m_m // -Hid /3 width=8/
+ elim (lift_trans_le … HW12 … HWU2) -W2 // >minus_plus <plus_minus_m_m
+ /3 width=8 by ssta_ldef, ex2_intro/
| lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
elim (le_inv_plus_l … Hid) -Hid #Hdie #ei
- elim (lift_split … HWU2 d (i-e+1)) -HWU2 // [3: /2 width=1/ ]
- [ #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /2 width=1/ /3 width=8/
+ elim (lift_split … HWU2 d (i-e+1)) -HWU2 // [3: /2 width=1 by le_S/ ]
+ [ #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m
+ /3 width=8 by ssta_ldef, le_S, ex2_intro/
| <le_plus_minus_comm //
]
]
-| #G #L2 #K2 #W2 #U2 #l #i #HLK2 #HW2l #HWU2 #L1 #d #e #HL21 #X #H
+| #G #L2 #K2 #W2 #U2 #l #i #HLK2 #HW2l #HWU2 #L1 #s #d #e #HL21 #X #H
elim (lift_inv_lref2 … H) * #Hid #H destruct
[ elim (ldrop_conf_lt … HL21 … HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12
lapply (da_inv_lift … HW2l … HK21 … HW12) -K2
- elim (lift_trans_le … HW12 … HWU2) -W2 // >minus_plus <plus_minus_m_m // -Hid /3 width=8/
+ elim (lift_trans_le … HW12 … HWU2) -W2 // >minus_plus <plus_minus_m_m
+ /3 width=8 by ssta_ldec, ex2_intro/
| lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
elim (le_inv_plus_l … Hid) -Hid #Hdie #ei
- elim (lift_split … HWU2 d (i-e+1)) -HWU2 // [3: /2 width=1/ ]
- [ #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /2 width=1/ /3 width=8/
+ elim (lift_split … HWU2 d (i-e+1)) -HWU2 // [3: /2 width=1 by le_S/ ]
+ [ #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m
+ /3 width=8 by ssta_ldec, le_S, ex2_intro/
| <le_plus_minus_comm //
]
]
-| #a #I #G #L2 #V2 #T2 #U2 #_ #IHTU2 #L1 #d #e #HL21 #X #H
+| #a #I #G #L2 #V2 #T2 #U2 #_ #IHTU2 #L1 #s #d #e #HL21 #X #H
elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
- elim (IHTU2 (L1.ⓑ{I}V1) … HT12) -IHTU2 -HT12 /2 width=1/ -HL21 /3 width=5/
-| #G #L2 #V2 #T2 #U2 #_ #IHTU2 #L1 #d #e #HL21 #X #H
+ elim (IHTU2 (L1.ⓑ{I}V1) … HT12) -IHTU2 -HT12
+ /3 width=5 by ssta_bind, ldrop_skip, lift_bind, ex2_intro/
+| #G #L2 #V2 #T2 #U2 #_ #IHTU2 #L1 #s #d #e #HL21 #X #H
elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
- elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=5/
-| #G #L2 #W2 #T2 #U2 #_ #IHTU2 #L1 #d #e #HL21 #X #H
+ elim (IHTU2 … HL21 … HT12) -L2 -HT12
+ /3 width=5 by ssta_appl, lift_flat, ex2_intro/
+| #G #L2 #W2 #T2 #U2 #_ #IHTU2 #L1 #s #d #e #HL21 #X #H
elim (lift_inv_flat2 … H) -H #W1 #T1 #_ #HT12 #H destruct
- elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=3/
+ elim (IHTU2 … HL21 … HT12) -L2 -HT12
+ /3 width=3 by ssta_cast, ex2_intro/
]
qed-.
∀l. ⦃G, L⦄ ⊢ T ▪[h, g] l → ⦃G, L⦄ ⊢ U ▪[h, g] l-1.
#h #g #G #L #T #U #H elim H -G -L -T -U
[ #G #L #k #l #H
- lapply (da_inv_sort … H) -H /3 width=1/
+ lapply (da_inv_sort … H) -H /3 width=1 by da_sort, deg_next/
| #G #L #K #V #U #W #i #HLK #_ #HWU #IHVW #l #H
elim (da_inv_lref … H) -H * #K0 #V0 [| #l0] #HLK0 #HV0
lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct
- lapply (ldrop_fwd_drop2 … HLK) -HLK /3 width=7/
+ lapply (ldrop_fwd_drop2 … HLK) -HLK /3 width=8 by da_lift/
| #G #L #K #W #U #l0 #i #HLK #_ #HWU #l #H -l0
elim (da_inv_lref … H) -H * #K0 #V0 [| #l1] #HLK0 #HV0 [| #H0 ]
lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct
- lapply (ldrop_fwd_drop2 … HLK) -HLK /3 width=7/
+ lapply (ldrop_fwd_drop2 … HLK) -HLK /3 width=8 by da_lift/
| #a #I #G #L #V #T #U #_ #IHTU #l #H
- lapply (da_inv_bind … H) -H /3 width=1/
+ lapply (da_inv_bind … H) -H /3 width=1 by da_bind/
| #G #L #V #T #U #_ #IHTU #l #H
- lapply (da_inv_flat … H) -H /3 width=1/
+ lapply (da_inv_flat … H) -H /3 width=1 by da_flat/
| #G #L #W #T #U #_ #IHTU #l #H
- lapply (da_inv_flat … H) -H /2 width=1/
+ lapply (da_inv_flat … H) -H /2 width=1 by /
]
qed-.
[ /2 width=2/
| #G #L #K #V #U #W #i #HLK #_ #HWU * #T #HWT
lapply (ldrop_fwd_drop2 … HLK) -HLK #HLK
- elim (lift_total T 0 (i+1)) /3 width=10/
+ elim (lift_total T 0 (i+1)) /3 width=11 by ssta_lift, ex_intro/
| #G #L #K #W #U #l #i #HLK #HWl #HWU
elim (da_ssta … HWl) -HWl #T #HWT
lapply (ldrop_fwd_drop2 … HLK) -HLK #HLK
- elim (lift_total T 0 (i+1)) /3 width=10/
-| #a #I #G #L #V #T #U #_ * /3 width=2/
-| #G #L #V #T #U #_ * #T0 #HUT0 /3 width=2/
-| #G #L #W #T #U #_ * /2 width=2/
+ elim (lift_total T 0 (i+1)) /3 width=11 by ssta_lift, ex_intro/
+| #a #I #G #L #V #T #U #_ * /3 width=2 by ssta_bind, ex_intro/
+| #G #L #V #T #U #_ * #T0 #HUT0 /3 width=2 by ssta_appl, ex_intro/
+| #G #L #W #T #U #_ * /2 width=2 by ex_intro/
]
qed-.
projects / helm.git / commitdiff