(* *)
(**************************************************************************)
-include "basic_2/static/sta_lift.ma".
+include "basic_2/substitution/drop_drop.ma".
include "basic_2/unfold/lstas.ma".
(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
(* Properties on relocation *************************************************)
-lemma lstas_lift: ∀h,G,l. l_liftable (llstar … (sta h G) l).
-/3 width=10 by l_liftable_llstar, sta_lift/ qed.
+(* Basic_1: was just: sty0_lift *)
+lemma lstas_lift: ∀h,G,l. l_liftable (lstas h G l).
+#h #G #l #L1 #T1 #U1 #H elim H -G -L1 -T1 -U1 -l
+[ #G #L1 #l #k #L2 #s #d #e #HL21 #X1 #H1 #X2 #H2
+ >(lift_inv_sort1 … H1) -X1
+ >(lift_inv_sort1 … H2) -X2 //
+| #G #L1 #K1 #V1 #W1 #W #i #l #HLK1 #_ #HW1 #IHVW1 #L2 #s #d #e #HL21 #X #H #U2 #HWU2
+ elim (lift_inv_lref1 … H) * #Hid #H destruct
+ [ elim (lift_trans_ge … HW1 … HWU2) -W // #W2 #HW12 #HWU2
+ elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+ elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
+ /3 width=9 by lstas_ldef/
+ | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W /2 width=1 by le_S/ #HW1U2
+ lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 /3 width=9 by lstas_ldef, drop_inv_gen/
+ ]
+| #G #L1 #K1 #V1 #W1 #i #HLK1 #_ #IHVW1 #L2 #s #d #e #HL21 #X #H #U2 #HWU2
+ >(lift_mono … HWU2 … H) -U2
+ elim (lift_inv_lref1 … H) * #Hid #H destruct
+ [ elim (lift_total W1 (d-i-1) e) #W2 #HW12
+ elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+ elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
+ /3 width=10 by lstas_zero/
+ | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1
+ /3 width=10 by lstas_zero, drop_inv_gen/
+ ]
+| #G #L1 #K1 #W1 #V1 #W #i #l #HLK1 #_ #HW1 #IHWV1 #L2 #s #d #e #HL21 #X #H #U2 #HWU2
+ elim (lift_inv_lref1 … H) * #Hid #H destruct
+ [ elim (lift_trans_ge … HW1 … HWU2) -W // <minus_plus #W #HW1 #HWU2
+ elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+ elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #W2 #HK21 #HW12 #H destruct
+ /3 width=9 by lstas_succ/
+ | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W /2 width=1 by le_S/ #HW1U2
+ lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 /3 width=9 by lstas_succ, drop_inv_gen/
+ ]
+| #a #I #G #L1 #V1 #T1 #U1 #l #_ #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=6 by lstas_bind, drop_skip/
+| #G #L1 #V1 #T1 #U1 #l #_ #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=6 by lstas_appl/
+| #G #L1 #W1 #T1 #U1 #l #_ #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 lstas_cast/
+]
+qed.
(* Inversion lemmas on relocation *******************************************)
-lemma lstas_inv_lift1: ∀h,G,l. l_deliftable_sn (llstar … (sta h G) l).
-/3 width=6 by l_deliftable_sn_llstar, sta_inv_lift1/ qed-.
+(* Note: apparently this was missing in basic_1 *)
+lemma lstas_inv_lift1: ∀h,G,l. l_deliftable_sn (lstas h G l).
+#h #G #l #L2 #T2 #U2 #H elim H -G -L2 -T2 -U2 -l
+[ #G #L2 #l #k #L1 #s #d #e #_ #X #H
+ >(lift_inv_sort2 … H) -X /2 width=3 by lstas_sort, lift_sort, ex2_intro/
+| #G #L2 #K2 #V2 #W2 #W #i #l #HLK2 #HVW2 #HW2 #IHVW2 #L1 #s #d #e #HL21 #X #H
+ elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HVW2 | -IHVW2 ]
+ [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #V1 #HLK1 #HK21 #HV12
+ elim (IHVW2 … HK21 … HV12) -K2 -V2 #W1 #HW12 #HVW1
+ elim (lift_trans_le … HW12 … HW2) -W2 // >minus_plus <plus_minus_m_m /3 width=8 by lstas_ldef, ex2_intro/
+ | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
+ elim (le_inv_plus_l … Hid) -Hid #Hdie #ei
+ elim (lift_split … HW2 d (i-e+1)) -HW2 /2 width=1 by le_S_S, le_S/
+ #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /3 width=8 by lstas_ldef, le_S, ex2_intro/
+ ]
+| #G #L2 #K2 #W2 #V2 #i #HLK2 #HWV2 #IHWV2 #L1 #s #d #e #HL21 #X #H
+ elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HWV2 | -IHWV2 ]
+ [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12
+ elim (IHWV2 … HK21 … HW12) -K2
+ /3 width=5 by lstas_zero, lift_lref_lt, ex2_intro/
+ | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2
+ /3 width=5 by lstas_zero, lift_lref_ge_minus, ex2_intro/
+ ]
+| #G #L2 #K2 #W2 #V2 #W #i #l #HLK2 #HWV2 #HW2 #IHWV2 #L1 #s #d #e #HL21 #X #H
+ elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HWV2 | -IHWV2 ]
+ [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12
+ elim (IHWV2 … HK21 … HW12) -K2 #V1 #HV12 #HWV1
+ elim (lift_trans_le … HV12 … HW2) -W2 // >minus_plus <plus_minus_m_m /3 width=8 by lstas_succ, ex2_intro/
+ | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
+ elim (le_inv_plus_l … Hid) -Hid #Hdie #ei
+ elim (lift_split … HW2 d (i-e+1)) -HW2 /2 width=1 by le_S_S, le_S/
+ #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /3 width=8 by lstas_succ, le_S, ex2_intro/
+ ]
+| #a #I #G #L2 #V2 #T2 #U2 #l #_ #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 /3 width=5 by lstas_bind, drop_skip, lift_bind, ex2_intro/
+| #G #L2 #V2 #T2 #U2 #l #_ #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 by lstas_appl, lift_flat, ex2_intro/
+| #G #L2 #W2 #T2 #U2 #l #_ #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 by lstas_cast, ex2_intro/
+]
+qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lstas_inv_lref1: ∀h,G,L,U,i,l. ⦃G, L⦄ ⊢ #i •*[h, l+1] U →
- (∃∃K,V,W. ⇩[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, l+1] W &
- ⇧[0, i+1] W ≡ U
- ) ∨
- (∃∃K,W,V,V0. ⇩[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •[h] V0 &
- ⦃G, K⦄ ⊢ W •*[h, l] V & ⇧[0, i+1] V ≡ U
- ).
-#h #G #L #U #i #l #H elim (lstas_inv_step_sn … H) -H
-#X #H #HXU elim (sta_inv_lref1 … H) -H
-* #K #V #W #HLK #HVW #HWX
-lapply (drop_fwd_drop2 … HLK) #H0LK
-elim (lstas_inv_lift1 … HXU … H0LK … HWX) -H0LK -X
-/4 width=8 by lstas_step_sn, ex4_4_intro, ex3_3_intro, or_introl, or_intror/
+lemma lstas_split_aux: ∀h,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l] T2 → ∀l1,l2. l = l1 + l2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, l1] T & ⦃G, L⦄ ⊢ T •*[h, l2] T2.
+#h #G #L #T1 #T2 #l #H elim H -G -L -T1 -T2 -l
+[ #G #L #l #k #l1 #l2 #H destruct
+ >commutative_plus >iter_plus /2 width=3 by lstas_sort, ex2_intro/
+| #G #L #K #V1 #V2 #U2 #i #l #HLK #_ #VU2 #IHV12 #l1 #l2 #H destruct
+ elim (IHV12 l1 l2) -IHV12 // #V
+ elim (lift_total V 0 (i+1))
+ lapply (drop_fwd_drop2 … HLK)
+ /3 width=12 by lstas_lift, lstas_ldef, ex2_intro/
+| #G #L #K #W1 #W2 #i #HLK #HW12 #_ #l1 #l2 #H
+ elim (zero_eq_plus … H) -H #H1 #H2 destruct
+ /3 width=5 by lstas_zero, ex2_intro/
+| #G #L #K #W1 #W2 #U2 #i #l #HLK #HW12 #HWU2 #IHW12 #l1 @(nat_ind_plus … l1) -l1
+ [ #l2 normalize #H destruct
+ elim (IHW12 0 l) -IHW12 //
+ lapply (drop_fwd_drop2 … HLK)
+ /3 width=8 by lstas_succ, lstas_zero, ex2_intro/
+ | #l1 #_ #l2 <plus_plus_comm_23 #H lapply (injective_plus_l … H) -H #H
+ elim (IHW12 … H) -l #W
+ elim (lift_total W 0 (i+1))
+ lapply (drop_fwd_drop2 … HLK)
+ /3 width=12 by lstas_lift, lstas_succ, ex2_intro/
+ ]
+| #a #I #G #L #V #T #U #l #_ #IHTU #l1 #l2 #H
+ elim (IHTU … H) -l /3 width=3 by lstas_bind, ex2_intro/
+| #G #L #V #T #U #l #_ #IHTU #l1 #l2 #H
+ elim (IHTU … H) -l /3 width=3 by lstas_appl, ex2_intro/
+| #G #L #W #T #U #l #_ #IHTU #l1 #l2 #H
+ elim (IHTU … H) -l /3 width=3 by lstas_cast, ex2_intro/
+]
qed-.
-(* Advanced forward lemmas **************************************************)
-
-lemma lstas_fwd_correct: ∀h,G,L,T1,U1. ⦃G, L⦄ ⊢ T1 •[h] U1 →
- ∀T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l] T2 →
- ∃U2. ⦃G, L⦄ ⊢ T2 •[h] U2.
-#h #G #L #T1 #U1 #HTU1 #T2 #l #H @(lstas_ind_dx … H) -l -T2 /2 width=3 by ex_intro/ -HTU1
-#l #T #T2 #_ #HT2 #_ -T1 -U1 -l
-elim (sta_fwd_correct … HT2) -T /2 width=2 by ex_intro/
-qed-.
+lemma lstas_split: ∀h,G,L,T1,T2,l1,l2. ⦃G, L⦄ ⊢ T1 •*[h, l1 + l2] T2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, l1] T & ⦃G, L⦄ ⊢ T •*[h, l2] T2.
+/2 width=3 by lstas_split_aux/ qed-.
(* Advanced properties ******************************************************)
-lemma lstas_total: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U →
- ∀l. ∃U0. ⦃G, L⦄ ⊢ T •*[h, l] U0.
-#h #G #L #T #U #HTU #l @(nat_ind_plus … l) -l /2 width=2 by ex_intro/
-#l * #U0 #HTU0 elim (lstas_fwd_correct … HTU … HTU0) -U
-/3 width=4 by lstas_step_dx, ex_intro/
+lemma lstas_lstas: ∀h,G,L,T,T1,l1. ⦃G, L⦄ ⊢ T •*[h, l1] T1 →
+ ∀l2. ∃T2. ⦃G, L⦄ ⊢ T •*[h, l2] T2.
+#h #G #L #T #T1 #l1 #H elim H -G -L -T -T1 -l1
+[ /2 width=2 by lstas_sort, ex_intro/
+| #G #L #K #V #V1 #U1 #i #l1 #HLK #_ #HVU1 #IHV1 #l2
+ elim (IHV1 l2) -IHV1 #V2
+ elim (lift_total V2 0 (i+1))
+ /3 width=7 by ex_intro, lstas_ldef/
+| #G #L #K #W #W1 #i #HLK #HW1 #IHW1 #l2
+ @(nat_ind_plus … l2) -l2 /3 width=5 by lstas_zero, ex_intro/
+ #l2 #_ elim (IHW1 l2) -IHW1 #W2
+ elim (lift_total W2 0 (i+1))
+ /3 width=7 by lstas_succ, ex_intro/
+| #G #L #K #W #W1 #U1 #i #l #HLK #_ #_ #IHW1 #l2
+ @(nat_ind_plus … l2) -l2
+ [ elim (IHW1 0) -IHW1 /3 width=5 by lstas_zero, ex_intro/
+ | #l2 #_ elim (IHW1 l2) -IHW1
+ #W2 elim (lift_total W2 0 (i+1)) /3 width=7 by ex_intro, lstas_succ/
+ ]
+| #a #I #G #L #V #T #U #l #_ #IHTU #l2
+ elim (IHTU l2) -IHTU /3 width=2 by lstas_bind, ex_intro/
+| #G #L #V #T #U #l #_ #IHTU #l2
+ elim (IHTU l2) -IHTU /3 width=2 by lstas_appl, ex_intro/
+| #G #L #W #T #U #l #_ #IHTU #l2
+ elim (IHTU l2) -IHTU /3 width=2 by lstas_cast, ex_intro/
+]
qed-.
-
-lemma lstas_ldef: ∀h,G,L,K,V,i. ⇩[i] L ≡ K.ⓓV →
- ∀W,l. ⦃G, K⦄ ⊢ V •*[h, l+1] W →
- ∀U. ⇧[0, i+1] W ≡ U → ⦃G, L⦄ ⊢ #i •*[h, l+1] U.
-#h #G #L #K #V #i #HLK #W #l #HVW #U #HWU
-lapply (drop_fwd_drop2 … HLK)
-elim (lstas_inv_step_sn … HVW) -HVW #W0
-elim (lift_total W0 0 (i+1)) /3 width=12 by lstas_step_sn, sta_ldef, lstas_lift/
-qed.
-
-lemma lstas_ldec: ∀h,G,L,K,W,i. ⇩[i] L ≡ K.ⓛW → ∀V0. ⦃G, K⦄ ⊢ W •[h] V0 →
- ∀V,l. ⦃G, K⦄ ⊢ W •*[h, l] V →
- ∀U. ⇧[0, i+1] V ≡ U → ⦃G, L⦄ ⊢ #i •*[h, l+1] U.
-#h #G #L #K #W #i #HLK #V0 #HWV0 #V #l #HWV #U #HVU
-lapply (drop_fwd_drop2 … HLK) #H
-elim (lift_total W 0 (i+1)) /3 width=12 by lstas_step_sn, sta_ldec, lstas_lift/
-qed.
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