(* *)
(**************************************************************************)
-include "basic_2/relocation/drops.ma".
+include "static_2/relocation/drops_drops.ma".
+include "static_2/s_computation/fqup_weight.ma".
+include "static_2/s_computation/fqup_drops.ma".
include "basic_2/rt_transition/cpg.ma".
-(* CONTEXT-SENSITIVE GENERIC PARALLEL RT-TRANSITION FOR TERMS ***************)
+(* BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *****************)
+
+(* Advanced properties ******************************************************)
+
+lemma cpg_delta_drops: ∀Rt,c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≘ K.ⓓV → ⦃G, K⦄ ⊢ V ⬈[Rt, c, h] V2 →
+ ⬆*[↑i] V2 ≘ T2 → ⦃G, L⦄ ⊢ #i ⬈[Rt, c, h] T2.
+#Rt #c #h #G #K #V #V2 #i elim i -i
+[ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_delta/
+| #i #IH #L0 #T0 #H0 #HV2 #HVT2
+ elim (drops_inv_succ … H0) -H0 #I #L #HLK #H destruct
+ elim (lifts_split_trans … HVT2 (𝐔❴↑i❵) (𝐔❴1❵) ?) -HVT2 /3 width=3 by cpg_lref/
+]
+qed.
+
+lemma cpg_ell_drops: ∀Rt,c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≘ K.ⓛV → ⦃G, K⦄ ⊢ V ⬈[Rt,c, h] V2 →
+ ⬆*[↑i] V2 ≘ T2 → ⦃G, L⦄ ⊢ #i ⬈[Rt, c+𝟘𝟙, h] T2.
+#Rt #c #h #G #K #V #V2 #i elim i -i
+[ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_ell/
+| #i #IH #L0 #T0 #H0 #HV2 #HVT2
+ elim (drops_inv_succ … H0) -H0 #I #L #HLK #H destruct
+ elim (lifts_split_trans … HVT2 (𝐔❴↑i❵) (𝐔❴1❵) ?) -HVT2 /3 width=3 by cpg_lref/
+]
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cpg_inv_lref1_drops: ∀Rt,c,h,G,i,L,T2. ⦃G, L⦄ ⊢ #i ⬈[Rt,c, h] T2 →
+ ∨∨ T2 = #i ∧ c = 𝟘𝟘
+ | ∃∃cV,K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 &
+ ⬆*[↑i] V2 ≘ T2 & c = cV
+ | ∃∃cV,K,V,V2. ⬇*[i] L ≘ K.ⓛV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 &
+ ⬆*[↑i] V2 ≘ T2 & c = cV + 𝟘𝟙.
+#Rt #c #h #G #i elim i -i
+[ #L #T2 #H elim (cpg_inv_zero1 … H) -H * /3 width=1 by or3_intro0, conj/
+ /4 width=8 by drops_refl, ex4_4_intro, or3_intro2, or3_intro1/
+| #i #IH #L #T2 #H elim (cpg_inv_lref1 … H) -H * /3 width=1 by or3_intro0, conj/
+ #I #K #V2 #H #HVT2 #H0 destruct elim (IH … H) -IH -H
+ [ * #H1 #H2 destruct lapply (lifts_inv_lref1_uni … HVT2) -HVT2 #H destruct /3 width=1 by or3_intro0, conj/ ] *
+ #cV #L #W #W2 #HKL #HW2 #HWV2 #H destruct
+ lapply (lifts_trans … HWV2 … HVT2 ??) -V2
+ /4 width=8 by drops_drop, ex4_4_intro, or3_intro2, or3_intro1/
+]
+qed-.
+
+lemma cpg_inv_atom1_drops: ∀Rt,c,h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ⬈[Rt, c, h] T2 →
+ ∨∨ T2 = ⓪{I} ∧ c = 𝟘𝟘
+ | ∃∃s. T2 = ⋆(next h s) & I = Sort s & c = 𝟘𝟙
+ | ∃∃cV,i,K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 &
+ ⬆*[↑i] V2 ≘ T2 & I = LRef i & c = cV
+ | ∃∃cV,i,K,V,V2. ⬇*[i] L ≘ K.ⓛV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 &
+ ⬆*[↑i] V2 ≘ T2 & I = LRef i & c = cV + 𝟘𝟙.
+#Rt #c #h * #n #G #L #T2 #H
+[ elim (cpg_inv_sort1 … H) -H *
+ /3 width=3 by or4_intro0, or4_intro1, ex3_intro, conj/
+| elim (cpg_inv_lref1_drops … H) -H *
+ /3 width=10 by or4_intro0, or4_intro2, or4_intro3, ex5_5_intro, conj/
+| elim (cpg_inv_gref1 … H) -H
+ /3 width=1 by or4_intro0, conj/
+]
+qed-.
(* Properties with generic slicing for local environments *******************)
-lemma cpg_delift: ∀h,r,I,G,K,V,T1,L,l. ⬇*[i] L ≡ (K.ⓑ{I}V) →
- ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡[h, 𝟘𝟘] T2 & ⬆*[↑1] T ≡ T2.
-#h #r #I #G #K #V #T1 elim T1 -T1
-[ * #i #L #l /2 width=4 by cpg_atom, lift_sort, lift_gref, ex2_2_intro/
- elim (lt_or_eq_or_gt i l) #Hil [1,3: /4 width=4 by cpg_atom, lift_lref_ge_minus, lift_lref_lt, ylt_inj, yle_inj, ex2_2_intro/ ]
- destruct
- elim (lift_total V 0 (i+1)) #W #HVW
- elim (lift_split … HVW i i) /3 width=7 by cpg_delta, ex2_2_intro/
-| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #l #HLK
- elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
- [ elim (IHU1 (L. ⓑ{I} W1) (l+1)) -IHU1 /3 width=9 by cpg_bind, drop_drop, lift_bind, ex2_2_intro/
- | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpg_flat, lift_flat, ex2_2_intro/
+(* Note: it should use drops_split_trans_pair2 *)
+lemma cpg_lifts_sn: ∀Rt. reflexive … Rt →
+ ∀c,h,G. d_liftable2_sn … lifts (cpg Rt h c G).
+#Rt #HRt #c #h #G #K #T generalize in match c; -c
+@(fqup_wf_ind_eq (Ⓣ) … G K T) -G -K -T #G0 #K0 #T0 #IH #G #K * *
+[ #s #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct -IH
+ lapply (lifts_inv_sort1 … H1) -H1 #H destruct
+ elim (cpg_inv_sort1 … H2) -H2 * #H1 #H2 destruct
+ /2 width=3 by cpg_atom, cpg_ess, lifts_sort, ex2_intro/
+| #i1 #HG #HK #HT #c #T2 #H2 #b #f #L #HLK #X1 #H1 destruct
+ elim (cpg_inv_lref1_drops … H2) -H2 *
+ [ #H1 #H2 destruct /3 width=3 by cpg_refl, ex2_intro/ ]
+ #cV #K0 #V #V2 #HK0 #HV2 #HVT2 #H destruct
+ elim (lifts_inv_lref1 … H1) -H1 #i2 #Hf #H destruct
+ lapply (drops_trans … HLK … HK0 ??) -HLK [3,6: |*: // ] #H
+ elim (drops_split_trans … H) -H [1,6: |*: /2 width=6 by after_uni_dx/ ] #Y #HL0 #HY
+ lapply (drops_tls_at … Hf … HY) -HY #HY
+ elim (drops_inv_skip2 … HY) -HY #Z #L0 #HLK0 #HZ #H destruct
+ elim (liftsb_inv_pair_sn … HZ) -HZ #W #HVW #H destruct
+ elim (IH … HV2 … HLK0 … HVW) -IH /2 width=2 by fqup_lref/ -K -K0 -V #W2 #HVW2 #HW2
+ elim (lifts_total W2 (𝐔❴↑i2❵)) #U2 #HWU2
+ lapply (lifts_trans … HVW2 … HWU2 ??) -HVW2 [3,6: |*: // ] #HVU2
+ lapply (lifts_conf … HVT2 … HVU2 f ?) -V2 [1,3: /2 width=3 by after_uni_succ_sn/ ]
+ /4 width=8 by cpg_ell_drops, cpg_delta_drops, drops_inv_gen, ex2_intro/
+| #l #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct -IH
+ lapply (lifts_inv_gref1 … H1) -H1 #H destruct
+ elim (cpg_inv_gref1 … H2) -H2 #H1 #H2 destruct
+ /2 width=3 by cpg_atom, lifts_gref, ex2_intro/
+| #p #I #V1 #T1 #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct
+ elim (lifts_inv_bind1 … H1) -H1 #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (cpg_inv_bind1 … H2) -H2 *
+ [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
+ elim (IH … HV12 … HLK … HVW1) -HV12 //
+ elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
+ /3 width=5 by cpg_bind, lifts_bind, ex2_intro/
+ | #cT #T2 #HT12 #HXT2 #H1 #H2 #H3 destruct
+ elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ] #U2 #HTU2 #HU12
+ lapply (lifts_trans … HXT2 … HTU2 ??) -T2 [3: |*: // ] #HXU2
+ elim (lifts_split_trans … HXU2 f (𝐔❴↑O❵)) [2: /2 width=1 by after_uni_one_dx/ ]
+ /3 width=5 by cpg_zeta, ex2_intro/
+ ]
+| * #V1 #T1 #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct
+ elim (lifts_inv_flat1 … H1) -H1 #W1 #U1 #HVW1 #HTU1 #H destruct
+ [ elim (cpg_inv_appl1 … H2) -H2 *
+ [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
+ elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
+ elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
+ /3 width=5 by cpg_appl, lifts_flat, ex2_intro/
+ | #cV #cY #cT #a #V2 #Y1 #Y2 #T0 #T2 #HV12 #HY12 #HT12 #H1 #H2 #H3 destruct
+ elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct
+ elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
+ elim (IH … HY12 … HLK … HYZ1) -HY12 //
+ elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
+ /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/
+ | #cV #cY #cT #a #V2 #V20 #Y1 #Y2 #T0 #T2 #HV12 #HV20 #HY12 #HT12 #H1 #H2 #H3 destruct
+ elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct
+ elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 // #W2 #HVW2 #HW12
+ elim (IH … HY12 … HLK … HYZ1) -HY12 //
+ elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
+ elim (lifts_total W2 (𝐔❴1❵)) #W20 #HW20
+ lapply (lifts_trans … HVW2 … HW20 ??) -HVW2 [3: |*: // ] #H
+ lapply (lifts_conf … HV20 … H (⫯f) ?) -V2 /2 width=3 by after_uni_one_sn/
+ /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/
+ ]
+ | elim (cpg_inv_cast1 … H2) -H2 *
+ [ #cV #cT #V2 #T2 #HV12 #HT12 #HcVT #H1 #H2 destruct
+ elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
+ elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
+ /3 width=5 by cpg_cast, lifts_flat, ex2_intro/
+ | #cT #HT12 #H destruct
+ elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
+ /3 width=3 by cpg_eps, ex2_intro/
+ | #cV #HV12 #H destruct
+ elim (IH … HV12 … HLK … HVW1) -IH -HV12 -HLK -HVW1 //
+ /3 width=3 by cpg_ee, ex2_intro/
+ ]
]
]
qed-.
-lemma cpg_inv_lref1: ∀h,r,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, r] T2 →
- T2 = #i ∨
- ∃∃I,K,V,V2. ⬇[i] L ≡ K. ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, r] V2 &
- ⬆[O, i+1] V2 ≡ T2.
-#h #r #G #L #T2 #i #H
-elim (cpg_inv_atom1 … H) -H /2 width=1 by or_introl/ *
-[ #s #d #_ #_ #H destruct
-| #I #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=7 by ex3_4_intro, or_intror/
+lemma cpg_lifts_bi: ∀Rt. reflexive … Rt →
+ ∀c,h,G. d_liftable2_bi … lifts (cpg Rt h c G).
+/3 width=12 by cpg_lifts_sn, d_liftable2_sn_bi, lifts_mono/ qed-.
+
+(* Inversion lemmas with generic slicing for local environments *************)
+
+lemma cpg_inv_lifts_sn: ∀Rt. reflexive … Rt →
+ ∀c,h,G. d_deliftable2_sn … lifts (cpg Rt h c G).
+#Rt #HRt #c #h #G #L #U generalize in match c; -c
+@(fqup_wf_ind_eq (Ⓣ) … G L U) -G -L -U #G0 #L0 #U0 #IH #G #L * *
+[ #s #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct -IH
+ lapply (lifts_inv_sort2 … H1) -H1 #H destruct
+ elim (cpg_inv_sort1 … H2) -H2 * #H1 #H2 destruct
+ /2 width=3 by cpg_atom, cpg_ess, lifts_sort, ex2_intro/
+| #i2 #HG #HL #HU #c #U2 #H2 #b #f #K #HLK #X1 #H1 destruct
+ elim (cpg_inv_lref1_drops … H2) -H2 *
+ [ #H1 #H2 destruct /3 width=3 by cpg_refl, ex2_intro/ ]
+ #cW #L0 #W #W2 #HL0 #HW2 #HWU2 #H destruct
+ elim (lifts_inv_lref2 … H1) -H1 #i1 #Hf #H destruct
+ lapply (drops_split_div … HLK (𝐔❴i1❵) ???) -HLK [4,8: * |*: // ] #Y0 #HK0 #HLY0
+ lapply (drops_conf … HL0 … HLY0 ??) -HLY0 [3,6: |*: /2 width=6 by after_uni_dx/ ] #HLY0
+ lapply (drops_tls_at … Hf … HLY0) -HLY0 #HLY0
+ elim (drops_inv_skip1 … HLY0) -HLY0 #Z #K0 #HLK0 #HZ #H destruct
+ elim (liftsb_inv_pair_dx … HZ) -HZ #V #HVW #H destruct
+ elim (IH … HW2 … HLK0 … HVW) -IH /2 width=2 by fqup_lref/ -L -L0 -W #V2 #HVW2 #HV2
+ lapply (lifts_trans … HVW2 … HWU2 ??) -W2 [3,6: |*: // ] #HVU2
+ elim (lifts_split_trans … HVU2 ? f) -HVU2 [1,4: |*: /2 width=4 by after_uni_succ_sn/ ]
+ /4 width=8 by cpg_ell_drops, cpg_delta_drops, drops_inv_F, ex2_intro/
+| #l #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct -IH
+ lapply (lifts_inv_gref2 … H1) -H1 #H destruct
+ elim (cpg_inv_gref1 … H2) -H2 #H1 #H2 destruct
+ /2 width=3 by cpg_atom, lifts_gref, ex2_intro/
+| #p #I #W1 #U1 #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct
+ elim (lifts_inv_bind2 … H1) -H1 #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (cpg_inv_bind1 … H2) -H2 *
+ [ #cW #cU #W2 #U2 #HW12 #HU12 #H1 #H2 destruct
+ elim (IH … HW12 … HLK … HVW1) -HW12 //
+ elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
+ /3 width=5 by cpg_bind, lifts_bind, ex2_intro/
+ | #cU #U2 #HU12 #HXU2 #H1 #H2 #H3 destruct
+ elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ] #T2 #HTU2 #HT12
+ elim (lifts_div4_one … HTU2 … HXU2) -U2 /3 width=5 by cpg_zeta, ex2_intro/
+ ]
+| * #W1 #U1 #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct
+ elim (lifts_inv_flat2 … H1) -H1 #V1 #T1 #HVW1 #HTU1 #H destruct
+ [ elim (cpg_inv_appl1 … H2) -H2 *
+ [ #cW #cU #W2 #U2 #HW12 #HU12 #H1 #H2 destruct
+ elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
+ elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
+ /3 width=5 by cpg_appl, lifts_flat, ex2_intro/
+ | #cW #cZ #cU #a #W2 #Z1 #Z2 #U0 #U2 #HW12 #HZ12 #HU12 #H1 #H2 #H3 destruct
+ elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct
+ elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
+ elim (IH … HZ12 … HLK … HYZ1) -HZ12 //
+ elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
+ /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/
+ | #cW #cZ #cU #a #W2 #W20 #Z1 #Z2 #U0 #U2 #HW12 #HW20 #HZ12 #HU12 #H1 #H2 #H3 destruct
+ elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct
+ elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 // #V2 #HVW2 #HV12
+ elim (IH … HZ12 … HLK … HYZ1) -HZ12 //
+ elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
+ lapply (lifts_trans … HVW2 … HW20 ??) -W2 [3: |*: // ] #H
+ elim (lifts_split_trans … H ? (⫯f)) -H [ |*: /2 width=3 by after_uni_one_sn/ ]
+ /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/
+ ]
+ | elim (cpg_inv_cast1 … H2) -H2 *
+ [ #cW #cU #W2 #U2 #HW12 #HU12 #HcWU #H1 #H2 destruct
+ elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
+ elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
+ /3 width=5 by cpg_cast, lifts_flat, ex2_intro/
+ | #cU #HU12 #H destruct
+ elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
+ /3 width=3 by cpg_eps, ex2_intro/
+ | #cW #HW12 #H destruct
+ elim (IH … HW12 … HLK … HVW1) -IH -HW12 -HLK -HVW1 //
+ /3 width=3 by cpg_ee, ex2_intro/
+ ]
+ ]
]
qed-.
+
+lemma cpg_inv_lifts_bi: ∀Rt. reflexive … Rt →
+ ∀c,h,G. d_deliftable2_bi … lifts (cpg Rt h c G).
+/3 width=12 by cpg_inv_lifts_sn, d_deliftable2_sn_bi, lifts_inj/ qed-.
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