(* Basic_1: includes: pr0_lift pr2_lift *)
(* Basic_2A1: includes: cpr_lift *)
-lemma cpm_lifts_sn: ∀n,h,G. d_liftable2_sn … lifts (λL. cpm h G L n).
-#n #h #G #K #T1 #T2 * #c #Hc #HT12 #b #f #L #HLK #U1 #HTU1
+lemma cpm_lifts_sn: ∀h,n,G. d_liftable2_sn … lifts (λL. cpm h G L n).
+#h #n #G #K #T1 #T2 * #c #Hc #HT12 #b #f #L #HLK #U1 #HTU1
elim (cpg_lifts_sn … HT12 … HLK … HTU1) -K -T1
/3 width=5 by ex2_intro/
qed-.
-lemma cpm_lifts_bi: ∀n,h,G. d_liftable2_bi … lifts (λL. cpm h G L n).
-#n #h #G #K #T1 #T2 * /3 width=11 by cpg_lifts_bi, ex2_intro/
+lemma cpm_lifts_bi: ∀h,n,G. d_liftable2_bi … lifts (λL. cpm h G L n).
+#h #n #G #K #T1 #T2 * /3 width=11 by cpg_lifts_bi, ex2_intro/
qed-.
(* Inversion lemmas with generic slicing for local environments *************)
(* Basic_1: includes: pr0_gen_lift pr2_gen_lift *)
(* Basic_2A1: includes: cpr_inv_lift1 *)
-lemma cpm_inv_lifts_sn: ∀n,h,G. d_deliftable2_sn … lifts (λL. cpm h G L n).
-#n #h #G #L #U1 #U2 * #c #Hc #HU12 #b #f #K #HLK #T1 #HTU1
+lemma cpm_inv_lifts_sn: ∀h,n,G. d_deliftable2_sn … lifts (λL. cpm h G L n).
+#h #n #G #L #U1 #U2 * #c #Hc #HU12 #b #f #K #HLK #T1 #HTU1
elim (cpg_inv_lifts_sn … HU12 … HLK … HTU1) -L -U1
/3 width=5 by ex2_intro/
qed-.
-lemma cpm_inv_lifts_bi: ∀n,h,G. d_deliftable2_bi … lifts (λL. cpm h G L n).
-#n #h #G #L #U1 #U2 * /3 width=11 by cpg_inv_lifts_bi, ex2_intro/
+lemma cpm_inv_lifts_bi: ∀h,n,G. d_deliftable2_bi … lifts (λL. cpm h G L n).
+#h #n #G #L #U1 #U2 * /3 width=11 by cpg_inv_lifts_bi, ex2_intro/
qed-.
(* Advanced properties ******************************************************)
(* Basic_1: includes: pr2_delta1 *)
(* Basic_2A1: includes: cpr_delta *)
-lemma cpm_delta_drops: ∀n,h,G,L,K,V,V2,W2,i.
- â¬\87*[i] L â\89\98 K.â\93\93V â\86\92 â¦\83G, Kâ¦\84 â\8a¢ V â\9e¡[n, h] V2 →
- â¬\86*[â\86\91i] V2 â\89\98 W2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ #i â\9e¡[n, h] W2.
-#n #h #G #L #K #V #V2 #W2 #i #HLK *
+lemma cpm_delta_drops: ∀h,n,G,L,K,V,V2,W2,i.
+ â\87©[i] L â\89\98 K.â\93\93V â\86\92 â\9dªG,Kâ\9d« â\8a¢ V â\9e¡[h,n] V2 →
+ â\87§[â\86\91i] V2 â\89\98 W2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ #i â\9e¡[h,n] W2.
+#h #n #G #L #K #V #V2 #W2 #i #HLK *
/3 width=8 by cpg_delta_drops, ex2_intro/
qed.
-lemma cpm_ell_drops: ∀n,h,G,L,K,V,V2,W2,i.
- â¬\87*[i] L â\89\98 K.â\93\9bV â\86\92 â¦\83G, Kâ¦\84 â\8a¢ V â\9e¡[n, h] V2 →
- â¬\86*[â\86\91i] V2 â\89\98 W2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ #i â\9e¡[â\86\91n, h] W2.
-#n #h #G #L #K #V #V2 #W2 #i #HLK *
+lemma cpm_ell_drops: ∀h,n,G,L,K,V,V2,W2,i.
+ â\87©[i] L â\89\98 K.â\93\9bV â\86\92 â\9dªG,Kâ\9d« â\8a¢ V â\9e¡[h,n] V2 →
+ â\87§[â\86\91i] V2 â\89\98 W2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ #i â\9e¡[h,â\86\91n] W2.
+#h #n #G #L #K #V #V2 #W2 #i #HLK *
/3 width=8 by cpg_ell_drops, isrt_succ, ex2_intro/
qed.
(* Advanced inversion lemmas ************************************************)
-lemma cpm_inv_atom1_drops: ∀n,h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡[n, h] T2 →
- ∨∨ T2 = ⓪{I} ∧ n = 0
- | ∃∃s. T2 = ⋆(next h s) & I = Sort s & n = 1
- | â\88\83â\88\83K,V,V2,i. â¬\87*[i] L â\89\98 K.â\93\93V & â¦\83G, Kâ¦\84 â\8a¢ V â\9e¡[n, h] V2 &
- â¬\86*[↑i] V2 ≘ T2 & I = LRef i
- | â\88\83â\88\83m,K,V,V2,i. â¬\87*[i] L â\89\98 K.â\93\9bV & â¦\83G, Kâ¦\84 â\8a¢ V â\9e¡[m, h] V2 &
- â¬\86*[↑i] V2 ≘ T2 & I = LRef i & n = ↑m.
-#n #h #I #G #L #T2 * #c #Hc #H elim (cpg_inv_atom1_drops … H) -H *
+lemma cpm_inv_atom1_drops: ∀h,n,I,G,L,T2. ❪G,L❫ ⊢ ⓪[I] ➡[h,n] T2 →
+ ∨∨ T2 = ⓪[I] ∧ n = 0
+ | ∃∃s. T2 = ⋆(⫯[h]s) & I = Sort s & n = 1
+ | â\88\83â\88\83K,V,V2,i. â\87©[i] L â\89\98 K.â\93\93V & â\9dªG,Kâ\9d« â\8a¢ V â\9e¡[h,n] V2 &
+ â\87§[↑i] V2 ≘ T2 & I = LRef i
+ | â\88\83â\88\83m,K,V,V2,i. â\87©[i] L â\89\98 K.â\93\9bV & â\9dªG,Kâ\9d« â\8a¢ V â\9e¡[h,m] V2 &
+ â\87§[↑i] V2 ≘ T2 & I = LRef i & n = ↑m.
+#h #n #I #G #L #T2 * #c #Hc #H elim (cpg_inv_atom1_drops … H) -H *
[ #H1 #H2 destruct lapply (isrt_inv_00 … Hc) -Hc
/3 width=1 by or4_intro0, conj/
| #s #H1 #H2 #H3 destruct lapply (isrt_inv_01 … Hc) -Hc
]
qed-.
-lemma cpm_inv_lref1_drops: ∀n,h,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[n, h] T2 →
+lemma cpm_inv_lref1_drops: ∀h,n,G,L,T2,i. ❪G,L❫ ⊢ #i ➡[h,n] T2 →
∨∨ T2 = #i ∧ n = 0
- | â\88\83â\88\83K,V,V2. â¬\87*[i] L â\89\98 K.â\93\93V & â¦\83G, Kâ¦\84 â\8a¢ V â\9e¡[n, h] V2 &
- â¬\86*[↑i] V2 ≘ T2
- | â\88\83â\88\83m,K,V,V2. â¬\87*[i] L â\89\98 K. â\93\9bV & â¦\83G, Kâ¦\84 â\8a¢ V â\9e¡[m, h] V2 &
- â¬\86*[↑i] V2 ≘ T2 & n = ↑m.
-#n #h #G #L #T2 #i * #c #Hc #H elim (cpg_inv_lref1_drops … H) -H *
+ | â\88\83â\88\83K,V,V2. â\87©[i] L â\89\98 K.â\93\93V & â\9dªG,Kâ\9d« â\8a¢ V â\9e¡[h,n] V2 &
+ â\87§[↑i] V2 ≘ T2
+ | â\88\83â\88\83m,K,V,V2. â\87©[i] L â\89\98 K. â\93\9bV & â\9dªG,Kâ\9d« â\8a¢ V â\9e¡[h,m] V2 &
+ â\87§[↑i] V2 ≘ T2 & n = ↑m.
+#h #n #G #L #T2 #i * #c #Hc #H elim (cpg_inv_lref1_drops … H) -H *
[ #H1 #H2 destruct lapply (isrt_inv_00 … Hc) -Hc
/3 width=1 by or3_intro0, conj/
| #cV #K #V1 #V2 #HLK #HV12 #HVT2 #H destruct
(* Advanced forward lemmas **************************************************)
-fact cpm_fwd_plus_aux (n) (h): ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
+fact cpm_fwd_plus_aux (h) (n): ∀G,L,T1,T2. ❪G,L❫ ⊢ T1 ➡[h,n] T2 →
∀n1,n2. n1+n2 = n →
- â\88\83â\88\83T. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[n1, h] T & â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡[n2, h] T2.
-#n #h #G #L #T1 #T2 #H @(cpm_ind … H) -G -L -T1 -T2 -n
+ â\88\83â\88\83T. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[h,n1] T & â\9dªG,Lâ\9d« â\8a¢ T â\9e¡[h,n2] T2.
+#h #n #G #L #T1 #T2 #H @(cpm_ind … H) -G -L -T1 -T2 -n
[ #I #G #L #n1 #n2 #H
elim (plus_inv_O3 … H) -H #H1 #H2 destruct
/2 width=3 by ex2_intro/
]
| #n #G #K #V1 #V2 #W2 #_ #IH #HVW2 #n1 #n2 #H destruct
elim IH [|*: // ] -IH #V #HV1 #HV2
- elim (lifts_total V ð\9d\90\94â\9d´â\86\91Oâ\9dµ) #W #HVW
+ elim (lifts_total V ð\9d\90\94â\9d¨â\86\91Oâ\9d©) #W #HVW
/5 width=11 by cpm_lifts_bi, cpm_delta, drops_refl, drops_drop, ex2_intro/
| #n #G #K #V1 #V2 #W2 #HV12 #IH #HVW2 #x1 #n2 #H
elim (plus_inv_S3_sn … H) -H *
[ #H1 #H2 destruct -IH /3 width=3 by cpm_ell, ex2_intro/
| #n1 #H1 #H2 destruct -HV12
elim (IH n1) [|*: // ] -IH #V #HV1 #HV2
- elim (lifts_total V ð\9d\90\94â\9d´â\86\91Oâ\9dµ) #W #HVW
+ elim (lifts_total V ð\9d\90\94â\9d¨â\86\91Oâ\9d©) #W #HVW
/5 width=11 by cpm_lifts_bi, cpm_ell, drops_refl, drops_drop, ex2_intro/
]
| #n #I #G #K #T2 #U2 #i #_ #IH #HTU2 #n1 #n2 #H destruct
elim IH [|*: // ] -IH #T #HT1 #HT2
- elim (lifts_total T ð\9d\90\94â\9d´â\86\91Oâ\9dµ) #U #HTU
+ elim (lifts_total T ð\9d\90\94â\9d¨â\86\91Oâ\9d©) #U #HTU
/5 width=11 by cpm_lifts_bi, cpm_lref, drops_refl, drops_drop, ex2_intro/
| #n #p #I #G #L #V1 #V2 #T1 #T2 #HV12 #_ #_ #IHT #n1 #n2 #H destruct
elim IHT [|*: // ] -IHT #T #HT1 #HT2
]
qed-.
-lemma cpm_fwd_plus (h) (G) (L): â\88\80n1,n2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[n1+n2, h] T2 →
- â\88\83â\88\83T. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[n1, h] T & â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡[n2, h] T2.
+lemma cpm_fwd_plus (h) (G) (L): â\88\80n1,n2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[h,n1+n2] T2 →
+ â\88\83â\88\83T. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[h,n1] T & â\9dªG,Lâ\9d« â\8a¢ T â\9e¡[h,n2] T2.
/2 width=3 by cpm_fwd_plus_aux/ qed-.