(* 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¡ K.ⓓV → ⦃G, K⦄ ⊢ V ➡[n, h] V2 →
- â¬\86*[⫯i] V2 â\89¡ W2 → ⦃G, L⦄ ⊢ #i ➡[n, h] W2.
+ â¬\87*[i] L â\89\98 K.ⓓV → ⦃G, K⦄ ⊢ V ➡[n, h] V2 →
+ â¬\86*[â\86\91i] V2 â\89\98 W2 → ⦃G, L⦄ ⊢ #i ➡[n, h] W2.
#n #h #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¡ K.ⓛV → ⦃G, K⦄ ⊢ V ➡[n, h] V2 →
- â¬\86*[⫯i] V2 â\89¡ W2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ #i â\9e¡[⫯n, h] W2.
+ â¬\87*[i] L â\89\98 K.ⓛV → ⦃G, K⦄ ⊢ V ➡[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 *
/3 width=8 by cpg_ell_drops, isrt_succ, ex2_intro/
qed.
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¡ K.ⓓV & ⦃G, K⦄ ⊢ V ➡[n, h] V2 &
- â¬\86*[⫯i] V2 â\89¡ T2 & I = LRef i
- | â\88\83â\88\83m,K,V,V2,i. â¬\87*[i] L â\89¡ K.ⓛV & ⦃G, K⦄ ⊢ V ➡[m, h] V2 &
- â¬\86*[⫯i] V2 â\89¡ T2 & I = LRef i & n = ⫯m.
+ | â\88\83â\88\83K,V,V2,i. â¬\87*[i] L â\89\98 K.ⓓV & ⦃G, K⦄ ⊢ V ➡[n, h] V2 &
+ â¬\86*[â\86\91i] V2 â\89\98 T2 & I = LRef i
+ | â\88\83â\88\83m,K,V,V2,i. â¬\87*[i] L â\89\98 K.ⓛV & ⦃G, K⦄ ⊢ V ➡[m, h] V2 &
+ â¬\86*[â\86\91i] V2 â\89\98 T2 & I = LRef i & n = â\86\91m.
#n #h #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/
lemma cpm_inv_lref1_drops: ∀n,h,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[n, h] T2 →
∨∨ T2 = #i ∧ n = 0
- | â\88\83â\88\83K,V,V2. â¬\87*[i] L â\89¡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡[n, h] V2 &
- â¬\86*[⫯i] V2 â\89¡ T2
- | â\88\83â\88\83m,K,V,V2. â¬\87*[i] L â\89¡ K. ⓛV & ⦃G, K⦄ ⊢ V ➡[m, h] V2 &
- â¬\86*[⫯i] V2 â\89¡ T2 & n = ⫯m.
+ | â\88\83â\88\83K,V,V2. â¬\87*[i] L â\89\98 K.ⓓV & ⦃G, K⦄ ⊢ V ➡[n, h] V2 &
+ â¬\86*[â\86\91i] V2 â\89\98 T2
+ | â\88\83â\88\83m,K,V,V2. â¬\87*[i] L â\89\98 K. ⓛV & ⦃G, K⦄ ⊢ V ➡[m, h] V2 &
+ â¬\86*[â\86\91i] V2 â\89\98 T2 & n = â\86\91m.
#n #h #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/
(* Basic_1: includes: pr0_lift pr2_lift *)
(* Basic_2A1: includes: cpr_lift *)
-lemma cpm_lifts: ∀n,h,G. d_liftable2 (cpm n h G).
+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
-elim (cpg_lifts … HT12 … HLK … HTU1) -K -T1
+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/
+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_lifts1: ∀n,h,G. d_deliftable2_sn (cpm n h G).
+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
-elim (cpg_inv_lifts1 … HU12 … HLK … HTU1) -L -U1
+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/
+qed-.