(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
+(* Properties with generic slicing for local environments *******************)
+
+(* 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
+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_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_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-.
+
(* 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.
- ⬇*[i] L ≘ K.ⓓV → ⦃G, K⦄ ⊢ V ➡[n, h] V2 →
- ⬆*[↑i] V2 ≘ W2 → ⦃G, L⦄ ⊢ #i ➡[n, h] W2.
+ ⬇*[i] L ≘ K.ⓓV → ⦃G,K⦄ ⊢ V ➡[n,h] V2 →
+ ⬆*[↑i] V2 ≘ 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.
- ⬇*[i] L ≘ K.ⓛV → ⦃G, K⦄ ⊢ V ➡[n, h] V2 →
- ⬆*[↑i] V2 ≘ W2 → ⦃G, L⦄ ⊢ #i ➡[↑n, h] W2.
+ ⬇*[i] L ≘ K.ⓛV → ⦃G,K⦄ ⊢ V ➡[n,h] V2 →
+ ⬆*[↑i] V2 ≘ W2 → ⦃G,L⦄ ⊢ #i ➡[↑n,h] W2.
#n #h #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 →
+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
- | ∃∃K,V,V2,i. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ➡[n, h] V2 &
+ | ∃∃K,V,V2,i. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V ➡[n,h] V2 &
⬆*[↑i] V2 ≘ T2 & I = LRef i
- | ∃∃m,K,V,V2,i. ⬇*[i] L ≘ K.ⓛV & ⦃G, K⦄ ⊢ V ➡[m, h] V2 &
+ | ∃∃m,K,V,V2,i. ⬇*[i] L ≘ K.ⓛV & ⦃G,K⦄ ⊢ V ➡[m,h] V2 &
⬆*[↑i] V2 ≘ T2 & I = LRef i & n = ↑m.
#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
]
qed-.
-lemma cpm_inv_lref1_drops: ∀n,h,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[n, h] T2 →
+lemma cpm_inv_lref1_drops: ∀n,h,G,L,T2,i. ⦃G,L⦄ ⊢ #i ➡[n,h] T2 →
∨∨ T2 = #i ∧ n = 0
- | ∃∃K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ➡[n, h] V2 &
+ | ∃∃K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V ➡[n,h] V2 &
⬆*[↑i] V2 ≘ T2
- | ∃∃m,K,V,V2. ⬇*[i] L ≘ K. ⓛV & ⦃G, K⦄ ⊢ V ➡[m, h] V2 &
+ | ∃∃m,K,V,V2. ⬇*[i] L ≘ K. ⓛV & ⦃G,K⦄ ⊢ V ➡[m,h] V2 &
⬆*[↑i] V2 ≘ T2 & n = ↑m.
#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
]
qed-.
-(* Properties with generic slicing for local environments *******************)
-
-(* 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
-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 *************)
+(* Advanced forward lemmas **************************************************)
-(* 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
-elim (cpg_inv_lifts_sn … HU12 … HLK … HTU1) -L -U1
-/3 width=5 by ex2_intro/
+fact cpm_fwd_plus_aux (n) (h): ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
+ ∀n1,n2. n1+n2 = n →
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T & ⦃G,L⦄ ⊢ T ➡[n2,h] T2.
+#n #h #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/
+| #G #L #s #x1 #n2 #H
+ elim (plus_inv_S3_sn … H) -H *
+ [ #H1 #H2 destruct /2 width=3 by ex2_intro/
+ | #n1 #H1 #H elim (plus_inv_O3 … H) -H #H2 #H3 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 𝐔❴↑O❵) #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 𝐔❴↑O❵) #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 𝐔❴↑O❵) #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
+ /3 width=5 by cpm_bind, ex2_intro/
+| #n #G #L #V1 #V2 #T1 #T2 #HV12 #_ #_ #IHT #n1 #n2 #H destruct
+ elim IHT [|*: // ] -IHT #T #HT1 #HT2
+ /3 width=5 by cpm_appl, ex2_intro/
+| #n #G #L #U1 #U2 #T1 #T2 #_ #_ #IHU #IHT #n1 #n2 #H destruct
+ elim IHU [|*: // ] -IHU #U #HU1 #HU2
+ elim IHT [|*: // ] -IHT #T #HT1 #HT2
+ /3 width=5 by cpm_cast, ex2_intro/
+| #n #G #K #V #U1 #T1 #T2 #HTU1 #_ #IH #n1 #n2 #H destruct
+ elim IH [|*: // ] -IH #T #HT1 #HT2
+ /3 width=3 by cpm_zeta, ex2_intro/
+| #n #G #L #U #T1 #T2 #_ #IH #n1 #n2 #H destruct
+ elim IH [|*: // ] -IH #T #HT1 #HT2
+ /3 width=3 by cpm_eps, ex2_intro/
+| #n #G #L #U1 #U2 #T #HU12 #IH #x1 #n2 #H
+ elim (plus_inv_S3_sn … H) -H *
+ [ #H1 #H2 destruct -IH /3 width=4 by cpm_ee, cpm_cast, ex2_intro/
+ | #n1 #H1 #H2 destruct -HU12
+ elim (IH n1) [|*: // ] -IH #U #HU1 #HU2
+ /3 width=3 by cpm_ee, ex2_intro/
+ ]
+| #n #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #_ #_ #_ #IH #n1 #n2 #H destruct
+ elim IH [|*: // ] -IH #T #HT1 #HT2
+ /4 width=7 by cpm_beta, cpm_appl, cpm_bind, ex2_intro/
+| #n #p #G #L #V1 #V2 #U2 #W1 #W2 #T1 #T2 #HV12 #HW12 #_ #_ #_ #IH #HVU2 #n1 #n2 #H destruct
+ elim IH [|*: // ] -IH #T #HT1 #HT2
+ /4 width=7 by cpm_theta, cpm_appl, cpm_bind, 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-.
+lemma cpm_fwd_plus (h) (G) (L): ∀n1,n2,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n1+n2,h] T2 →
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T & ⦃G,L⦄ ⊢ T ➡[n2,h] T2.
+/2 width=3 by cpm_fwd_plus_aux/ qed-.
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