(* 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 (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 (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 (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 (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.
- ⬇*[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 *
+lemma cpm_delta_drops (h) (n) (G) (L):
+ ∀K,V,V2,W2,i.
+ ⇩[i] L ≘ K.ⓓV → ❪G,K❫ ⊢ V ➡[h,n] V2 →
+ ⇧[↑i] V2 ≘ W2 → ❪G,L❫ ⊢ #i ➡[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.
- ⬇*[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 *
+lemma cpm_ell_drops (h) (n) (G) (L):
+ ∀K,V,V2,W2,i.
+ ⇩[i] L ≘ K.ⓛV → ❪G,K❫ ⊢ V ➡[h,n] V2 →
+ ⇧[↑i] V2 ≘ W2 → ❪G,L❫ ⊢ #i ➡[h,↑n] 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
- | ∃∃K,V,V2,i. ⬇*[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ➡[n, h] V2 &
- ⬆*[⫯i] V2 ≡ T2 & I = LRef i
- | ∃∃k,K,V,V2,i. ⬇*[i] L ≡ K.ⓛV & ⦃G, K⦄ ⊢ V ➡[k, h] V2 &
- ⬆*[⫯i] V2 ≡ T2 & I = LRef i & n = ⫯k.
-#n #h #I #G #L #T2 * #c #Hc #H elim (cpg_inv_atom1_drops … H) -H *
+lemma cpm_inv_atom1_drops (h) (n) (G) (L):
+ ∀I,T2. ❪G,L❫ ⊢ ⓪[I] ➡[h,n] T2 →
+ ∨∨ ∧∧ T2 = ⓪[I] & n = 0
+ | ∃∃s. T2 = ⋆(⫯[h]s) & I = Sort s & n = 1
+ | ∃∃K,V,V2,i. ⇩[i] L ≘ K.ⓓV & ❪G,K❫ ⊢ V ➡[h,n] V2 & ⇧[↑i] V2 ≘ T2 & I = LRef i
+ | ∃∃m,K,V,V2,i. ⇩[i] L ≘ K.ⓛV & ❪G,K❫ ⊢ V ➡[h,m] V2 & ⇧[↑i] V2 ≘ T2 & I = LRef i & n = ↑m.
+#h #n #G #L #I #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
+| #s1 #s2 #H1 #H2 #H3 #H4 destruct lapply (isrt_inv_01 … Hc) -Hc
/4 width=3 by or4_intro1, ex3_intro, sym_eq/ (**) (* sym_eq *)
| #cV #i #K #V1 #V2 #HLK #HV12 #HVT2 #H1 #H2 destruct
/4 width=8 by ex4_4_intro, ex2_intro, or4_intro2/
]
qed-.
-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 &
- ⬆*[⫯i] V2 ≡ T2
- | ∃∃k,K,V,V2. ⬇*[i] L ≡ K. ⓛV & ⦃G, K⦄ ⊢ V ➡[k, h] V2 &
- ⬆*[⫯i] V2 ≡ T2 & n = ⫯k.
-#n #h #G #L #T2 #i * #c #Hc #H elim (cpg_inv_lref1_drops … H) -H *
+lemma cpm_inv_lref1_drops (h) (n) (G) (L):
+ ∀T2,i. ❪G,L❫ ⊢ #i ➡[h,n] T2 →
+ ∨∨ ∧∧ T2 = #i & n = 0
+ | ∃∃K,V,V2. ⇩[i] L ≘ K.ⓓV & ❪G,K❫ ⊢ V ➡[h,n] V2 & ⇧[↑i] V2 ≘ T2
+ | ∃∃m,K,V,V2. ⇩[i] L ≘ K. ⓛV & ❪G,K❫ ⊢ V ➡[h,m] V2 & ⇧[↑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
]
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 (cpm n h G).
-#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-.
+(* Advanced forward lemmas **************************************************)
-lemma cpm_lifts_bi: ∀n,h,G. d_liftable2_bi (cpm n h G).
-/3 width=9 by cpm_lifts_sn, d_liftable2_sn_bi/ 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 (cpm n h G).
-#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 (h) (n) (G) (L):
+ ∀T1,T2. ❪G,L❫ ⊢ T1 ➡[h,n] T2 →
+ ∀n1,n2. n1+n2 = n →
+ ∃∃T. ❪G,L❫ ⊢ T1 ➡[h,n1] T & ❪G,L❫ ⊢ T ➡[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/
+| #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 (cpm n h G).
-/3 width=9 by cpm_inv_lifts_sn, d_deliftable2_sn_bi/ qed-.
+lemma cpm_fwd_plus (h) (G) (L):
+ ∀n1,n2,T1,T2. ❪G,L❫ ⊢ T1 ➡[h,n1+n2] T2 →
+ ∃∃T. ❪G,L❫ ⊢ T1 ➡[h,n1] T & ❪G,L❫ ⊢ T ➡[h,n2] T2.
+/2 width=3 by cpm_fwd_plus_aux/ qed-.
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