--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/relocation/drops_ltc.ma".
+include "basic_2/rt_transition/cpt_drops.ma".
+include "basic_2/rt_computation/cpts.ma".
+
+(* T-BOUND CONTEXT-SENSITIVE PARALLEL T-COMPUTATION FOR TERMS ***************)
+
+(* Properties with generic slicing for local environments *******************)
+
+lemma cpts_lifts_sn (h) (n) (G):
+ d_liftable2_sn … lifts (λL. cpts h G L n).
+/3 width=6 by d2_liftable_sn_ltc, cpt_lifts_sn/ qed-.
+
+lemma cpts_lifts_bi (h) (n) (G):
+ d_liftable2_bi … lifts (λL. cpts h G L n).
+#h #n #G @d_liftable2_sn_bi
+/2 width=6 by cpts_lifts_sn, lifts_mono/
+qed-.
+
+(* Inversion lemmas with generic slicing for local environments *************)
+
+lemma cpts_inv_lifts_sn (h) (n) (G):
+ d_deliftable2_sn … lifts (λL. cpts h G L n).
+/3 width=6 by d2_deliftable_sn_ltc, cpt_inv_lifts_sn/ qed-.
+
+lemma cpts_inv_lifts_bi (h) (n) (G):
+ d_deliftable2_bi … lifts (λL. cpts h G L n).
+#h #n #G @d_deliftable2_sn_bi
+/2 width=6 by cpts_inv_lifts_sn, lifts_inj/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma cpts_delta (h) (n) (G):
+ ∀K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬆*[h,n] V2 →
+ ∀W2. ⇧*[1] V2 ≘ W2 → ⦃G,K.ⓓV1⦄ ⊢ #0 ⬆*[h,n] W2.
+#h #n #G #K #V1 #V2 #H @(cpts_ind_dx … H) -V2
+[ /3 width=3 by cpt_cpts, cpt_delta/
+| #n1 #n2 #V #V2 #_ #IH #HV2 #W2 #HVW2
+ elim (lifts_total V (𝐔❴1❵)) #W #HVW
+ /5 width=11 by cpts_step_dx, cpt_lifts_bi, drops_refl, drops_drop/
+]
+qed.
+
+lemma cpts_ell (h) (n) (G):
+ ∀K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬆*[h,n] V2 →
+ ∀W2. ⇧*[1] V2 ≘ W2 → ⦃G,K.ⓛV1⦄ ⊢ #0 ⬆*[h,↑n] W2.
+#h #n #G #K #V1 #V2 #H @(cpts_ind_dx … H) -V2
+[ /3 width=3 by cpt_cpts, cpt_ell/
+| #n1 #n2 #V #V2 #_ #IH #HV2 #W2 #HVW2
+ elim (lifts_total V (𝐔❴1❵)) #W #HVW >plus_S1
+ /5 width=11 by cpts_step_dx, cpt_lifts_bi, drops_refl, drops_drop/
+]
+qed.
+
+lemma cpts_lref (h) (n) (I) (G):
+ ∀K,T,i. ⦃G,K⦄ ⊢ #i ⬆*[h,n] T →
+ ∀U. ⇧*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬆*[h,n] U.
+#h #n #I #G #K #T #i #H @(cpts_ind_dx … H) -T
+[ /3 width=3 by cpt_cpts, cpt_lref/
+| #n1 #n2 #T #T2 #_ #IH #HT2 #U2 #HTU2
+ elim (lifts_total T (𝐔❴1❵)) #U #TU
+ /5 width=11 by cpts_step_dx, cpt_lifts_bi, drops_refl, drops_drop/
+]
+qed.
+
+lemma cpts_cast_sn (h) (n) (G) (L):
+ ∀U1,U2. ⦃G,L⦄ ⊢ U1 ⬆*[h,n] U2 →
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬆[h,n] T2 → ⦃G,L⦄ ⊢ ⓝU1.T1 ⬆*[h,n] ⓝU2.T2.
+#h #n #G #L #U1 #U2 #H @(cpts_ind_sn … H) -U1 -n
+[ /3 width=3 by cpt_cpts, cpt_cast/
+| #n1 #n2 #U1 #U #HU1 #_ #IH #T1 #T2 #H
+ elim (cpt_fwd_plus … H) -H #T #HT1 #HT2
+ /3 width=3 by cpts_step_sn, cpt_cast/
+]
+qed.
+
+lemma cpts_delta_drops (h) (n) (G):
+ ∀L,K,V,i. ⇩*[i] L ≘ K.ⓓV →
+ ∀V2. ⦃G,K⦄ ⊢ V ⬆*[h,n] V2 →
+ ∀W2. ⇧*[↑i] V2 ≘ W2 → ⦃G,L⦄ ⊢ #i ⬆*[h,n] W2.
+#h #n #G #L #K #V #i #HLK #V2 #H @(cpts_ind_dx … H) -V2
+[ /3 width=6 by cpt_cpts, cpt_delta_drops/
+| #n1 #n2 #V1 #V2 #_ #IH #HV12 #W2 #HVW2
+ lapply (drops_isuni_fwd_drop2 … HLK) -HLK // #HLK
+ elim (lifts_total V1 (𝐔❴↑i❵)) #W1 #HVW1
+ /3 width=11 by cpt_lifts_bi, cpts_step_dx/
+]
+qed.
+
+lemma cpts_ell_drops (h) (n) (G):
+ ∀L,K,W,i. ⇩*[i] L ≘ K.ⓛW →
+ ∀W2. ⦃G,K⦄ ⊢ W ⬆*[h,n] W2 →
+ ∀V2. ⇧*[↑i] W2 ≘ V2 → ⦃G,L⦄ ⊢ #i ⬆*[h,↑n] V2.
+#h #n #G #L #K #W #i #HLK #W2 #H @(cpts_ind_dx … H) -W2
+[ /3 width=6 by cpt_cpts, cpt_ell_drops/
+| #n1 #n2 #W1 #W2 #_ #IH #HW12 #V2 #HWV2
+ lapply (drops_isuni_fwd_drop2 … HLK) -HLK // #HLK
+ elim (lifts_total W1 (𝐔❴↑i❵)) #V1 #HWV1 >plus_S1
+ /3 width=11 by cpt_lifts_bi, cpts_step_dx/
+]
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cpts_inv_lref_sn_drops (h) (n) (G) (L) (i):
+ ∀X2. ⦃G,L⦄ ⊢ #i ⬆*[h,n] X2 →
+ ∨∨ ∧∧ X2 = #i & n = 0
+ | ∃∃K,V,V2. ⇩*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V ⬆*[h,n] V2 & ⇧*[↑i] V2 ≘ X2
+ | ∃∃m,K,V,V2. ⇩*[i] L ≘ K.ⓛV & ⦃G,K⦄ ⊢ V ⬆*[h,m] V2 & ⇧*[↑i] V2 ≘ X2 & n = ↑m.
+#h #n #G #L #i #X2 #H @(cpts_ind_dx … H) -X2
+[ /3 width=1 by or3_intro0, conj/
+| #n1 #n2 #T #T2 #_ #IH #HT2 cases IH -IH *
+ [ #H1 #H2 destruct
+ elim (cpt_inv_lref_sn_drops … HT2) -HT2 *
+ [ /3 width=1 by or3_intro0, conj/
+ | /4 width=6 by cpt_cpts, or3_intro1, ex3_3_intro/
+ | /4 width=8 by cpt_cpts, or3_intro2, ex4_4_intro/
+ ]
+ | #K #V0 #V #HLK #HV0 #HVT
+ lapply (drops_isuni_fwd_drop2 … HLK) // #H0LK
+ elim (cpt_inv_lifts_sn … HT2 … H0LK … HVT) -H0LK -T
+ /4 width=6 by cpts_step_dx, ex3_3_intro, or3_intro1/
+ | #m #K #V0 #V #HLK #HV0 #HVT #H destruct
+ lapply (drops_isuni_fwd_drop2 … HLK) // #H0LK
+ elim (cpt_inv_lifts_sn … HT2 … H0LK … HVT) -H0LK -T
+ /4 width=8 by cpts_step_dx, ex4_4_intro, or3_intro2/
+ ]
+]
+qed-.
+
+lemma cpts_inv_delta_sn (h) (n) (G) (K) (V):
+ ∀X2. ⦃G,K.ⓓV⦄ ⊢ #0 ⬆*[h,n] X2 →
+ ∨∨ ∧∧ X2 = #0 & n = 0
+ | ∃∃V2. ⦃G,K⦄ ⊢ V ⬆*[h,n] V2 & ⇧*[1] V2 ≘ X2.
+#h #n #G #K #V #X2 #H
+elim (cpts_inv_lref_sn_drops … H) -H *
+[ /3 width=1 by or_introl, conj/
+| #Y #X #V2 #H #HV2 #HVT2
+ lapply (drops_fwd_isid … H ?) -H [ // ] #H destruct
+ /3 width=3 by ex2_intro, or_intror/
+| #m #Y #X #V2 #H #HV2 #HVT2
+ lapply (drops_fwd_isid … H ?) -H [ // ] #H destruct
+]
+qed-.
+
+lemma cpts_inv_ell_sn (h) (n) (G) (K) (V):
+ ∀X2. ⦃G,K.ⓛV⦄ ⊢ #0 ⬆*[h,n] X2 →
+ ∨∨ ∧∧ X2 = #0 & n = 0
+ | ∃∃m,V2. ⦃G,K⦄ ⊢ V ⬆*[h,m] V2 & ⇧*[1] V2 ≘ X2 & n = ↑m.
+#h #n #G #K #V #X2 #H
+elim (cpts_inv_lref_sn_drops … H) -H *
+[ /3 width=1 by or_introl, conj/
+| #Y #X #V2 #H #HV2 #HVT2
+ lapply (drops_fwd_isid … H ?) -H [ // ] #H destruct
+| #m #Y #X #V2 #H #HV2 #HVT2 #H0 destruct
+ lapply (drops_fwd_isid … H ?) -H [ // ] #H destruct
+ /3 width=5 by ex3_2_intro, or_intror/
+]
+qed-.
+
+lemma cpts_inv_lref_sn (h) (n) (I) (G) (K) (i):
+ ∀X2. ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬆*[h,n] X2 →
+ ∨∨ ∧∧ X2 = #↑i & n = 0
+ | ∃∃T2. ⦃G,K⦄ ⊢ #i ⬆*[h,n] T2 & ⇧*[1] T2 ≘ X2.
+#h #n #I #G #K #i #X2 #H
+elim (cpts_inv_lref_sn_drops … H) -H *
+[ /3 width=1 by or_introl, conj/
+| #L #V #V2 #H #HV2 #HVU2
+ lapply (drops_inv_drop1 … H) -H #HLK
+ elim (lifts_split_trans … HVU2 (𝐔❴↑i❵) (𝐔❴1❵)) -HVU2
+ [| // ] #T2 #HVT2 #HTU2
+ /4 width=6 by cpts_delta_drops, ex2_intro, or_intror/
+| #m #L #V #V2 #H #HV2 #HVU2 #H0 destruct
+ lapply (drops_inv_drop1 … H) -H #HLK
+ elim (lifts_split_trans … HVU2 (𝐔❴↑i❵) (𝐔❴1❵)) -HVU2
+ [| // ] #T2 #HVT2 #HTU2
+ /4 width=6 by cpts_ell_drops, ex2_intro, or_intror/
+]
+qed-.
+
+lemma cpts_inv_succ_sn (h) (n) (G) (L):
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬆*[h,↑n] T2 →
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ⬆*[h,1] T & ⦃G,L⦄ ⊢ T ⬆*[h,n] T2.
+#h #n #G #L #T1 #T2
+@(insert_eq_0 … (↑n)) #m #H
+@(cpts_ind_sn … H) -T1 -m
+[ #H destruct
+| #x1 #n2 #T1 #T #HT1 #HT2 #IH #H
+ elim (plus_inv_S3_sn x1 n2) [1,2: * |*: // ] -H
+ [ #H1 #H2 destruct -HT2
+ elim IH -IH // #T0 #HT0 #HT02
+ /3 width=3 by cpts_step_sn, ex2_intro/
+ | #n1 #H1 #H2 destruct -IH
+ elim (cpt_fwd_plus … 1 n1 … T1 T) [|*: // ] -HT1 #T0 #HT10 #HT0
+ /3 width=5 by cpts_step_sn, cpt_cpts, ex2_intro/
+ ]
+]
+qed-.
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