-(**************************************************************************)
-(* ___ *)
-(* ||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 "basic_2/unfold/tpss.ma".
-include "basic_2/reducibility/tpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-(* Basic_1: includes: pr2_delta1 *)
-definition cpr: lenv → relation term ≝
- λL,T1,T2. ∃∃T. T1 ➡ T & L ⊢ T ▶* [0, |L|] T2.
-
-interpretation
- "context-sensitive parallel reduction (term)"
- 'PRed L T1 T2 = (cpr L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma cpr_intro: ∀L,T1,T,T2,d,e. T1 ➡ T → L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ➡ T2.
-/3 width=5/ qed-.
-
-(* Basic_1: was by definition: pr2_free *)
-lemma cpr_tpr: ∀T1,T2. T1 ➡ T2 → ∀L. L ⊢ T1 ➡ T2.
-/2 width=3/ qed.
-
-lemma cpr_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ➡ T2.
-/3 width=5/ qed.
-
-lemma cpr_refl: ∀L,T. L ⊢ T ➡ T.
-/2 width=1/ qed.
-
-(* Note: new property *)
-(* Basic_1: was only: pr2_thin_dx *)
-lemma cpr_flat: ∀I,L,V1,V2,T1,T2.
- L ⊢ V1 ➡ V2 → L ⊢ T1 ➡ T2 → L ⊢ ⓕ{I} V1. T1 ➡ ⓕ{I} V2. T2.
-#I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 * /3 width=5/
-qed.
-
-lemma cpr_cast: ∀L,V,T1,T2.
- L ⊢ T1 ➡ T2 → L ⊢ ⓝV. T1 ➡ T2.
-#L #V #T1 #T2 * /3 width=3/
-qed.
-
-(* Note: it does not hold replacing |L1| with |L2| *)
-(* Basic_1: was only: pr2_change *)
-lemma cpr_lsubr_trans: ∀L1,T1,T2. L1 ⊢ T1 ➡ T2 →
- ∀L2. L2 ⊑ [0, |L1|] L1 → L2 ⊢ T1 ➡ T2.
-#L1 #T1 #T2 * #T #HT1 #HT2 #L2 #HL12
-lapply (tpss_lsubr_trans … HT2 … HL12) -HT2 -HL12 /3 width=4/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Basic_1: was: pr2_gen_csort *)
-lemma cpr_inv_atom: ∀T1,T2. ⋆ ⊢ T1 ➡ T2 → T1 ➡ T2.
-#T1 #T2 * #T #HT normalize #HT2
-<(tpss_inv_refl_O2 … HT2) -HT2 //
-qed-.
-
-(* Basic_1: was: pr2_gen_sort *)
-lemma cpr_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ➡ T2 → T2 = ⋆k.
-#L #T2 #k * #X #H
->(tpr_inv_atom1 … H) -H #H
->(tpss_inv_sort1 … H) -H //
-qed-.
-
-(* Basic_1: was: pr2_gen_cast *)
-lemma cpr_inv_cast1: ∀L,V1,T1,U2. L ⊢ ⓝV1. T1 ➡ U2 → (
- ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
- U2 = ⓝV2. T2
- ) ∨ L ⊢ T1 ➡ U2.
-#L #V1 #T1 #U2 * #X #H #HU2
-elim (tpr_inv_cast1 … H) -H /3 width=3/
-* #V #T #HV1 #HT1 #H destruct
-elim (tpss_inv_flat1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma cpr_fwd_bind1_minus: ∀I,L,V1,T1,T. L ⊢ -ⓑ{I}V1.T1 ➡ T → ∀b.
- ∃∃V2,T2. L ⊢ ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
- T = -ⓑ{I}V2.T2.
-#I #L #V1 #T1 #T * #X #H1 #H2 #b
-elim (tpr_fwd_bind1_minus … H1 b) -H1 #V0 #T0 #HT10 #H destruct
-elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/
+lemma cpr_delift: ∀G,K,V,T1,L,l. ⬇[l] L ≡ (K.ⓓV) →
+ ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡ T2 & ⬆[l, 1] T ≡ T2.
+#G #K #V #T1 elim T1 -T1
+[ * /2 width=4 by cpr_atom, lift_sort, lift_gref, ex2_2_intro/
+ #i #L #l #HLK elim (lt_or_eq_or_gt i l)
+ #Hil [1,3: /4 width=4 by lift_lref_ge_minus, lift_lref_lt, ylt_inj, yle_inj, ex2_2_intro/ ]
+ destruct
+ elim (lift_total V 0 (i+1)) #W #HVW
+ elim (lift_split … HVW i i) /3 width=6 by cpr_delta, ex2_2_intro/
+| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #l #HLK
+ elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
+ [ elim (IHU1 (L. ⓑ{I}W1) (l+1)) -IHU1 /3 width=9 by drop_drop, cpr_bind, lift_bind, ex2_2_intro/
+ | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpr_flat, lift_flat, ex2_2_intro/
+ ]
+]
qed-.
-lemma cpr_fwd_shift1: ∀L,L1,T1,T. L ⊢ L1 @@ T1 ➡ T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#L #L1 #T1 #T * #X #H1 #H2
-elim (tpr_fwd_shift1 … H1) -H1 #L0 #T0 #HL10 #H destruct
-elim (tpss_fwd_shift1 … H2) -H2 #L2 #T2 #HL02 #H destruct /2 width=4/
+fact lstas_cpr_aux: ∀h,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*[h, d] T2 →
+ d = 0 → ⦃G, L⦄ ⊢ T1 ➡ T2.
+#h #G #L #T1 #T2 #d #H elim H -G -L -T1 -T2 -d
+/3 width=1 by cpr_eps, cpr_flat, cpr_bind/
+[ #G #L #K #V1 #V2 #W2 #i #d #HLK #_ #HVW2 #IHV12 #H destruct
+ /3 width=6 by cpr_delta/
+| #G #L #K #V1 #V2 #W2 #i #d #_ #_ #_ #_ <plus_n_Sm #H destruct
+]
qed-.
-(* Basic_1: removed theorems 6:
- pr2_head_2 pr2_cflat pr2_gen_cflat clear_pr2_trans
- pr2_gen_ctail pr2_ctail
- Basic_1: removed local theorems 3:
- pr2_free_free pr2_free_delta pr2_delta_delta
-*)
+lemma lstas_cpr: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*[h, 0] T2 → ⦃G, L⦄ ⊢ T1 ➡ T2.
+/2 width=4 by lstas_cpr_aux/ qed.
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