(* Properties with generic slicing for local environments *******************)
(* Basic_2A1: uses: snv_lift *)
-lemma csv_lifts (a) (h): ∀G. d_liftable1 (cnv a h G).
+lemma cnv_lifts (a) (h): ∀G. d_liftable1 (cnv a h G).
#a #h #G #K #T
@(fqup_wf_ind_eq (Ⓣ) … G K T) -G -K -T #G0 #K0 #T0 #IH #G #K * * [|||| * ]
[ #s #HG #HK #HT #_ #b #f #L #_ #X #H2 destruct
--- /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 "basic_2/rt_computation/fpbg.ma".
+include "basic_2/rt_computation/cpms_fpbs.ma".
+include "basic_2/dynamic/cnv.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+(* Inductive premises for the preservation results **************************)
+
+definition IH_cnv_cpm_trans_lpr (a) (h): relation3 genv lenv term ≝
+ λG,L1,T1. ⦃G, L1⦄ ⊢ T1 ![a,h] →
+ ∀n,T2. ⦃G, L1⦄ ⊢ T1 ➡[n,h] T2 →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L2⦄ ⊢ T2 ![a,h].
+
+definition IH_cnv_cpms_trans_lpr (a) (h): relation3 genv lenv term ≝
+ λG,L1,T1. ⦃G, L1⦄ ⊢ T1 ![a,h] →
+ ∀n,T2. ⦃G, L1⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L2⦄ ⊢ T2 ![a,h].
+
+definition IH_cnv_cpm_conf_lpr (a) (h): relation3 genv lenv term ≝
+ λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
+ ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
+
+definition IH_cnv_cpms_strip_lpr (a) (h): relation3 genv lenv term ≝
+ λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
+ ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
+
+definition IH_cnv_cpms_conf_lpr (a) (h): relation3 genv lenv term ≝
+ λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
+ ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡*[n2,h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
+
+(* Auxiliary properties for preservation ************************************)
+
+fact cnv_cpms_trans_lpr_aux (a) (h) (o):
+ ∀G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_trans_lpr a h G1 L1 T1.
+#a #h #o #G0 #L0 #T0 #IH #G1 #L1 #T1 #H01 #HT1 #n #T2 #H
+@(cpms_ind_dx … H) -n -T2
+/4 width=7 by cpms_fwd_fpbs, fpbg_fpbs_trans/
+qed-.
+(*
+fact cnv_cpms_strip_lpr_aux (a) (h) (o):
+ ∀G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpm_conf_lpr a h G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_strip_lpr a h G1 L1 T1.
+#a #h #o #G0 #L0 #T0 #IH0 #G #L #T #H0 #HT #n1 #T1 #H
+generalize in match HT; generalize in match H0; -H0 -HT
+@(cpms_ind_sn … H) -n1 -T [ /2 width=8 by/ ]
+#n1 #n2 #T #X #HTX #HXT1 #IH #H0 #HT #n2 #T2 #HT2 #L1 #HL1 #L2 #HL2
+elim (IH0 … HTX … HT2 … HL1 … HL2) // -L -T #T0 #HXT0 #HT20
+
+ @(IH … 0 T … HT2 … HL1 … HL2) // -L -IH
+ #T0 #HT20 #HT0
+*)
--- /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 "basic_2/rt_computation/cpms_fpbg.ma".
+include "basic_2/rt_computation/cprs_cprs.ma".
+include "basic_2/rt_computation/lprs_cpms.ma".
+include "basic_2/dynamic/cnv_drops.ma".
+(*
+include "basic_2/dynamic/snv_aaa.ma".
+*)
+include "basic_2/dynamic/cnv_etc.ma".
+(*
+include "basic_2/dynamic/lsubsv_snv.ma".
+*)
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+(* Properties with context-free parallel reduction for local environments *****)
+
+fact cnv_cpm_trans_lpr_aux (a) (h) (o): a = Ⓕ →
+ ∀G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
+ ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_cnv_cpm_trans_lpr a h G1 L1 T1.
+#a #h #o #Ha #G0 #L0 #T0 (* #IH4 #IH3 *) #IH2 #IH1 #G1 #L1 * * [|||| * ]
+[ #s #HG0 #HL0 #HT0 #H1 #x #X #H2 #L2 #_ destruct (* -IH4 -IH3 *) -IH2 -IH1 -H1
+ elim (cpm_inv_sort1 … H2) -H2 * #H1 #H2 destruct //
+| #i #HG0 #HL0 #HT0 #H1 #x #X #H2 #L2 #HL12 destruct (* -IH4 -IH3 *) -IH2
+ elim (cnv_inv_lref_drops … H1) -H1 #I #K1 #V1 #HLK1 #HV1
+ elim (lpr_drops_conf … HLK1 … HL12) -HL12 // #Y #H #HLK2
+ elim (lpr_inv_pair_sn … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+ lapply (fqup_lref (Ⓣ) … G1 … HLK1) #HKL
+ elim (cpm_inv_lref1_drops … H2) -H2 *
+ [ #H1 #H2 destruct -HLK1 /4 width=7 by fqup_fpbg, cnv_lref_drops/
+ | #K0 #V0 #W0 #H #HVW0 #W0 -HV12
+ lapply (drops_mono … H … HLK1) -HLK1 -H #H destruct
+ lapply (drops_isuni_fwd_drop2 … HLK2) -HLK2 /4 width=7 by fqup_fpbg, cnv_lifts/
+ | #n #K0 #V0 #W0 #H #HVW0 #W0 #H destruct -HV12
+ lapply (drops_mono … H … HLK1) -HLK1 -H #H destruct
+ lapply (drops_isuni_fwd_drop2 … HLK2) -HLK2 /4 width=7 by fqup_fpbg, cnv_lifts/
+ ]
+| #l #HG0 #HL0 #HT0 #H1 #x #X #H2 #L2 #HL12 destruct (* -IH4 -IH3 *) -IH2 -IH1
+ elim (cnv_inv_gref … H1)
+| #p #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #x #X #H2 #L2 #HL12 destruct (* -IH4 -IH3 *) -IH2
+ elim (cnv_inv_bind … H1) -H1 #HV1 #HT1
+ elim (cpm_inv_bind1 … H2) -H2 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct /4 width=9 by fqup_fpbg, cnv_bind, lpr_pair/
+ | #T2 #HT12 #HXT2 #H1 #H2 destruct -HV1
+ /4 width=11 by fqup_fpbg, cnv_inv_lifts, lpr_pair, drops_refl, drops_drop/
+ ]
+| #V1 #T1 #HG0 #HL0 #HT0 #H1 #x #X #H2 #L2 #HL12 destruct
+ elim (cnv_inv_appl … H1) -H1 #n #p #W1 #U1 #Hn #HV1 #HT1 #HVW1 #HTU1
+ elim (cpm_inv_appl1 … H2) -H2 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct (* -IH4 *)
+ lapply (IH1 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HV2
+ lapply (IH1 … HT12 … HL12) /2 width=1 by fqup_fpbg/ #HT2
+ elim (IH2 … HVW1 … V2 … L2 … L2) [|*: /2 width=2 by fqup_fpbg, cpm_cpms/ ] -HVW1 -HV12
+ <minus_n_O <minus_O_n #XW1 #HXW1 #HXV2
+ elim (IH2 … HTU1 … T2 … L2 … L2) [|*: /2 width=2 by fqup_fpbg, cpm_cpms/ ] -HTU1 -HT12
+ #X #H #HTU2 (* -IH3 *) -IH2 -IH1 -L1 -V1 -T1
+ elim (cpms_inv_abst_sn … H) -H #W2 #U2 #HW12 #_ #H destruct
+ elim (cprs_conf … HXW1 … HW12) -W1 #W1 #HXW1 #HW21
+ lapply (cpms_trans … HXV2 … HXW1) -XW1 <plus_n_O #HV2W1
+ lapply (cpms_trans … HTU2 … (ⓛ{p}W1.U2) ?)
+ [3:|*: /2 width=2 by cpms_bind/ ] -W2 <plus_n_O #HTU2
+ @(cnv_appl … HV2W1 HTU2) // #H destruct
+ | #q #V2 #W10 #W20 #T10 #T20 #HV12 #HW120 #HT120 #H1 #H2 destruct
+ elim (cnv_inv_bind … HT1) -HT1 #HW10 #HT10
+ elim (cpms_inv_abst_sn … HTU1) -HTU1 #W30 #T30 #HW130 #_ #H destruct -T30
+ lapply (IH1 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HV2
+ lapply (IH1 … HW120 … HL12) /2 width=1 by fqup_fpbg/ #HW20
+ lapply (IH1 … HT10 … HT120 … (L2.ⓛW20) ?) /2 width=1 by fqup_fpbg, lpr_pair/ -HT10 -HT120 #HT20
+ elim (IH2 … HVW1 … V2 … L2 … L2) [|*: /2 width=2 by fqup_fpbg, cpm_cpms/ ] -HVW1 -HV12
+ <minus_n_O <minus_O_n #XW1 #HXW1 #HXV2
+ elim (IH2 … HW130 … W20 … L2 … L2) [|*: /2 width=2 by fqup_fpbg, cpm_cpms/ ] -HW130 -HW120
+ <minus_n_O #XW30 #HXW30 #HW230
+ elim (cprs_conf … HXW1 … HXW30) -W30 #W30 #HXW1 #HXW30
+ lapply (cpms_trans … HXV2 … HXW1) -XW1 <plus_n_O #HV2W30
+ lapply (cprs_trans … HW230 … HXW30) -XW30 #HW230
+ @cnv_bind [ /2 width=3 by cnv_cast/ ]
+(*
+ @(lsubsv_snv_trans … HT20) -HT20
+ @(lsubsv_beta … (d-1)) //
+ @shnv_cast [1,2: // ] #d0 #Hd0
+ lapply (scpes_le_aux … IH4 IH1 IH2 IH3 … HW20d … HV2d … d0 … HVW20) -IH4 -IH3 -IH2 -IH1 -HW20d -HV2d -HVW20
+ /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs, le_S_S/
+*)
+ | #q #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct (* -IH4 *)
+ elim (cnv_inv_bind … HT1) -HT1 #HW0 #HT0
+ elim (cpms_inv_abbr_abst … HTU1) -HTU1 #X #HTU0 #HX #H destruct
+ elim (lifts_inv_bind1 … HX) -HX #W3 #U3 #HW13 #_ #H destruct
+ lapply (IH1 … HW02 … HL12) /2 width=1 by fqup_fpbg/ #HW2
+ lapply (IH1 … HV10 … HL12) /2 width=1 by fqup_fpbg/ #HV0
+ lapply (IH1 … HT02 (L2.ⓓW2) ?) /2 width=1 by fqup_fpbg, lpr_pair/ #HT2
+ elim (IH2 … HVW1 … V0 … L2 … L2) [|*: /2 width=2 by fqup_fpbg, cpm_cpms/ ] -HVW1 -HV10
+ <minus_n_O <minus_O_n #XW1 #HXW1 #HXV0
+ elim (IH2 … HTU0 … T2 … (L2.ⓓW2) … (L2.ⓓW2)) [|*: /2 width=2 by fqup_fpbg, cpm_cpms, lpr_pair/ ] -HT02 -HTU0 -HW02
+ #X #H #HTU2 -IH2 -IH1
+ elim (cpms_inv_abst_sn … H) -H #W #U2 #HW3 #_ #H destruct -U3
+ lapply (cnv_lifts … HV0 (Ⓣ) … (L2.ⓓW2) … HV02) /3 width=1 by drops_refl, drops_drop/ -HV0 #HV2
+ elim (cpms_lifts_sn … HXV0 (Ⓣ) … (L2.ⓓW2) … HV02) /3 width=1 by drops_refl, drops_drop/ -V0 #XW2 #HXW12 #HXVW2
+ lapply (cpms_lifts_bi … HXW1 (Ⓣ) … (L2.ⓓW2) … HW13 … HXW12) /3 width=1 by drops_refl, drops_drop/ -W1 -XW1 #HXW32
+ elim (cprs_conf … HXW32 … HW3) -W3 #W3 #HXW23 #HW3
+ lapply (cpms_trans … HXVW2 … HXW23) -XW2 <plus_n_O #H1
+ lapply (cpms_trans … HTU2 ? (ⓛ{p}W3.U2) ?) [3:|*:/2 width=2 by cpms_bind/ ] -W #H2
+ @cnv_bind // @(cnv_appl … H1 H2) // #H destruct
+ ]
+| #W1 #T1 #HG0 #HL0 #HT0 #H1 #x #X #H2 #L2 #HL12 destruct (* -IH4 *)
+ elim (cnv_inv_cast … H1) -H1 #U1 #HW1 #HT1 #HWU1 #HTU1
+ elim (cpm_inv_cast1 … H2) -H2
+ [ * #W2 #T2 #HW12 #HT12 #H destruct
+ lapply (IH1 … HW12 … HL12) /2 width=1 by fqup_fpbg/ #HW2
+ lapply (IH1 … HT12 … HL12) /2 width=1 by fqup_fpbg/ #HT2
+ lapply (cnv_cpms_trans_lpr_aux … IH1 … HTU1 L1 ?) /2 width=1 by fqup_fpbg/ #HU1
+ elim (IH2 … HWU1 … W2 … L1 … HL12) [|*: /2 width=2 by fqup_fpbg, cpm_cpms/ ] -HW12
+ <minus_n_O <minus_O_n #XW1 #HXUW1 #HXW21
+ elim (IH2 … HTU1 … T2 … L1 … HL12) [|*: /2 width=2 by fqup_fpbg, cpm_cpms/ ] -HTU1 -HT12
+ #XT1 #HXUT1 #HXT21
+ elim (IH2 … HXUW1 … HXUT1 … HL12 … HL12) [|*: /2 width=4 by fqup_cpms_fwd_fpbg/ ] -HXUW1 -HXUT1 -HWU1
+ >eq_minus_O // #W0 #H1 #H2
+ lapply (cprs_trans … HXW21 … H1) -XW1 #H1
+ lapply (cpms_trans … HXT21 … H2) -XT1 <arith_l #H2
+ /2 width=3 by cnv_cast/
+ | #HX (* -IH3 *) -IH2 -HW1 -U1
+ lapply (IH1 … HX … HL12) /2 width=1 by fqup_fpbg/
+ | * #n #HX #H destruct -IH2 -HT1 -U1
+ lapply (IH1 … HX … HL12) /2 width=1 by fqup_fpbg/
+ ]
+]
+(*
+qed-.
+*)
cnv.ma cnv_drops.ma cnv_fqus.ma cnv_aaa.ma cnv_fsb.ma
+cnv_etc.ma cnv_lpr.ma
+++ /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 "basic_2/dynamic/snv_lift.ma".
-include "basic_2/dynamic/snv_aaa.ma".
-include "basic_2/dynamic/snv_scpes.ma".
-include "basic_2/dynamic/lsubsv_snv.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Properties on context-free parallel reduction for local environments *****)
-
-fact snv_cpr_lpr_aux: ∀h,o,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_lstas h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
- ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_snv_cpr_lpr h o G1 L1 T1.
-#h #o #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
-[ #s #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #_ destruct -IH4 -IH3 -IH2 -IH1 -H1
- >(cpr_inv_sort1 … H2) -X //
-| #i #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2
- elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
- elim (lpr_drop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- lapply (fqup_lref … G1 … HLK1) #HKL
- elim (cpr_inv_lref1 … H2) -H2
- [ #H destruct -HLK1 /4 width=10 by fqup_fpbg, snv_lref/
- | * #K0 #V0 #W0 #H #HVW0 #W0 -HV12
- lapply (drop_mono … H … HLK1) -HLK1 -H #H destruct
- lapply (drop_fwd_drop2 … HLK2) -HLK2 /4 width=8 by fqup_fpbg, snv_lift/
- ]
-| #p #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2 -IH1
- elim (snv_inv_gref … H1)
-| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2
- elim (snv_inv_bind … H1) -H1 #HV1 #HT1
- elim (cpr_inv_bind1 … H2) -H2 *
- [ #V2 #T2 #HV12 #HT12 #H destruct /4 width=8 by fqup_fpbg, snv_bind, lpr_pair/
- | #T2 #HT12 #HXT2 #H1 #H2 destruct -HV1
- /4 width=10 by fqup_fpbg, snv_inv_lift, lpr_pair, drop_drop/
- ]
-| #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct
- elim (snv_inv_appl … H1) -H1 #a #W1 #U1 #d0 #HV1 #HT1 #HVW1 #HTU1
- elim (cpr_inv_appl1 … H2) -H2 *
- [ #V2 #T2 #HV12 #HT12 #H destruct -IH4
- lapply (IH1 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HV2
- lapply (IH1 … HT12 … HL12) /2 width=1 by fqup_fpbg/ #HT2
- elim (scpds_cpr_lpr_aux … IH2 IH3 … HVW1 … HV12 … HL12) /2 width=1 by fqup_fpbg/ -HVW1 -HV12 #XV #HVW2 #HXV
- elim (scpds_cpr_lpr_aux … IH2 IH3 … HTU1 … HT12 … HL12) /2 width=1 by fqup_fpbg/ -HTU1 -HT12 #X #HTU2 #H
- elim (cprs_inv_abst1 … H) -H #XW #U2 #HXW #_ #H destruct -IH1 -IH3 -IH2 -L1
- elim (cprs_conf … HXV … HXW) -W1 #W2 #HXV #HXW
- lapply (scpds_cprs_trans … HVW2 … HXV) -XV
- lapply (scpds_cprs_trans … (ⓛ{a}W2.U2) … HTU2 ?)
- /2 width=7 by snv_appl, cprs_bind/
- | #b #V2 #W10 #W20 #T10 #T20 #HV12 #HW120 #HT120 #H1 #H2 destruct
- elim (snv_inv_bind … HT1) -HT1 #HW10 #HT10
- elim (scpds_inv_abst1 … HTU1) -HTU1 #W30 #T30 #HW130 #_ #H destruct -T30 -d0
- elim (snv_fwd_da … HV1) #d #HV1d
- elim (snv_fwd_da … HW10) #d0 #HW10d
- lapply (cprs_scpds_div … HW130 … HW10d … 1 HVW1) -W30 #HVW10
- elim (da_scpes_aux … IH4 IH1 IH2 … HW10d … HV1d … HVW10) /2 width=1 by fqup_fpbg/
- #_ #Hd <minus_n_O #H destruct >(plus_minus_k_k d 1) in HV1d; // -Hd #HV1d
- lapply (scpes_cpr_lpr_aux … IH2 IH3 … HVW10 … HW120 … HV12 … HL12) /2 width=1 by fqup_fpbg/ -HVW10 #HVW20
- lapply (IH2 … HV1d … HV12 … HL12) /2 width=1 by fqup_fpbg/ -HV1d #HV2d
- lapply (IH2 … HW10d … HW120 … HL12) /2 width=1 by fqup_fpbg/ -HW10d #HW20d
- lapply (IH1 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HV2
- lapply (IH1 … HW120 … HL12) /2 width=1 by fqup_fpbg/ -HW10 #HW20
- lapply (IH1 … HT10 … HT120 … (L2.ⓛW20) ?) /2 width=1 by fqup_fpbg, lpr_pair/ -HT10 #HT20
- @snv_bind /2 width=1 by snv_cast_scpes/
- @(lsubsv_snv_trans … HT20) -HT20
- @(lsubsv_beta … (d-1)) //
- @shnv_cast [1,2: // ] #d0 #Hd0
- lapply (scpes_le_aux … IH4 IH1 IH2 IH3 … HW20d … HV2d … d0 … HVW20) -IH4 -IH3 -IH2 -IH1 -HW20d -HV2d -HVW20
- /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs, le_S_S/
- | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -IH4
- elim (snv_inv_bind … HT1) -HT1 #HW0 #HT0
- elim (scpds_inv_abbr_abst … HTU1) -HTU1 #X #HTU0 #HX #H destruct
- elim (lift_inv_bind1 … HX) -HX #W3 #U3 #HW13 #_ #H destruct
- elim (scpds_cpr_lpr_aux … IH2 IH3 … HVW1 … HV10 … HL12) /2 width=1 by fqup_fpbg/ -HVW1 #XV #HXV0 #HXVW1
- elim (scpds_cpr_lpr_aux … IH2 IH3 … HTU0 … HT02 (L2.ⓓW2)) /2 width=1 by fqup_fpbg, lpr_pair/ -HTU0 #X #HXT2 #H
- elim (cprs_inv_abst1 … H) -H #W #U2 #HW3 #_ #H destruct -U3
- lapply (IH1 … HW02 … HL12) /2 width=1 by fqup_fpbg/ #HW2
- lapply (IH1 … HV10 … HL12) /2 width=1 by fqup_fpbg/ #HV0
- lapply (IH1 … HT02 (L2.ⓓW2) ?) /2 width=1 by fqup_fpbg, lpr_pair/ -L1 #HT2
- lapply (snv_lift … HV0 (L2.ⓓW2) (Ⓕ) … HV02) /2 width=1 by drop_drop/ -HV0 #HV2
- elim (lift_total XV 0 1) #XW #HXVW
- lapply (scpds_lift … HXV0 (L2.ⓓW2) (Ⓕ) … HV02 … HXVW) /2 width=1 by drop_drop/ -V0 #HXWV2
- lapply (cprs_lift … HXVW1 (L2.ⓓW2) (Ⓕ) … HW13 … HXVW) /2 width=1 by drop_drop/ -W1 -XV #HXW3
- elim (cprs_conf … HXW3 … HW3) -W3 #W3 #HXW3 #HW3
- lapply (scpds_cprs_trans … HXWV2 … HXW3) -XW
- lapply (scpds_cprs_trans … (ⓛ{a}W3.U2) … HXT2 ?) /2 width=1 by cprs_bind/ -W
- /3 width=6 by snv_appl, snv_bind/
- ]
-| #W1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4
- elim (snv_inv_cast … H1) -H1 #U1 #HW1 #HT1 #HWU1 #HTU1
- elim (cpr_inv_cast1 … H2) -H2
- [ * #W2 #T2 #HW12 #HT12 #H destruct
- elim (snv_fwd_da … HW1) #d #HW1d
- lapply (scpds_div … HWU1 … HTU1) -U1 -d #HWT1
- lapply (scpes_cpr_lpr_aux … IH2 IH3 … HWT1 … HW12 … HT12 … HL12) /2 width=1 by fqup_fpbg/
- lapply (IH1 … HW12 … HL12) /2 width=1 by fqup_fpbg/
- lapply (IH1 … HT12 … HL12) /2 width=1 by fqup_fpbg/ -L1
- /2 width=1 by snv_cast_scpes/
- | #H -IH3 -IH2 -HW1 -U1
- lapply (IH1 … H … HL12) /2 width=1 by fqup_fpbg/
- ]
-]
-qed-.
(*
-lemma cprs_fpbs: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → ⦃G, L, T1⦄ ≥[h, o] ⦃G, L, T2⦄.
-/3 width=1 by cprs_cpxs, cpxs_fpbs/ qed.
-
lemma lprs_fpbs: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1, T⦄ ≥[h, o] ⦃G, L2, T⦄.
/3 width=1 by lprs_lpxs, lpxs_fpbs/ qed.
--- /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 "basic_2/rt_computation/fpbg_fqup.ma".
+include "basic_2/rt_computation/cpms_fpbs.ma".
+
+(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
+
+(* Forward lemmas with proper parallel rst-computation for closures *********)
+
+lemma fqup_cpms_fwd_fpbg (h) (o): ∀G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T⦄ →
+ ∀n,T2. ⦃G2, L2⦄ ⊢ T ➡*[n,h] T2 → ⦃G1, L1, T1⦄ >[h,o] ⦃G2, L2, T2⦄.
+/3 width=5 by cpms_fwd_fpbs, fqup_fpbg,fpbg_fpbs_trans/ qed-.
--- /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 "basic_2/rt_computation/fpbs_cpxs.ma".
+include "basic_2/rt_computation/cpms_cpxs.ma".
+
+(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
+
+(* Forward lemmas with parallel rst-computation for closures ****************)
+
+(* Basic_2A1: uses: cprs_fpbs *)
+lemma cpms_fwd_fpbs (n) (h) (o): ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G, L, T1⦄ ≥[h,o] ⦃G, L, T2⦄.
+/3 width=2 by cpms_fwd_cpxs, cpxs_fpbs/ qed-.
| ∃∃s. T2 = ⋆(next h s) & J = Sort s & n = 1
| ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓓV1 & J = LRef 0
- | ∃∃k,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[k, h] V2 & ⬆*[1] V2 ≘ T2 &
- L = K.ⓛV1 & J = LRef 0 & n = ↑k
+ | ∃∃m,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[m, h] V2 & ⬆*[1] V2 ≘ T2 &
+ L = K.ⓛV1 & J = LRef 0 & n = ↑m
| ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T & ⬆*[1] T ≘ T2 &
L = K.ⓘ{I} & J = LRef (↑i).
#n #h #J #G #L #T2 * #c #Hc #H elim (cpg_inv_atom1 … H) -H *
| #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 #H3 destruct
/4 width=6 by or5_intro2, ex4_3_intro, ex2_intro/
| #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 #H3 destruct
- elim (isrt_inv_plus_SO_dx … Hc) -Hc // #k #Hc #H destruct
+ elim (isrt_inv_plus_SO_dx … Hc) -Hc // #m #Hc #H destruct
/4 width=9 by or5_intro3, ex5_4_intro, ex2_intro/
| #I #K #V2 #i #HV2 #HVT2 #H1 #H2 destruct
/4 width=8 by or5_intro4, ex4_4_intro, ex2_intro/
∨∨ T2 = #0 ∧ n = 0
| ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓓV1
- | ∃∃k,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[k, h] V2 & ⬆*[1] V2 ≘ T2 &
- L = K.ⓛV1 & n = ↑k.
+ | ∃∃m,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[m, h] V2 & ⬆*[1] V2 ≘ T2 &
+ L = K.ⓛV1 & n = ↑m.
#n #h #G #L #T2 * #c #Hc #H elim (cpg_inv_zero1 … H) -H *
[ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or3_intro0, conj/
| #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 destruct
/4 width=8 by or3_intro1, ex3_3_intro, ex2_intro/
| #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 destruct
- elim (isrt_inv_plus_SO_dx … Hc) -Hc // #k #Hc #H destruct
+ elim (isrt_inv_plus_SO_dx … Hc) -Hc // #m #Hc #H destruct
/4 width=8 by or3_intro2, ex4_4_intro, ex2_intro/
]
qed-.
∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 & ⦃G, L⦄ ⊢ U1 ➡[n, h] T2 &
U2 = ⓝV2.T2
| ⦃G, L⦄ ⊢ U1 ➡[n, h] U2
- | ∃∃k. ⦃G, L⦄ ⊢ V1 ➡[k, h] U2 & n = ↑k.
+ | ∃∃m. ⦃G, L⦄ ⊢ V1 ➡[m, h] U2 & n = ↑m.
#n #h #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_cast1 … H) -H *
[ #cV #cT #V2 #T2 #HV12 #HT12 #HcVT #H1 #H2 destruct
elim (isrt_inv_max … Hc) -Hc #nV #nT #HcV #HcT #H destruct
| #cU #U12 #H destruct
/4 width=3 by isrt_inv_plus_O_dx, or3_intro1, ex2_intro/
| #cU #H12 #H destruct
- elim (isrt_inv_plus_SO_dx … Hc) -Hc // #k #Hc #H destruct
+ elim (isrt_inv_plus_SO_dx … Hc) -Hc // #m #Hc #H destruct
/4 width=3 by or3_intro2, ex2_intro/
]
qed-.
}
]
[ { "t-bound context-sensitive parallel rt-computation" * } {
- [ [ "for terms" ] "cpms" + "( ⦃?,?⦄ ⊢ ? ➡*[?,?] ? )" "cpms_drops" + "cpms_lsubr" + "cpms_aaa" + "cpms_lpr" + "cpms_cpxs" + "cpms_cpms" * ]
+ [ [ "for terms" ] "cpms" + "( ⦃?,?⦄ ⊢ ? ➡*[?,?] ? )" "cpms_drops" + "cpms_lsubr" + "cpms_aaa" + "cpms_lpr" + "cpms_cpxs" + "cpms_fpbs" + "cpms_fpbg" + "cpms_cpms" * ]
}
]
[ { "unbound context-sensitive parallel rst-computation" * } {
lemma minus_plus_m_m_commutative: ∀n,m:nat. n = m + n - m.
// qed-.
+lemma plus_n_2: ∀n. n + 2 = n + 1 + 1.
+// qed.
+
+lemma arith_l: ∀x. 1 = 1-x+(x-(x-1)).
+* // #x >minus_S_S >minus_S_S <minus_O_n <minus_n_O //
+qed.
+
(* Note: uses minus_minus_comm, minus_plus_m_m, commutative_plus, plus_minus *)
lemma plus_minus_minus_be: ∀x,y,z. y ≤ z → z ≤ x → (x - z) + (z - y) = x - y.
#x #z #y #Hzy #Hyx >plus_minus // >commutative_plus >plus_minus //
fact plus_minus_minus_be_aux: ∀i,x,y,z. y ≤ z → z ≤ x → i = z - y → x - z + i = x - y.
/2 width=1 by plus_minus_minus_be/ qed-.
-lemma plus_n_2: ∀n. n + 2 = n + 1 + 1.
-// qed.
-
lemma le_plus_minus: ∀m,n,p. p ≤ n → m + n - p = m + (n - p).
/2 by plus_minus/ qed-.
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