(* *)
(**************************************************************************)
-include "basic_2/unfold/sstas_sstas.ma".
-include "basic_2/equivalence/cpcs_cpcs.ma".
-include "basic_2/dynamic/snv_sstas.ma".
-include "basic_2/dynamic/ygt.ma".
+include "basic_2/unfold/lstas_lstas.ma".
+include "basic_2/computation/fpbs_lift.ma".
+include "basic_2/computation/fpbg_fleq.ma".
+include "basic_2/equivalence/cpes_cpds.ma".
+include "basic_2/dynamic/snv.ma".
(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
(* Inductive premises for the preservation results **************************)
-definition IH_snv_cpr_lpr: ∀h:sh. sd h → relation2 lenv term ≝
- λh,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[h, g] →
- ∀T2. L1 ⊢ T1 ➡ T2 → ∀L2. L1 ⊢ ➡ L2 → ⦃h, L2⦄ ⊢ T2 ¡[h, g].
+definition IH_snv_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
+ λh,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, g].
-definition IH_ssta_cpr_lpr: ∀h:sh. sd h → relation2 lenv term ≝
- λh,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[h, g] →
- ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[h, g] ⦃l, U1⦄ →
- ∀T2. L1 ⊢ T1 ➡ T2 → ∀L2. L1 ⊢ ➡ L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[h, g] ⦃l, U2⦄ & L2 ⊢ U1 ⬌* U2.
+definition IH_da_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
+ λh,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀l. ⦃G, L1⦄ ⊢ T1 ▪[h, g] l →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ⦃G, L2⦄ ⊢ T2 ▪[h, g] l.
-definition IH_snv_ssta: ∀h:sh. sd h → relation2 lenv term ≝
- λh,g,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] →
- ∀U,l. ⦃G, L⦄ ⊢ T •[h, g] ⦃l+1, U⦄ → ⦃G, L⦄ ⊢ U ¡[h, g].
+definition IH_lstas_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
+ λh,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀l1,l2. l2 ≤ l1 → ⦃G, L1⦄ ⊢ T1 ▪[h, g] l1 →
+ ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, l2] U1 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, l2] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
-definition IH_snv_lsubsv: ∀h:sh. sd h → relation2 lenv term ≝
- λh,g,L2,T. ⦃h, L2⦄ ⊢ T ¡[h, g] →
- ∀L1. h ⊢ L1 ¡⊑[h, g] L2 → ⦃h, L1⦄ ⊢ T ¡[h, g].
+definition IH_snv_lstas: ∀h:sh. sd h → relation3 genv lenv term ≝
+ λh,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] →
+ ∀l1,l2. l2 ≤ l1 → ⦃G, L⦄ ⊢ T ▪[h, g] l1 →
+ ∀U. ⦃G, L⦄ ⊢ T •*[h, l2] U → ⦃G, L⦄ ⊢ U ¡[h, g].
(* Properties for the preservation results **********************************)
-fact snv_cprs_lpr_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[h, g] →
- ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 → ⦃h, L2⦄ ⊢ T2 ¡[h, g].
-#h #g #L0 #T0 #IH #L1 #T1 #HLT0 #HT1 #T2 #H
-elim H -T2 [ /2 width=6/ ] -HT1
-/4 width=6 by ygt_yprs_trans, cprs_yprs/
+fact snv_cprs_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, g].
+#h #g #G0 #L0 #T0 #IH #G #L1 #T1 #HLT0 #HT1 #T2 #H
+@(cprs_ind … H) -T2 /4 width=6 by fpbg_fpbs_trans, cprs_fpbs/
qed-.
-fact ssta_cprs_lpr_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
- ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[h, g] →
- ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[h, g] ⦃l, U1⦄ →
- ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[h, g] ⦃l, U2⦄ & L2 ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 #l #HTU1 #T2 #H
-elim H -T2 [ /2 width=7/ ]
-#T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12
-elim (IHT1 L1) // -IHT1 #U #HTU #HU1
-elim (IH1 … HTU … HTT2 … HL12) -IH1 -HTU -HTT2
-[2: /3 width=9 by snv_cprs_lpr_aux/
-|3: /5 width=6 by ygt_yprs_trans, cprs_yprs/
-] -L0 -T0 -T1 -T #U2 #HTU2 #HU2
-lapply (lpr_cpcs_conf … HL12 … HU1) -L1 #HU1
-lapply (cpcs_trans … HU1 … HU2) -U /2 width=3/
+fact da_cprs_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀l. ⦃G, L1⦄ ⊢ T1 ▪[h, g] l →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ▪[h, g] l.
+#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #l #Hl #T2 #H
+@(cprs_ind … H) -T2 /4 width=10 by snv_cprs_lpr_aux, fpbg_fpbs_trans, cprs_fpbs/
qed-.
-fact ssta_cpcs_lpr_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
- ∀L1,T1,T2. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T2⦄ →
- ⦃h, L1⦄ ⊢ T1 ¡[h, g] → ⦃h, L1⦄ ⊢ T2 ¡[h, g] →
- ∀U1,l1. ⦃h, L1⦄ ⊢ T1 •[h, g] ⦃l1, U1⦄ →
- ∀U2,l2. ⦃h, L1⦄ ⊢ T2 •[h, g] ⦃l2, U2⦄ →
- L1 ⊢ T1 ⬌* T2 → ∀L2. L1 ⊢ ➡ L2 →
- l1 = l2 ∧ L2 ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #T2 #H01 #H02 #HT1 #HT2 #U1 #l1 #HTU1 #U2 #l2 #HTU2 #H #L2 #HL12
-elim (cpcs_inv_cprs … H) -H #T #H1 #H2
-elim (ssta_cprs_lpr_aux … H01 HT1 … HTU1 … H1 … HL12) -T1 /2 width=1/ #W1 #H1 #HUW1
-elim (ssta_cprs_lpr_aux … H02 HT2 … HTU2 … H2 … HL12) -T2 /2 width=1/ #W2 #H2 #HUW2 -L0 -T0
-elim (ssta_mono … H1 … H2) -h -T #H1 #H2 destruct
-lapply (cpcs_canc_dx … HUW1 … HUW2) -W2 /2 width=1/
+fact da_cpcs_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+ ∀G,L,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, g] →
+ ∀T2. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T2⦄ → ⦃G, L⦄ ⊢ T2 ¡[h, g] →
+ ∀l1. ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 → ∀l2. ⦃G, L⦄ ⊢ T2 ▪[h, g] l2 →
+ ⦃G, L⦄ ⊢ T1 ⬌* T2 → l1 = l2.
+#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L #T1 #HLT01 #HT1 #T2 #HLT02 #HT2 #l1 #Hl1 #l2 #Hl2 #H
+elim (cpcs_inv_cprs … H) -H /4 width=18 by da_cprs_lpr_aux, da_mono/
qed-.
-fact snv_sstas_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- ∀L,T. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L, T⦄ → ⦃G, L⦄ ⊢ T ¡[h, g] →
- ∀U. ⦃G, L⦄ ⊢ T •*[h, g] U → ⦃G, L⦄ ⊢ U ¡[h, g].
-#h #g #L0 #T0 #IH #L #T #H01 #HT #U #H
-@(sstas_ind … H) -U // -HT /4 width=5 by ygt_yprs_trans, sstas_yprs/
+fact sta_cpr_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀l. ⦃G, L1⦄ ⊢ T1 ▪[h, g] l+1 →
+ ∀U1. ⦃G, L1⦄ ⊢ T1 •[h] U1 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ∃∃U2. ⦃G, L2⦄ ⊢ T2 •[h] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+#h #g #G0 #L0 #T0 #IH #G #L1 #T1 #H01 #HT1 #l #Hl #U1 #HTU1 #T2 #HT12 #L2 #HL12
+elim (IH … H01 … 1 … Hl U1 … HT12 … HL12) -H01 -Hl -HT12 -HL12
+/3 width=3 by lstas_inv_SO, sta_lstas, ex2_intro/
+qed-.
+
+fact lstas_cprs_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀l1,l2. l2 ≤ l1 → ⦃G, L1⦄ ⊢ T1 ▪[h, g] l1 →
+ ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, l2] U1 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, l2] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #l1 #l2 #Hl21 #Hl1 #U1 #HTU1 #T2 #H
+@(cprs_ind … H) -T2 [ /2 width=10 by/ ]
+#T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12
+elim (IHT1 L1) // -IHT1 #U #HTU #HU1
+elim (IH1 … Hl21 … HTU … HTT2 … HL12) -IH1 -HTU -HTT2
+[2: /3 width=12 by da_cprs_lpr_aux/
+|3: /3 width=10 by snv_cprs_lpr_aux/
+|4: /3 width=5 by fpbg_fpbs_trans, cprs_fpbs/
+] -G0 -L0 -T0 -T1 -T -l1
+/4 width=5 by lpr_cpcs_conf, cpcs_trans, ex2_intro/
qed-.
-fact snv_sstas_lpr_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- ∀L1,T. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T⦄ → ⦃h, L1⦄ ⊢ T ¡[h, g] →
- ∀U. ⦃h, L1⦄ ⊢ T •*[h, g] U → ∀L2. L1 ⊢ ➡ L2 → ⦃h, L2⦄ ⊢ U ¡[h, g].
-/4 width=7 by snv_sstas_aux, ygt_yprs_trans, sstas_yprs/
+fact lstas_cpcs_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀l,l1. l ≤ l1 → ⦃G, L1⦄ ⊢ T1 ▪[h, g] l1 → ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, l] U1 →
+ ∀T2. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T2⦄ → ⦃G, L1⦄ ⊢ T2 ¡[h, g] →
+ ∀l2. l ≤ l2 → ⦃G, L1⦄ ⊢ T2 ▪[h, g] l2 → ∀U2. ⦃G, L1⦄ ⊢ T2 •*[h, l] U2 →
+ ⦃G, L1⦄ ⊢ T1 ⬌* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #l #l1 #Hl1 #HTl1 #U1 #HTU1 #T2 #H02 #HT2 #l2 #Hl2 #HTl2 #U2 #HTU2 #H #L2 #HL12
+elim (cpcs_inv_cprs … H) -H #T #H1 #H2
+elim (lstas_cprs_lpr_aux … H01 HT1 … Hl1 HTl1 … HTU1 … H1 … HL12) -T1 /2 width=1 by/ #W1 #H1 #HUW1
+elim (lstas_cprs_lpr_aux … H02 HT2 … Hl2 HTl2 … HTU2 … H2 … HL12) -T2 /2 width=1 by/ #W2 #H2 #HUW2 -L0 -T0
+lapply (lstas_mono … H1 … H2) -h -T -l #H destruct /2 width=3 by cpcs_canc_dx/
qed-.
-fact sstas_cprs_lpr_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
- ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[h, g] →
- ∀U1. ⦃h, L1⦄ ⊢ T1 •*[h, g] U1 → ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[h, g] U2 & L2 ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 #H
-@(sstas_ind … H) -U1 [ /3 width=5 by lpr_cprs_conf, ex2_intro/ ]
-#U1 #W1 #l1 #HTU1 #HUW1 #IHTU1 #T2 #HT12 #L2 #HL12
-elim (IHTU1 … HT12 … HL12) -IHTU1 #U2 #HTU2 #HU12
-lapply (snv_cprs_lpr_aux … IH2 … HT1 … HT12 … HL12) // #HT2
-elim (snv_sstas_fwd_correct … HTU2) // #W2 #l2 #HUW2
-elim (IH1 … HUW1 U1 … HL12) -HUW1 //
-[2: /3 width=7 by snv_sstas_aux/
-|3: /3 width=4 by ygt_yprs_trans, sstas_yprs/
-] #W #HU1W #HW1
-elim (ssta_cpcs_lpr_aux … IH2 IH1 … HU1W … HUW2 … HU12 L2) // -IH1 -HU1W -HU12
-[2: /4 width=8 by snv_sstas_aux, ygt_yprs_trans, cprs_lpr_yprs/
-|3: /3 width=10 by snv_sstas_lpr_aux/
-|4,5: /4 width=5 by ygt_yprs_trans, cprs_lpr_yprs, sstas_yprs/
-] -L0 -T0 -L1 -T1 -HT2 #H #HW12 destruct
-lapply (cpcs_trans … HW1 … HW12) -W /3 width=4/
+fact snv_sta_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_lstas h g G1 L1 T1) →
+ ∀G,L,T. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L⦄ ⊢ T ¡[h, g] →
+ ∀l. ⦃G, L⦄ ⊢ T ▪[h, g] l+1 →
+ ∀U. ⦃G, L⦄ ⊢ T •[h] U → ⦃G, L⦄ ⊢ U ¡[h, g].
+/3 width=8 by lstas_inv_SO, sta_lstas/ qed-.
+
+fact lstas_cpds_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_lstas h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
+ ∀G,L,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, g] →
+ ∀l1,l2. l2 ≤ l1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 →
+ ∀U1. ⦃G, L⦄ ⊢ T1 •*[h, l2] U1 → ∀T2. ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2 →
+ ∃∃U2,l. l ≤ l2 & ⦃G, L⦄ ⊢ T2 •*[h, l] U2 & ⦃G, L⦄ ⊢ U1 •*⬌*[h, g] U2.
+#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G #L #T1 #H01 #HT1 #l1 #l2 #Hl21 #Hl1 #U1 #HTU1 #T2 * #T #l0 #l #Hl0 #H #HT1T #HTT2
+lapply (da_mono … H … Hl1) -H #H destruct
+lapply (lstas_da_conf … HTU1 … Hl1) #Hl12
+elim (le_or_ge l2 l) #Hl2
+[ lapply (lstas_conf_le … HTU1 … HT1T) -HT1T
+ /5 width=11 by cpds_cpes_dx, monotonic_le_minus_l, ex3_2_intro, ex4_3_intro/
+| lapply (lstas_da_conf … HT1T … Hl1) #Hl1l
+ lapply (lstas_conf_le … HT1T … HTU1) -HTU1 // #HTU1
+ elim (lstas_cprs_lpr_aux … IH3 IH2 IH1 … Hl1l … HTU1 … HTT2 L) -IH3 -IH2 -IH1 -Hl1l -HTU1 -HTT2
+ /3 width=8 by cpcs_cpes, fpbg_fpbs_trans, lstas_fpbs, monotonic_le_minus_l, ex3_2_intro/
+]
qed-.
-fact cpds_cprs_lpr_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
- ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[h, g] →
- ∀U1. ⦃h, L1⦄ ⊢ T1 •*➡*[h, g] U1 →
- ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*➡*[h, g] U2 & L2 ⊢ U1 ➡* U2.
-#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 * #W1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12
-elim (sstas_cprs_lpr_aux … IH3 IH2 IH1 … H01 … HTW1 … HT12 … HL12) // -L0 -T0 -T1 #W2 #HTW2 #HW12
+fact cpds_cpr_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀U1. ⦃G, L1⦄ ⊢ T1 •*➡*[h, g] U1 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*➡*[h, g] U2 & ⦃G, L2⦄ ⊢ U1 ➡* U2.
+#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #U1 * #W1 #l1 #l2 #Hl21 #Hl1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12
+elim (IH1 … H01 … HTW1 … HT12 … HL12) -IH1 // #W2 #HTW2 #HW12
+lapply (IH2 … H01 … Hl1 … HT12 … HL12) -L0 -T0 // -T1
lapply (lpr_cprs_conf … HL12 … HWU1) -L1 #HWU1
lapply (cpcs_canc_sn … HW12 HWU1) -W1 #H
-elim (cpcs_inv_cprs … H) -H /3 width=3/
-qed-.
-
-fact ssta_cpds_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
- ∀L,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, g] →
- ∀l,U1. ⦃G, L⦄ ⊢ T1 •[h, g] ⦃l+1, U1⦄ → ∀T2. ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2 →
- ∃∃U,U2. ⦃G, L⦄ ⊢ U1 •*[h, g] U & ⦃G, L⦄ ⊢ T2 •*[h, g] U2 & ⦃G, L⦄ ⊢ U ⬌* U2.
-#h #g #L0 #T0 #IH2 #IH1 #L #T1 #H01 #HT1 #l #U1 #HTU1 #T2 * #T #HT1T #HTT2
-elim (sstas_strip … HT1T … HTU1) #HU1T destruct [ -HT1T | -L0 -T0 -T1 ]
-[ elim (ssta_cprs_lpr_aux … IH2 IH1 … HTU1 … HTT2 L) // -L0 -T0 -T /3 width=5/
-| @(ex3_2_intro …T2 HU1T) // /2 width=1/
-]
+elim (cpcs_inv_cprs … H) -H /3 width=7 by ex4_3_intro, ex2_intro/
qed-.
projects / helm.git / blobdiff