(* NOTATION FOR THE "functional" COMPONENT ********************************)
+notation "hvbox( T . break ④ { term 46 I } break { term 46 T1 , break term 46 T2 , break term 46 T3 } )"
+ non associative with precedence 50
+ for @{ 'DxItem4 $T $I $T1 $T2 $T3 }.
+
notation "hvbox( ↑ [ term 46 d , break term 46 e ] break term 46 T )"
non associative with precedence 46
for @{ 'Lift $d $e $T }.
(* Properties for the preservation results **********************************)
-fact sstas_lsubsv_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
+fact lsubsv_sstas_aux: ∀h,g,L0,T0.
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
(∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
(∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
∀L2,T. h ⊢ ⦃L0, T0⦄ >[g] ⦃L2, T⦄ → ⦃h, L2⦄ ⊢ T ¡[g] →
lapply (IH1 … H … HL12) [ /3 width=4 by ygt_yprs_trans, sstas_yprs/ ] -H #HU2
elim (snv_fwd_ssta … HU1) #l1 #W1 #HUW1
elim (lsubsv_ssta_trans … HU2W … HL12) -HU2W #W2 #HUW2 #HW2
-elim (ssta_ltpr_cpcs_aux … IH4 IH3 … HU1 HU2 … HUW1 … HUW2 … HU12) -HU1 -HU2 -HUW2 -HU12 //
+elim (ssta_cpcs_lpr_aux … IH4 IH3 … HU1 HU2 … HUW1 … HUW2 … HU12 L1) -HU1 -HU2 -HUW2 -HU12 //
[2,3: /4 width=4 by ygt_yprs_trans, sstas_yprs, lsubsv_yprs/ ] -L2 -L0 -T0 -U2 #H #HW12 destruct
lapply (cpcs_trans … HW12 … HW2) -W2 /3 width=4/
qed-.
-fact dxprs_lsubsv_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
+fact lsubsv_dxprs_aux: ∀h,g,L0,T0.
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
(∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
(∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
∀L2,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L2, T1⦄ → ⦃h, L2⦄ ⊢ T1 ¡[g] →
∃∃T. ⦃h, L1⦄ ⊢ T1 •*➡*[g] T & L1 ⊢ T2 ➡* T.
#h #g #L0 #T0 #IH4 #IH3 #IH2 #IH1 #L2 #T1 #HLT0 #HT1 #L1 #HL12 #T2 * #T #HT1T #HTT2
lapply (lsubsv_cprs_trans … HL12 … HTT2) -HTT2 #HTT2
-elim (sstas_lsubsv_aux … IH4 IH3 IH2 IH1 … HLT0 … HL12 … HT1T) // -L2 -L0 -T0 #T0 #HT10 #HT0
+elim (lsubsv_sstas_aux … IH4 IH3 IH2 IH1 … HLT0 … HL12 … HT1T) // -L2 -L0 -T0 #T0 #HT10 #HT0
lapply (cpcs_cprs_strap1 … HT0 … HTT2) -T #HT02
elim (cpcs_inv_cprs … HT02) -HT02 /3 width=3/
qed-.
(* Properties concerning stratified native validity *************************)
fact snv_lsubsv_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
(∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
(∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
∀L1,T1. L0 = L1 → T0 = T1 → IH_snv_lsubsv h g L1 T1.
lapply (lsubsv_cprs_trans … HL12 … HW0) -HW0 #HW0
elim (lsubsv_ssta_trans … HVW … HL12) -HVW #W1 #HVW1 #HW1
lapply (cpcs_cprs_strap1 … HW1 … HW0) -W #HW10
- elim (dxprs_lsubsv_aux … IH4 IH3 IH2 IH1 … HL12 … HTU) -IH4 -IH3 -IH2 -HTU // /2 width=1/ #X #HTU #H
- elim (cprs_inv_abst1 Abst W0 … H) -H #W #U2 #HW0 #HU2 #H destruct
+ elim (lsubsv_dxprs_aux … IH4 IH3 IH2 IH1 … HL12 … HTU) -IH4 -IH3 -IH2 -HTU // /2 width=1/ #X #HTU #H
+ elim (cprs_fwd_abst1 … H Abst W0) -H #W #U2 #HW0 #HU2 #H destruct
lapply (cpcs_cprs_strap1 … HW10 … HW0) -W0 #H
elim (cpcs_inv_cprs … H) -H #W0 #HW10 #HW0
lapply (dxprs_cprs_trans … (ⓛ{a}W0.U2) HTU ?) [ /2 width=1/ ] -HTU -HW0
λh,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[g] →
∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
∀T2. L1 ⊢ T1 ➡ T2 → ∀L2. L1 ⊢ ➡ L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L1 ⊢ U1 ⬌* U2.
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L2 ⊢ U1 ⬌* U2.
definition IH_snv_ssta: ∀h:sh. sd h → relation2 lenv term ≝
- λh,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[g] →
- ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l+1, U1⦄ → ⦃h, L1⦄ ⊢ U1 ¡[g].
+ λh,g,L,T. ⦃h, L⦄ ⊢ T ¡[g] →
+ ∀U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l+1, U⦄ → ⦃h, L⦄ ⊢ U ¡[g].
definition IH_snv_lsubsv: ∀h:sh. sd h → relation2 lenv term ≝
λh,g,L2,T. ⦃h, L2⦄ ⊢ T ¡[g] →
(∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
- ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L1 ⊢ U1 ⬌* U2.
+ ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[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=5/ ]
+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 -HL12
+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 (cpcs_lpr_conf … HL12 … HU1) -L1 #HU1
lapply (cpcs_trans … HU1 … HU2) -U /2 width=3/
qed-.
-fact ssta_cpcs_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
- ∀L,T1,T2. h ⊢ ⦃L0, T0⦄ >[g] ⦃L, T1⦄ → h ⊢ ⦃L0, T0⦄ >[g] ⦃L, T2⦄ →
- ⦃h, L⦄ ⊢ T1 ¡[g] → ⦃h, L⦄ ⊢ T2 ¡[g] →
- ∀U1,l1. ⦃h, L⦄ ⊢ T1 •[g] ⦃l1, U1⦄ →
- ∀U2,l2. ⦃h, L⦄ ⊢ T2 •[g] ⦃l2, U2⦄ →
- L ⊢ T1 ⬌* T2 → l1 = l2 ∧ L ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH2 #IH1 #L #T1 #T2 #HLT01 #HLT02 #HT1 #HT2 #U1 #l1 #HTU1 #U2 #l2 #HTU2 #H
+fact ssta_cpcs_lpr_aux: ∀h,g,L0,T0.
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+ ∀L1,T1,T2. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T2⦄ →
+ ⦃h, L1⦄ ⊢ T1 ¡[g] → ⦃h, L1⦄ ⊢ T2 ¡[g] →
+ ∀U1,l1. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l1, U1⦄ →
+ ∀U2,l2. ⦃h, L1⦄ ⊢ T2 •[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 … HLT01 HT1 … HTU1 … H1 L) -T1 // /2 width=1/ #W1 #H1 #HUW1
-elim (ssta_cprs_lpr_aux … HLT02 HT2 … HTU2 … H2 L) -T2 // /2 width=1/ #W2 #H2 #HUW2 -L0 -T0
+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/
qed-.
fact snv_sstas_aux: ∀h,g,L0,T0.
(∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
- ∀U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 → ⦃h, L1⦄ ⊢ U1 ¡[g].
-#h #g #L0 #T0 #IH #L1 #T1 #HLT0 #HT1 #U1 #H
-@(sstas_ind … H) -U1 // -HT1 /4 width=5 by ygt_yprs_trans, sstas_yprs/
+ ∀L,T. h ⊢ ⦃L0, T0⦄ >[g] ⦃L, T⦄ → ⦃h, L⦄ ⊢ T ¡[g] →
+ ∀U. ⦃h, L⦄ ⊢ T •*[g] U → ⦃h, L⦄ ⊢ U ¡[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/
+qed-.
+
+fact snv_sstas_lpr_aux: ∀h,g,L0,T0.
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+ ∀L1,T. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T⦄ → ⦃h, L1⦄ ⊢ T ¡[g] →
+ ∀U. ⦃h, L1⦄ ⊢ T •*[g] U → ∀L2. L1 ⊢ ➡ L2 → ⦃h, L2⦄ ⊢ U ¡[g].
+/4 width=7 by snv_sstas_aux, ygt_yprs_trans, sstas_yprs/
qed-.
fact sstas_cprs_lpr_aux: ∀h,g,L0,T0.
(∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
(∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
- ∀T2. L1 ⊢ T1 ➡* T2 → ∀U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 → ∀L2. L1 ⊢ ➡ L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L1 ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #L2 #HL12 #T2 #HT12 #U1 #H
-@(sstas_ind … H) -U1 [ /3 width=3/ ]
-#U1 #W1 #l1 #HTU1 #HUW1 * #U2 #HTU2 #HU12
-lapply (snv_ltpr_cprs_aux … IH2 … HT1 … HT12) // #HT2
+ ∀U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 → ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[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 cprs_lpr_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 (ssta_ltpr_cpcs_aux … IH2 IH1 … HUW1 … HUW2 … HU12) -IH2 -IH1 -HUW1 -HU12 //
-[2: /4 width=8 by snv_sstas_aux, ygt_yprs_trans, ltpr_cprs_yprs/
-|3: /3 width=7 by snv_sstas_aux, ygt_yprs_trans, cprs_yprs/
-|4: /4 width=5 by ygt_yprs_trans, ltpr_cprs_yprs, sstas_yprs/
-|5: /3 width=4 by ygt_yprs_trans, cprs_yprs, sstas_yprs/
-] -L0 -T0 -T1 -HT2 #H #HW12 destruct /3 width=4/
+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/
qed-.
-fact dxprs_ltpr_cprs_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
- ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
- ∀U1. ⦃h, L1⦄ ⊢ T1 •*➡*[g] U1 →
- ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡* T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*➡*[g] U2 & L2 ⊢ U1 ➡* U2.
-#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 * #W1 #HTW1 #HWU1 #L2 #HL12 #T2 #HT12
-elim (sstas_ltpr_cprs_aux … IH3 IH2 IH1 … H01 … HT12 … HTW1) // -L0 -T0 -T1 #W2 #HTW2 #HW12
-lapply (ltpr_cprs_conf … HL12 … HWU1) -L1 #HWU1
+fact dxprs_cprs_lpr_aux: ∀h,g,L0,T0.
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+ ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
+ ∀U1. ⦃h, L1⦄ ⊢ T1 •*➡*[g] U1 →
+ ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*➡*[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
+lapply (cprs_lpr_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_dxprs_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
- ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
- ∀l,U1. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l+1, U1⦄ → ∀T2. ⦃h, L1⦄ ⊢ T1 •*➡*[g] T2 →
- ∃∃U,U2. ⦃h, L1⦄ ⊢ U1 •*[g] U & ⦃h, L1⦄ ⊢ T2 •*[g] U2 & L1 ⊢ U ⬌* U2.
-#h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #H01 #HT1 #l #U1 #HTU1 #T2 * #T #HT1T #HTT2
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+ ∀L,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L, T1⦄ → ⦃h, L⦄ ⊢ T1 ¡[g] →
+ ∀l,U1. ⦃h, L⦄ ⊢ T1 •[g] ⦃l+1, U1⦄ → ∀T2. ⦃h, L⦄ ⊢ T1 •*➡*[g] T2 →
+ ∃∃U,U2. ⦃h, L⦄ ⊢ U1 •*[g] U & ⦃h, L⦄ ⊢ T2 •*[g] U2 & 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_ltpr_cprs_aux … IH2 IH1 … HTU1 L1 … HTT2) // -L0 -T0 -T /3 width=5/
+[ elim (ssta_cprs_lpr_aux … IH2 IH1 … HTU1 … HTT2 L) // -L0 -T0 -T /3 width=5/
| @(ex3_2_intro …T2 HU1T) // /2 width=1/
]
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/computation/dxprs_dxprs.ma".
+include "basic_2/dynamic/snv_lift.ma".
+include "basic_2/dynamic/snv_cpcs.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties on context-free parallel reduction for local environments *****)
+
+fact snv_cpr_lpr_aux: ∀h,g,L0,T0.
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+ ∀L1,T1. L0 = L1 → T0 = T1 → IH_snv_cpr_lpr h g L1 T1.
+#h #g #L0 #T0 #IH4 #IH3 #IH2 #IH1 #L1 * * [||||*]
+[ #k #HL0 #HT0 #H1 #X #H2 #L2 #_ destruct -IH4 -IH3 -IH2 -IH1 -H1
+ >(cpr_inv_sort1 … H2) -X //
+| #i #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2
+ elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
+ elim (lpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
+ elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+ lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKL
+ elim (cpr_inv_lref1 … H2) -H2
+ [ #H destruct -HLK1
+ lapply (IH1 … HV12 … HK12) -IH1 -HV12 -HK12 // -HV1 [ /2 width=1/ ] -HKL /2 width=5/
+ | * #K0 #V0 #W0 #H #HVW0 #W0 -HV12
+ lapply (ldrop_mono … H … HLK1) -HLK1 -H #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
+ lapply (IH1 … HVW0 … HK12) -IH1 -HVW0 -HK12 // -HV1 [ /2 width=1/ ] -HKL /3 width=7/
+ ]
+| #p #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2 -IH1
+ elim (snv_inv_gref … H1)
+| #a #I #V1 #T1 #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
+ lapply (IH1 … HV12 … HL12) // -HV1 [ /2 width=1/ ] #HV2
+ lapply (IH1 … HT12 (L2.ⓑ{I}V2) ?) // -IH1 /2 width=1/
+ | #T2 #HT12 #HXT2 #H1 #H2 destruct -HV1
+ lapply (IH1 … HT1 … HT12 (L2.ⓓV1) ?) -IH1 [1,2: /2 width=1/ ] -L1 -T1 #HT2
+ lapply (snv_inv_lift … HT2 L2 … HXT2) -T2 // /2 width=1/
+ ]
+| #V1 #T1 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct
+ elim (snv_inv_appl … H1) -H1 #a #W10 #W1 #U1 #l #HV1 #HT1 #HVW1 #HW10 #HTU1
+ elim (cpr_inv_appl1 … H2) -H2 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct -IH4
+ lapply (IH1 … HV1 … HV12 … HL12) [ /2 width=1/ ] #HV2
+ lapply (IH1 … HT1 … HT12 … HL12) [ /2 width=1/ ] #HT2
+ elim (IH3 … HVW1 … HV12 … HL12) -HVW1 -HV12 // -HV1 [2: /2 width=1/ ] #W2 #HVW2 #HW12
+ elim (dxprs_cprs_lpr_aux … IH2 IH1 IH3 … HTU1 … T2 … HL12) // [2,3: /2 width=1/ ] -IH2 -IH1 -IH3 -HT1 -HT12 -HTU1 #X #HTU2 #H
+ elim (cprs_fwd_abst1 … H Abst W1) -H #W20 #U2 #HW120 #_ #H destruct
+ lapply (cprs_lpr_conf … HL12 … HW10) -L1 #HW10
+ lapply (cpcs_cprs_strap1 … HW10 … HW120) -W1 #HW120
+ lapply (cpcs_canc_sn … HW12 HW120) -W10 #HW20
+ elim (cpcs_inv_cprs … HW20) -HW20 #W0 #HW20 #HW200
+ lapply (dxprs_cprs_trans … (ⓛ{a}W0.U2) HTU2 ?) [ /2 width=1/ ] -HW200 -HTU2 /2 width=8/
+ | #b #V2 #W20 #T20 #T2 #HV12 #HT202 #H1 #H2 destruct
+ elim (snv_inv_bind … HT1) -HT1 #HW20 #HT20
+ elim (dxprs_inv_abst1 … HTU1) -HTU1 #W30 #T30 #HW230 #_ #H destruct -T30
+ lapply (cprs_div … HW10 … HW230) -W30 #HW120
+ lapply (cpcs_lpr_conf … HL12 … HW120) -HW120 #HW120
+ elim (IH3 … HVW1 … HV12 … HL12) // [2: /2 width=1/ ] -HVW1 #W2 #HVW2 #HW102
+ lapply (cpcs_canc_sn … HW102 … HW120) -W10 #HW220
+ lapply (IH1 … HV12 … HL12) // [ /2 width=1/ ] #HV2 -HV1
+ lapply (IH1 … HW20 … W20 … HL12) // [ /2 width=1/ ] -HW20 #HW20
+ lapply (IH1 … HT20 … HT202 … (L2.ⓛW20) ?) [1,2: /2 width=1/ ] -HT20 #HT2
+ lapply (IH2 … HVW2) //
+ [ @(ygt_yprs_trans … L1 L1 … V1) (**) (* auto /4 width=5/ is a bit slow even with trace *)
+ [ /2 width=1 by fw_ygt/
+ | /3 width=1 by cprs_lpr_yprs, cpr_cprs/
+ ]
+ ] #HW2
+ elim (snv_fwd_ssta … HW20) #l0 #U20 #HWU20
+ elim (ssta_fwd_correct … HVW2) <minus_plus_m_m #U2 #HWU2
+ elim (ssta_cpcs_lpr_aux … IH1 IH3 … HWU2 … HWU20 … HW220 … L2) // -IH3
+ [2: /4 width=4 by ygt_yprs_trans, fw_ygt, ypr_yprs, ypr_lpr/
+ |3: @(ygt_yprs_trans … L1 L2 … V2) (**) (* auto not tried *)
+ [ @(ygt_yprs_trans … L1 L1 … V1)
+ [ /2 width=1 by fw_ygt/
+ | /3 width=1 by cprs_lpr_yprs, cpr_cprs/
+ ]
+ | /3 width=2 by ypr_ssta, ypr_yprs/
+ ]
+ ] #H #_ destruct -IH2 -U2
+ lapply (IH4 … HT2 (L2.ⓓV2) ?)
+ [ /2 width=6/
+ | @(ygt_yprs_trans … (L1.ⓛW20) … T2) (**) (* auto /5 width=5/ is too slow even with trace timeout=35 *)
+ [ /4 width=4 by ygt_yprs_trans, fw_ygt, ypr_yprs, ypr_cpr/
+ | /4 width=1 by ypr_yprs, ypr_lpr, lpr_pair/
+ ]
+ ] -L1 -V1 -W2 -T20 -U20 -W20 -l0 /2 width=1/
+ | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -IH4
+ elim (snv_inv_bind … HT1) -HT1 #HW0 #HT0
+ elim (dxprs_inv_abbr_abst … HTU1) -HTU1 #X #HTU0 #HX #H destruct
+ elim (lift_inv_bind1 … HX) -HX #W3 #U3 #HW13 #_ #H destruct
+ lapply (cprs_lpr_conf … HL12 … HW10) -HW10 #HW10
+ elim (dxprs_cprs_lpr_aux … IH2 IH1 IH3 … HTU0 T2 … (L2.ⓓW2) ?) // [2,3,4: /2 width=1/ ] -IH2 -HTU0 #X #HTU2 #H
+ elim (cprs_fwd_abst1 … H Abst W3) -H #W #U2 #HW1 #_ #H destruct -U3
+ elim (IH3 … HVW1 … HV10 … HL12) // /2 width=1/ -IH3 -HVW1 #X #H1 #H2
+ lapply (cpcs_canc_sn … H2 HW10) -W10 #H2
+ elim (lift_total X 0 1) #W20 #H3
+ lapply (ssta_lift … H1 (L2.ⓓW2) … HV02 … H3) /2 width=1/ -H1 #HVW20
+ lapply (cpcs_lift (L2.ⓓW2) … H3 … HW13 H2) /2 width=1/ -HW13 -H3 -H2 #HW320
+ lapply (cpcs_cprs_strap1 … HW320 … HW1) -W3 #HW20
+ elim (cpcs_inv_cprs … HW20) -HW20 #W3 #HW203 #HW3
+ lapply (dxprs_cprs_trans … (ⓛ{a}W3.U2) HTU2 ?) [ /2 width=1/ ] -HW3 -HTU2 #HTU2
+ lapply (IH1 … HW02 … HL12) // [ /2 width=1/ ] -HW0 #HW2
+ lapply (IH1 … HV10 … HL12) // [ /2 width=1/ ] -HV1 -HV10 #HV0
+ lapply (IH1 … HT02 (L2.ⓓW2) ?) // [1,2: /2 width=1/ ] -L1 #HT2
+ lapply (snv_lift … HV0 (L2.ⓓW2) … HV02) /2 width=1/ -HV0 -HV02 /3 width=8/
+ ]
+| #W1 #T1 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH2
+ elim (snv_inv_cast … H1) -H1 #U1 #l #HW1 #HT1 #HTU1 #HUW1
+ elim (cpr_inv_cast1 … H2) -H2
+ [ * #W2 #T2 #HW12 #HT12 #H destruct
+ lapply (cpcs_cprs_strap1 … HUW1 W2 ?) [ /2 width=1/ ] -HUW1 #H1
+ lapply (IH1 … HW12 … HL12) // /2 width=1/ -HW1 -HW12 #HW2
+ lapply (IH1 … HT12 … HL12) // /2 width=1/ -IH1 #HT2
+ elim (IH3 … HTU1 … HT12 … HL12) // /2 width=1/ -IH3 -HT1 -HT12 -HTU1 #U2 #HTU2 #HU12
+ lapply (cpcs_lpr_conf … HL12 … H1) -L1 #H1
+ lapply (cpcs_canc_sn … HU12 H1) -U1 /2 width=4/
+ | #H -IH3 -HW1 -HTU1 -HUW1
+ lapply (IH1 … H … HL12) // /2 width=1/
+ ]
+]
+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/computation/dxprs_dxprs.ma".
-include "basic_2/dynamic/snv_cpcs.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Properties on context-free parallel reduction for local environments *****)
-
-fact snv_ltpr_tpr_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
- ∀L1,T1. L0 = L1 → T0 = T1 → IH_snv_ltpr_tpr h g L1 T1.
-#h #g #L0 #T0 #IH4 #IH3 #IH2 #IH1 #L1 * * [||||*]
-[ #k #HL0 #HT0 #H1 #L2 #_ #X #H2 destruct -IH4 -IH3 -IH2 -IH1 -L1
- >(tpr_inv_atom1 … H2) -X //
-| #i #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct -IH4 -IH3 -IH2
- elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
- >(tpr_inv_atom1 … H2) -X
- elim (ltpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) -HLK1 #HLK1
- lapply (IH1 … HK12 … HV12) -IH1 -HV12 -HK12 // -HV1 [ /2 width=1/ ] -HLK1 /2 width=5/
-| #p #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct -IH4 -IH3 -IH2 -IH1
- elim (snv_inv_gref … H1)
-| #a #I #V1 #T1 #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct -IH4 -IH3 -IH2
- elim (snv_inv_bind … H1) -H1 #HV1 #HT1
- elim (tpr_inv_bind1 … H2) -H2 *
- [ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct
- lapply (tps_lsubr_trans … HT02 (L2.ⓑ{I}V2) ?) -HT02 /2 width=1/ #HT02
- lapply (IH1 … HL12 … HV12) // [ /2 width=1/ ] #HV2
- lapply (snv_ltpr_cpr_aux … HT1 (L2.ⓑ{I}V2) … T2 ?) -HT1
- [ /3 width=5 by cpr_intro, tps_tpss/ | /2 width=1/ | /3 width=1/ ] -IH1 -T0 /2 width=1/
- | #T2 #HT12 #HXT2 #H1 #H2 destruct
- lapply (IH1 … HT1 (L2.ⓓV1) … HT12) -IH1 -HT1 -HT12 [1,2: /2 width=1/ ] -HL12 #HT2
- lapply (snv_inv_lift … HT2 L2 … HXT2) -T2 // /2 width=1/
- ]
-| #V1 #T1 #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct
- elim (snv_inv_appl … H1) -H1 #a #W10 #W1 #U1 #l #HV1 #HT1 #HVW1 #HW10 #HTU1
- elim (tpr_inv_appl1 … H2) -H2 *
- [ #V2 #T2 #HV12 #HT12 #H destruct -IH4
- lapply (IH1 … HV1 … HL12 … HV12) [ /2 width=1/ ] #HV2
- lapply (IH1 … HT1 … HL12 … HT12) [ /2 width=1/ ] #HT2
- elim (IH3 … HVW1 … HL12 … HV12) -HVW1 -HV12 // -HV1 [2: /2 width=1/ ] #W2 #HVW2 #HW12
- elim (dxprs_ltpr_cprs_aux … IH2 IH1 IH3 … HTU1 … HL12 T2) // [2: /3 width=1/ |3: /2 width=1/ ] -IH2 -IH1 -IH3 -HT1 -HT12 -HTU1 #X #HTU2 #H
- elim (cprs_inv_abst1 Abst W1 … H) -H #W20 #U2 #HW120 #_ #H destruct
- lapply (ltpr_cprs_conf … HL12 … HW10) -L1 #HW10
- lapply (cpcs_cprs_strap1 … HW10 … HW120) -W1 #HW120
- lapply (cpcs_canc_sn … HW12 HW120) -W10 #HW20
- elim (cpcs_inv_cprs … HW20) -HW20 #W0 #HW20 #HW200
- lapply (dxprs_cprs_trans … (ⓛ{a}W0.U2) HTU2 ?) [ /2 width=1/ ] -HW200 -HTU2 /2 width=8/
- | #b #V2 #W20 #T20 #T2 #HV12 #HT202 #H1 #H2 destruct
- elim (snv_inv_bind … HT1) -HT1 #HW20 #HT20
- elim (dxprs_inv_abst1 … HTU1) -HTU1 #W30 #T30 #HW230 #_ #H destruct -T30
- lapply (cprs_div … HW10 … HW230) -W30 #HW120
- elim (snv_fwd_ssta … HW20) #l0 #U20 #HWU20
- elim (ssta_fwd_correct … HVW1) <minus_plus_m_m #U10 #HWU10
- elim (ssta_ltpr_cpcs_aux … IH1 IH3 … HW20 … HWU10 … HWU20) // -IH3 -HWU10
- [2: /3 width=5/ |3: /2 width=1/
- |4: /4 width=4 by ygt_yprs_trans, fw_ygt, sstas_yprs, ssta_sstas/
- ] #H #_ -IH2 -U10 destruct
- lapply (IH4 … HT20 (L1.ⓓV1) ?) [ /2 width=6/ | /2 width=1/ ] -U20 -W10 -l0 -IH4 -HT20 -HW20 #HT20
- lapply (IH1 … HL12 … HV12) // [ /2 width=1/ ] #HV2
- lapply (IH1 … HT20 … (L2.ⓓV2) … HT202) [1,2: /2 width=1/ ] -L1 -V1 -W20 -T20 /2 width=1/
- | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct -IH4
- elim (snv_inv_bind … HT1) -HT1 #HW0 #HT0
- elim (dxprs_inv_abbr_abst … HTU1) -HTU1 #X #HTU0 #HX #H destruct
- elim (lift_inv_bind1 … HX) -HX #W3 #U3 #HW13 #_ #H destruct
- lapply (ltpr_cprs_conf … HL12 … HW10) -HW10 #HW10
- elim (dxprs_ltpr_cprs_aux … IH2 IH1 IH3 … HTU0 (L2.ⓓW2) … T2 ?) // [2: /3 width=1/ |3,4: /2 width=1/ ] -IH2 -HTU0 #X #HTU2 #H
- elim (cprs_inv_abst1 Abst W3 … H) -H #W #U2 #HW1 #_ #H destruct -U3
- elim (IH3 … HVW1 … HL12 … HV10) // /2 width=1/ -IH3 -HVW1 #X #H1 #H2
- lapply (cpcs_canc_sn … H2 HW10) -W10 #H2
- elim (lift_total X 0 1) #W20 #H3
- lapply (ssta_lift … H1 (L2.ⓓW2) … HV02 … H3) /2 width=1/ -H1 #HVW20
- lapply (cpcs_lift (L2.ⓓW2) … H3 … HW13 H2) /2 width=1/ -HW13 -H3 -H2 #HW320
- lapply (cpcs_cprs_strap1 … HW320 … HW1) -W3 #HW20
- elim (cpcs_inv_cprs … HW20) -HW20 #W3 #HW203 #HW3
- lapply (dxprs_cprs_trans … (ⓛ{a}W3.U2) HTU2 ?) [ /2 width=1/ ] -HW3 -HTU2 #HTU2
- lapply (IH1 … HL12 … HW02) // [ /2 width=1/ ] -HW0 #HW2
- lapply (IH1 … HL12 … HV10) // [ /2 width=1/ ] -HV1 -HV10 #HV0
- lapply (IH1 … HT0 … (L2.ⓓW2) … HT02) [1,2: /2 width=1/ ] -L1 -HT02 -HW02 #HT2
- lapply (snv_lift … HV0 (L2.ⓓW2) … HV02) /2 width=1/ -HV0 -HV02 /3 width=8/
- ]
-| #W1 #T1 #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct -IH4 -IH2
- elim (snv_inv_cast … H1) -H1 #U1 #l #HW1 #HT1 #HTU1 #HUW1
- elim (tpr_inv_cast1 … H2) -H2
- [ * #W2 #T2 #HW12 #HT12 #H destruct
- lapply (cpcs_cprs_strap1 … HUW1 W2 ?) [ /3 width=1/ ] -HUW1 #H1
- lapply (IH1 … HL12 … HW12) // /2 width=1/ -HW1 -HW12 #HW2
- lapply (IH1 … HL12 … HT12) // /2 width=1/ -IH1 #HT2
- elim (IH3 … HTU1 … HL12 … HT12) // /2 width=1/ -IH3 -HT1 -HT12 -HTU1 #U2 #HTU2 #HU12
- lapply (ltpr_cpcs_conf … HL12 … H1) -L1 #H1
- lapply (cpcs_canc_sn … HU12 H1) -U1 /2 width=4/
- | #H -IH3 -HW1 -HTU1 -HUW1
- lapply (IH1 … HL12 … H) // /2 width=1/
- ]
-]
-qed-.
(* *)
(**************************************************************************)
+include "basic_2/dynamic/snv_lift.ma".
include "basic_2/dynamic/snv_cpcs.ma".
(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
(* Properties on stratified static type assignment for terms ****************)
fact snv_ssta_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
(∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
∀L1,T1. L0 = L1 → T0 = T1 → IH_snv_ssta h g L1 T1.
#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 * * [||||*]
elim (ssta_dxprs_aux … IH3 IH2 … HTU1 … HT10) -IH3 -IH2 // /2 width=2/ -T1 #U #X #HU1U #H #HU0
elim (sstas_inv_bind1 … H) -H #U0 #HTU0 #H destruct
elim (cpcs_inv_abst2 … HU0) -HU0 #W2 #U2 #HU2 #HU02
- elim (cprs_inv_abst … HU02 Abst W0) -HU02 #HW02 #_
+ elim (cprs_fwd_abst … HU02 Abst W0) -HU02 #HW02 #_
lapply (cprs_trans … HW10 … HW02) -W0 /3 width=10 by snv_appl, ex2_intro/ (**) (* auto is too slow without trace *)
| #W1 #T1 #HL0 #HT0 #H1 #X #l #H2 destruct -IH3 -IH2
elim (snv_inv_cast … H1) -H1 #U1 #l0 #HW1 #HT1 #HTU1 #HUW1
--- /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/equivalence/lsubss_ssta.ma".
+include "basic_2/dynamic/snv_cpcs.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties on sn parallel reduction for local environments ***************)
+
+fact ssta_cpr_lpr_aux: ∀h,g,L0,T0.
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+ ∀L1,T1. L0 = L1 → T0 = T1 → IH_ssta_cpr_lpr h g L1 T1.
+#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 * * [|||| *]
+[ #k #_ #_ #_ #X2 #l #H2 #X3 #H3 #L2 #HL12 -IH3 -IH2 -IH1
+ elim (ssta_inv_sort1 … H2) -H2 #Hkl #H destruct
+ >(cpr_inv_sort1 … H3) -X3 /4 width=6/
+| #i #HL0 #HT0 #H1 #X2 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
+ elim (snv_inv_lref … H1) -H1 #I0 #K0 #V0 #H #HV1
+ elim (ssta_inv_lref1 … H2) -H2 * #K1
+ #V1 #W1 [| #l0 ] #HLK1 #HVW1 #HWU1 [| #H destruct ]
+ lapply (ldrop_mono … H … HLK1) -H #H destruct
+ lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
+ elim (lpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
+ elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) #H2
+ elim (cpr_inv_lref1 … H3) -H3
+ [1,3: #H destruct -HLK1
+ |2,4: * #K0 #V0 #W0 #H #HVW0 #HW0
+ lapply (ldrop_mono … H … HLK1) -H -HLK1 #H destruct
+ ]
+ [ elim (IH1 … HVW1 … HV12 … HK12) -IH1 -HVW1 // [2: /2 width=1/ ] -K1 -V1 #W2 #HVW2 #HW12
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (cpcs_lift … H2 … HWU1 … HWU2 HW12) -H2 -W1 /3 width=6/
+ | elim (IH1 … HVW1 … HV12 … HK12) -IH1 -HVW1 // [2: /2 width=1/ ] #W2 #HVW2 #_
+ elim (lift_total V2 0 (i+1)) #U2 #HVU2
+ lapply (lpr_cpr_conf … HK12 … HV12) -HV12 #HV12
+ lapply (cpcs_lift … H2 … HWU1 … HVU2 HV12) -H2 -V1 /3 width=6/
+ | elim (IH1 … HVW1 … HVW0 … HK12) -IH1 -HVW1 // [2: /2 width=1/ ] -K1 -V1 #W2 #HVW2 #HW12
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (ssta_lift … HVW2 … H2 … HW0 … HWU2) -HVW2 -HW0
+ lapply (cpcs_lift … H2 … HWU1 … HWU2 HW12) -H2 -W1 /3 width=6/
+ ]
+| #p #_ #HT0 #H1 destruct -IH3 -IH2 -IH1
+ elim (snv_inv_gref … H1)
+| #a #I #V1 #T1 #HL0 #HT0 #H1 #X2 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
+ elim (snv_inv_bind … H1) -H1 #_ #HT1
+ elim (ssta_inv_bind1 … H2) -H2 #U1 #HTU1 #H destruct
+ elim (cpr_inv_bind1 … H3) -H3 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IH1 … HTU1 … HT12 (L2.ⓑ{I}V2)) // [2,3: /2 width=1/ ] -T1
+ lapply (lpr_cpr_conf … HL12 … HV12) -L1 /3 width=5/
+ | #T2 #HT12 #HT2 #H1 #H2 destruct
+ elim (IH1 … HTU1 … HT12 (L2.ⓓV1)) -IH1 -HTU1 // [2,3: /2 width=1/ ] -T1 -HL12 #U2 #HTU2 #HU12
+ elim (ssta_inv_lift1 … HTU2 … HT2) -T2 [3: /2 width=1/ |2: skip ] #U #HXU #HU2
+ lapply (cpcs_bind1 true … V1 V1 … HU12) // -HU12 #HU12
+ lapply (cpcs_cpr_strap1 … HU12 U ?) -HU12 /2 width=3/
+ ]
+| #V1 #T1 #HL0 #HT0 #H1 #X2 #l #H2 #X3 #H3 #L2 #HL12 destruct
+ elim (snv_inv_appl … H1) -H1 #a #W1 #W10 #U10 #l0 #HV1 #HT1 #HVW1 #HW10 #HTU10
+ elim (ssta_inv_appl1 … H2) -H2 #U1 #HTU1 #H destruct
+ elim (cpr_inv_appl1 … H3) -H3 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct -a -l0 -W1 -W10 -U10 -HV1 -IH3 -IH2
+ elim (IH1 … HTU1 … HT12 … HL12) -IH1 -HTU1 // [2: /2 width=1/ ] -T1 #U2 #HTU2 #HU12
+ lapply (lpr_cpr_conf … HL12 … HV12) -L1 /3 width=5/
+ | #b #V2 #W #T2 #T20 #HV12 #HT20 #H1 #H2 destruct
+ elim (snv_inv_bind … HT1) -HT1 #HW #HT2
+ elim (ssta_inv_bind1 … HTU1) -HTU1 #U2 #HTU2 #H destruct
+ elim (dxprs_inv_abst1 … HTU10) -HTU10 #W0 #U0 #HW0 #_ #H destruct
+ lapply (cprs_div … HW10 … HW0) -W0 #HW1W
+ elim (ssta_fwd_correct … HVW1) <minus_plus_m_m #X1 #HWX1
+ elim (snv_fwd_ssta … HW) #l1 #V #HWV
+ lapply (IH3 … HVW1) -IH3 // [ /2 width=1/ ] #HW1
+ elim (ssta_cpcs_lpr_aux … IH2 IH1 … HWX1 … HWV … L1) -IH2 -HWX1 //
+ [2: /2 width=1/
+ |3: /4 width=4 by ygt_yprs_trans, fw_ygt, sstas_yprs, ssta_sstas/
+ ] #H #_ destruct -X1
+ elim (IH1 … HVW1 … HV12 … HL12) -HVW1 // -HV1 [2: /2 width=1/ ] #W2 #HVW2 #HW12
+ elim (IH1 … HWV W … HL12) -HWV // -HW [2: /2 width=1/ ] #V0 #HWV0 #_
+ elim (IH1 … HTU2 … HT20 (L2.ⓛW)) -IH1 -HTU2 -HT20 // [2,3: /2 width=1/ ] -HT2 #U20 #HTU20 #HU20
+ lapply (cpcs_lpr_conf … HL12 … HW1W) #HW1W
+ lapply (lpr_cpr_conf … HL12 … HV12) -L1 #HV12
+ lapply (cpcs_ext_bind … HV12 … HU20 b Abbr) -HV12 -HU20 #HU20
+ lapply (cpcs_canc_sn … HW12 HW1W) -W1 #HW2
+ elim (lsubss_ssta_trans … HTU20 (L2.ⓓV2) ?) -HTU20
+ [ #U #HTU20 #HUU20 -HWV0 -W2
+ lapply (cpcs_bind1 b … V2 V2 … HUU20) // -HUU20 #HUU20
+ lapply (cpcs_canc_dx … HU20 … HUU20) -U20 #HU2
+ lapply (cpcs_cpr_strap2 … (ⓐV1.ⓛ{b}W.U2) … HU2) [ /2 width=1/ ] /3 width=3/
+ | -b -l -V -V1 -T2 -T20 -U0 -U2 -U20 /2 width=6/
+ ]
+ | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -a -l0 -W1 -W10 -HV1 -IH3 -IH2
+ elim (ssta_inv_bind1 … HTU1) -HTU1 #U0 #HTU0 #H destruct
+ elim (snv_inv_bind … HT1) -HT1 #_ #HT0
+ elim (IH1 … HTU0 … HT02 (L2.ⓓW2)) -IH1 -HTU0 // [2,3: /2 width=1/ ] -T0 #U2 #HTU2 #HU02
+ lapply (lpr_cpr_conf … HL12 … HV10) -HV10 #HV10
+ lapply (lpr_cpr_conf … HL12 … HW02) -L1 #HW02
+ lapply (cpcs_bind2 b … HW02 … HU02) -HW02 -HU02 #HU02
+ lapply (cpcs_flat … HV10 … HU02 Appl) -HV10 -HU02 #HU02
+ lapply (cpcs_cpr_strap1 … HU02 (ⓓ{b}W2.ⓐV2.U2) ?) [ /2 width=3/ ] -V0 /4 width=3/
+ ]
+| #U0 #T1 #HL0 #HT0 #H1 #X2 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
+ elim (snv_inv_cast … H1) -H1 #T0 #l0 #_ #HT1 #HT10 #_
+ lapply (ssta_inv_cast1 … H2) -H2 #HTU1
+ elim (ssta_mono … HT10 … HTU1) -HT10 #H1 #H2 destruct
+ elim (cpr_inv_cast1 … H3) -H3
+ [ * #U2 #T2 #_ #HT12 #H destruct
+ elim (IH1 … HTU1 … HT12 … HL12) -IH1 -HTU1 -HL12 // [2: /2 width=1/ ] -T1 -U0 /3 width=3/
+ | #HT1X3
+ elim (IH1 … HTU1 … HT1X3 … HL12) -IH1 -HTU1 -HL12 // [2: /2 width=1/ ] -U0 -T1 /2 width=3/
+ ]
+]
+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/equivalence/lsubss_ssta.ma".
-include "basic_2/dynamic/snv_cpcs.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Properties on context-free parallel reduction for local environments *****)
-
-fact ssta_ltpr_tpr_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
- ∀L1,T1. L0 = L1 → T0 = T1 → IH_ssta_ltpr_tpr h g L1 T1.
-#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 * * [|||| *]
-[ #k #_ #_ #_ #X2 #l #H2 #L2 #HL12 #X3 #H3 -IH3 -IH2 -IH1
- elim (ssta_inv_sort1 … H2) -H2 #Hkl #H destruct
- >(tpr_inv_atom1 … H3) -X3 /4 width=6/
-| #i #HL0 #HT0 #H1 #X2 #l #H2 #L2 #HL12 #X3 #H3 destruct -IH3 -IH2
- elim (snv_inv_lref … H1) -H1 #I0 #K0 #V0 #H #HV1
- elim (ssta_inv_lref1 … H2) -H2 * #K1
- >(tpr_inv_atom1 … H3) -X3
- [ #V1 #W1 #HLK1 #HVW1 #HWU1
- lapply (ldrop_mono … H … HLK1) -H #H destruct
- lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
- elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
- elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- elim (IH1 … HVW1 … HK12 … HV12) -IH1 -HVW1 -HV12 // [2: /2 width=1/ ] -V1 #W2 #HVW2 #HW12
- lapply (ldrop_fwd_ldrop2 … HLK2) #H2
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (cpcs_lift … H2 … HWU1 … HWU2 HW12) -H2 -W1 /3 width=6/
- | #V1 #W1 #l0 #HLK1 #HVW1 #HVU1 #H destruct
- lapply (ldrop_mono … H … HLK1) -H #H destruct
- lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
- elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 #X #H #HLK2
- elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- elim (IH1 … HVW1 … HK12 … HV12) -IH1 -HVW1 -HK12 // [2: /2 width=1/ ] -HV1 -HKV1 #W2 #HVW2 #_ -W1
- elim (lift_total V2 0 (i+1)) #U2 #HVU2
- lapply (tpr_lift … HV12 … HVU1 … HVU2) -V1 /4 width=6/
- ]
-| #p #_ #HT0 #H1 destruct -IH3 -IH2 -IH1
- elim (snv_inv_gref … H1)
-| #a #I #V1 #T1 #HL0 #HT0 #H1 #X2 #l #H2 #L2 #HL12 #X3 #H3 destruct -IH3 -IH2
- elim (snv_inv_bind … H1) -H1 #_ #HT1
- elim (ssta_inv_bind1 … H2) -H2 #U1 #HTU1 #H destruct
- elim (tpr_inv_bind1 … H3) -H3 *
- [ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct
- lapply (tps_lsubr_trans … HT02 (L2.ⓑ{I}V2) ?) -HT02 [ /2 width=1/ ] #HT02
- elim (ssta_ltpr_cpr_aux … HT1 … HTU1 (L2.ⓑ{I}V2) … T2) -HT1 -HTU1
- [2: /3 width=5 by cpr_intro, tps_tpss/ |3: /2 width=1/ |4: /3 width=1/ ] -IH1 -T0 -HL12 #U2 #HTU2 #HU12
- lapply (cpcs_bind2 … V1 … HU12 … a) -HU12 [ /2 width=1/ ] -HV12 /3 width=3/
- | #T2 #HT12 #HT2 #H1 #H2 destruct
- elim (IH1 … HTU1 (L2.ⓓV1) … T2) -IH1 -HTU1 // [2,3: /2 width=1/ ] -T1 -HL12 #U2 #HTU2 #HU12
- lapply (cpcs_bind1 … HU12 … V1 … true) // -HU12 #HU12
- elim (ssta_inv_lift1 … HTU2 … HT2) -T2 [3: /2 width=1/ |2: skip ] #U #HXU #HU2
- lapply (cpcs_cpr_strap1 … HU12 U ?) -HU12 [ /3 width=3/ ] -U2 /2 width=3/
- ]
-| #V1 #T1 #HL0 #HT0 #H1 #X2 #l #H2 #L2 #HL12 #X3 #H3 destruct
- elim (snv_inv_appl … H1) -H1 #a #W1 #W10 #U10 #l0 #HV1 #HT1 #HVW1 #HW10 #HTU10
- elim (ssta_inv_appl1 … H2) -H2 #U1 #HTU1 #H destruct
- elim (tpr_inv_appl1 … H3) -H3 *
- [ #V2 #T2 #HV12 #HT12 #H destruct -a -l0 -W1 -W10 -U10 -HV1 -IH3 -IH2
- elim (IH1 … HTU1 … HL12 … HT12) -IH1 -HTU1 -HL12 // [2: /2 width=1/ ] -T1 #U2 #HTU2 #HU12
- lapply (cpcs_flat … V1 V2 … HU12) -HU12 [ /2 width=1/ ] -HV12 /3 width=3/
- | #b #V2 #W #T2 #T20 #HV12 #HT20 #H1 #H2 destruct
- elim (snv_inv_bind … HT1) -HT1 #HW #HT2
- elim (ssta_inv_bind1 … HTU1) -HTU1 #U2 #HTU2 #H destruct
- elim (dxprs_inv_abst1 … HTU10) -HTU10 #W0 #U0 #HW0 #_ #H destruct
- lapply (cprs_div … HW10 … HW0) -W0 #HW1W
- elim (ssta_fwd_correct … HVW1) <minus_plus_m_m #X1 #HWX1
- elim (snv_fwd_ssta … HW) #l1 #V #HWV
- lapply (IH3 … HVW1) -IH3 // [ /2 width=1/ ] #HW1
- elim (ssta_ltpr_cpcs_aux … IH2 IH1 … HWX1 … HWV …) -IH2 -HWX1 //
- [2: /2 width=1/
- |3: /4 width=4 by ygt_yprs_trans, fw_ygt, sstas_yprs, ssta_sstas/
- ] #H #_ destruct -X1
- elim (IH1 … HVW1 … HL12 … HV12) -HVW1 // -HV1 [2: /2 width=1/ ] #W2 #HVW2 #HW12
- elim (IH1 … HWV … HL12 W) -HWV // -HW [2: /2 width=1/ ] #V0 #HWV0 #_
- elim (IH1 … HTU2 (L2.ⓛW) … HT20) -IH1 -HTU2 -HT20 // [2,3: /2 width=1/ ] -HT2 #U20 #HTU20 #HU20
- lapply (ltpr_cpcs_conf … HL12 … HW1W) -L1 #HW1W
- lapply (cpcs_canc_sn … HW12 HW1W) -W1 #HW2
- elim (lsubss_ssta_trans … HTU20 (L2.ⓓV2) ?) -HTU20
- [ #U #HTU20 #HUU20 -HWV0 -W2
- @(ex2_intro … (ⓓ{b}V2.U)) [ /2 width=1/ ] -h -l -l1 -V -V0 -T2 -T20 -U0
- @(cpcs_cprs_strap2 … (ⓓ{b}V2.U2)) [ /3 width=1/ ] -V1
- @(cpcs_canc_dx … (ⓓ{b}V2.U20)) /2 width=2/
- | -b -l -V -V1 -T2 -T20 -U0 -U2 -U20 /2 width=6/
- ]
- | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct -a -l0 -W1 -W10 -HV1 -IH3 -IH2
- elim (ssta_inv_bind1 … HTU1) -HTU1 #U0 #HTU0 #H destruct
- elim (snv_inv_bind … HT1) -HT1 #_ #HT0
- elim (IH1 … HTU0 (L2.ⓓW2) … HT02) -IH1 -HTU0 // [2,3: /2 width=1/ ] -T0 -HL12 #U2 #HTU2 #HU02
- lapply (cpcs_bind2 … W0 … HU02 b) -HU02 [ /2 width=1/ ] #HU02
- lapply (cpcs_flat … V1 V0 … HU02 Appl) [ /2 width=1/ ] -HV10 -HU02 #HU02
- lapply (cpcs_cpr_strap1 … HU02 (ⓓ{b}W2.ⓐV2.U2) ?) -HU02 [ /3 width=3/ ] -V0 -HW02 /4 width=3/
- ]
-| #U0 #T1 #HL0 #HT0 #H1 #X2 #l #H2 #L2 #HL12 #X3 #H3 destruct -IH3 -IH2
- elim (snv_inv_cast … H1) -H1 #T0 #l0 #_ #HT1 #HT10 #_
- lapply (ssta_inv_cast1 … H2) -H2 #HTU1
- elim (ssta_mono … HT10 … HTU1) -HT10 #H1 #H2 destruct
- elim (tpr_inv_cast1 … H3) -H3
- [ * #U2 #T2 #_ #HT12 #H destruct
- elim (IH1 … HTU1 … HL12 … HT12) -IH1 -HTU1 -HL12 // [2: /2 width=1/ ] -T1 -U0 /3 width=3/
- | #HT1X3
- elim (IH1 … HTU1 … HL12 … HT1X3) -IH1 -HTU1 -HL12 // [2: /2 width=1/ ] -U0 -T1 /2 width=3/
- ]
-]
-qed-.
lemma lsubsv_yprs: ∀h,g,L1,L2,T. h ⊢ L2 ¡⊑[g] L1 → h ⊢ ⦃L1, T⦄ ≥[g] ⦃L2, T⦄.
/3 width=1/ qed.
-lemma lpr_cprs_yprs: ∀h,g,L1,L2,T1,T2. L1 ⊢ ➡ L2 → L2 ⊢ T1 ➡* T2 →
+lemma cprs_lpr_yprs: ∀h,g,L1,L2,T1,T2. L1 ⊢ T1 ➡* T2 → L1 ⊢ ➡ L2 →
h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
-/3 width=4 by yprs_strap2, ypr_lpr, cprs_yprs/
+/3 width=4 by yprs_strap1, ypr_lpr, cprs_yprs/
qed.
(* Basic_1: was: pc3_wcpr0_t *)
(* Basic_1: note: pc3_wcpr0_t should be renamed *)
-lemma lpr_cprs_conf: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡* T2 → L2 ⊢ T1 ⬌* T2.
+lemma cprs_lpr_conf: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡* T2 → L2 ⊢ T1 ⬌* T2.
#L1 #L2 #HL12 #T1 #T2 #H @(cprs_ind … H) -T2 //
/3 width=5 by cpcs_trans, lpr_cpr_conf/
qed-.
(* Basic_1: was: pc3_wcpr0 *)
-lemma lpr_cpcs_conf: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T1,T2. L1 ⊢ T1 ⬌* T2 → L2 ⊢ T1 ⬌* T2.
+lemma cpcs_lpr_conf: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T1,T2. L1 ⊢ T1 ⬌* T2 → L2 ⊢ T1 ⬌* T2.
#L1 #L2 #HL12 #T1 #T2 #H
-elim (cpcs_inv_cprs … H) -H /3 width=5 by cpcs_canc_dx, lpr_cprs_conf/
+elim (cpcs_inv_cprs … H) -H /3 width=5 by cpcs_canc_dx, cprs_lpr_conf/
qed-.
(* *)
(**************************************************************************)
+notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPConv $L1 $T1 $L2 $T2 }.
+
+notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ ⬌ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPConvAlt $L1 $T1 $L2 $T2 }.
+
include "basic_2/reducibility/fpr.ma".
(* CONTEXT-FREE PARALLEL CONVERSION ON CLOSURES *****************************)
(* *)
(**************************************************************************)
+notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPConvStar $L1 $T1 $L2 $T2 }.
+
+notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ ⬌ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPConvStarAlt $L1 $T1 $L2 $T2 }.
+
include "basic_2/conversion/fpc.ma".
(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
(* *)
(**************************************************************************)
+notation "hvbox( ⦃ term 46 L1 ⦄ ⬌ ⬌ break ⦃ term 46 L2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPConvAlt $L1 $L2 }.
+
include "basic_2/reducibility/lfpr.ma".
(* FOCALIZED PARALLEL CONVERSION ON LOCAL ENVIRONMENTS **********************)
(* *)
(**************************************************************************)
+notation "hvbox( ⦃ term 46 L1 ⦄ ⬌ ⬌ * break ⦃ term 46 L2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'FocalizedPConvStarAlt $L1 $L2 }.
+
include "basic_2/conversion/lfpc.ma".
(* FOCALIZED PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *********************)
--- /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/unwind/sstas_sstas.ma".
+include "basic_2/equivalence/cpcs_ltpr.ma".
+include "basic_2/dynamic/snv_ltpss_dx.ma".
+include "basic_2/dynamic/snv_sstas.ma".
+include "basic_2/dynamic/ygt.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Inductive premises for the preservation results **************************)
+
+definition IH_snv_ltpr_tpr: ∀h:sh. sd h → relation2 lenv term ≝
+ λh,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[g] →
+ ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊢ T2 ¡[g].
+
+definition IH_ssta_ltpr_tpr: ∀h:sh. sd h → relation2 lenv term ≝
+ λh,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[g] →
+ ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+ ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L2 ⊢ U1 ⬌* U2.
+
+definition IH_snv_ssta: ∀h:sh. sd h → relation2 lenv term ≝
+ λh,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[g] →
+ ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l+1, U1⦄ → ⦃h, L1⦄ ⊢ U1 ¡[g].
+
+definition IH_snv_lsubsv: ∀h:sh. sd h → relation2 lenv term ≝
+ λh,g,L2,T. ⦃h, L2⦄ ⊢ T ¡[g] →
+ ∀L1. h ⊢ L1 ¡⊑[g] L2 → ⦃h, L1⦄ ⊢ T ¡[g].
+
+(* Properties for the preservation results **********************************)
+
+fact snv_ltpr_cpr_aux: ∀h,g,L1,T1. IH_snv_ltpr_tpr h g L1 T1 →
+ ⦃h, L1⦄ ⊢ T1 ¡[g] →
+ ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡ T2 → ⦃h, L2⦄ ⊢ T2 ¡[g].
+#h #g #L1 #T1 #IH #HT1 #L2 #HL12 #T2 * #T #HT1T #HTT2
+lapply (IH … HL12 … HT1T) -HL12 // -T1 #HT0
+lapply (snv_tpss_conf … HT0 … HTT2) -T //
+qed-.
+
+fact ssta_ltpr_cpr_aux: ∀h,g,L1,T1. IH_ssta_ltpr_tpr h g L1 T1 →
+ ⦃h, L1⦄ ⊢ T1 ¡[g] →
+ ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+ ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡ T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L2 ⊢ U1 ⬌* U2.
+#h #g #L1 #T1 #IH #HT1 #U1 #l #HTU1 #L2 #HL12 #T2 * #T #HT1T #HTT2
+elim (IH … HTU1 … HL12 … HT1T) // -L1 -T1 #U #HTU #HU1
+elim (ssta_tpss_conf … HTU … HTT2) -T #U2 #HTU2 #HU2
+lapply (cpcs_cpr_strap1 … HU1 U2 ?) /2 width=3/
+qed-.
+
+fact snv_ltpr_cprs_aux: ∀h,g,L0,T0.
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
+ ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
+ ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡* T2 → ⦃h, L2⦄ ⊢ T2 ¡[g].
+#h #g #L0 #T0 #IH #L1 #T1 #HLT0 #HT1 #L2 #HL12 #T2 #H
+@(cprs_ind … H) -T2 [ /2 width=6 by snv_ltpr_cpr_aux/ ] -HT1
+/5 width=6 by snv_ltpr_cpr_aux, ygt_yprs_trans, ltpr_cprs_yprs/
+qed-.
+
+fact ssta_ltpr_cprs_aux: ∀h,g,L0,T0.
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
+ ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
+ ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+ ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡* T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L2 ⊢ U1 ⬌* U2.
+#h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 #l #HTU1 #L2 #HL12 #T2 #H
+@(cprs_ind … H) -T2 [ /2 width=7 by ssta_ltpr_cpr_aux/ ]
+#T #T2 #HT1T #HTT2 * #U #HTU #HU1
+elim (ssta_ltpr_cpr_aux … HTU … HTT2) //
+[2: /3 width=9 by snv_ltpr_cprs_aux/
+|3: /5 width=6 by ygt_yprs_trans, ltpr_cprs_yprs/
+] -L0 -L1 -T0 -T1 -T #U2 #HTU2 #HU2
+lapply (cpcs_trans … HU1 … HU2) -U /2 width=3/
+qed-.
+
+fact ssta_ltpr_cpcs_aux: ∀h,g,L0,T0.
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
+ ∀L1,L2,T1,T2. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → h ⊢ ⦃L0, T0⦄ >[g] ⦃L2, T2⦄ →
+ ⦃h, L1⦄ ⊢ T1 ¡[g] → ⦃h, L2⦄ ⊢ T2 ¡[g] →
+ ∀U1,l1. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l1, U1⦄ →
+ ∀U2,l2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l2, U2⦄ →
+ L1 ➡ L2 → L2 ⊢ T1 ⬌* T2 →
+ l1 = l2 ∧ L2 ⊢ U1 ⬌* U2.
+#h #g #L0 #T0 #IH2 #IH1 #L1 #L2 #T1 #T2 #HLT01 #HLT02 #HT1 #HT2 #U1 #l1 #HTU1 #U2 #l2 #HTU2 #HL12 #H
+elim (cpcs_inv_cprs … H) -H #T #H1 #H2
+elim (ssta_ltpr_cprs_aux … HLT01 HT1 … HTU1 … H1) -T1 /2 width=1/ #W1 #H1 #HUW1
+elim (ssta_ltpr_cprs_aux … HLT02 HT2 … HTU2 … H2) -T2 /2 width=1/ #W2 #H2 #HUW2 -L1 -L0 -T0
+elim (ssta_mono … H1 … H2) -h -T #H1 #H2 destruct
+lapply (cpcs_canc_dx … HUW1 … HUW2) -W2 /2 width=1/
+qed-.
+
+fact snv_sstas_aux: ∀h,g,L0,T0.
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+ ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
+ ∀U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 → ⦃h, L1⦄ ⊢ U1 ¡[g].
+#h #g #L0 #T0 #IH #L1 #T1 #HLT0 #HT1 #U1 #H
+@(sstas_ind … H) -U1 // -HT1 /4 width=5 by ygt_yprs_trans, sstas_yprs/
+qed-.
+
+fact sstas_ltpr_cprs_aux: ∀h,g,L0,T0.
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
+ ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
+ ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡* T2 → ∀U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L2 ⊢ U1 ⬌* U2.
+#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #L2 #HL12 #T2 #HT12 #U1 #H
+@(sstas_ind … H) -U1 [ /3 width=3/ ]
+#U1 #W1 #l1 #HTU1 #HUW1 * #U2 #HTU2 #HU12
+lapply (snv_ltpr_cprs_aux … IH2 … HT1 … HT12) // #HT2
+elim (snv_sstas_fwd_correct … HTU2) // #W2 #l2 #HUW2
+elim (ssta_ltpr_cpcs_aux … IH2 IH1 … HUW1 … HUW2 … HU12) -IH2 -IH1 -HUW1 -HU12 //
+[2: /4 width=8 by snv_sstas_aux, ygt_yprs_trans, ltpr_cprs_yprs/
+|3: /3 width=7 by snv_sstas_aux, ygt_yprs_trans, cprs_yprs/
+|4: /4 width=5 by ygt_yprs_trans, ltpr_cprs_yprs, sstas_yprs/
+|5: /3 width=4 by ygt_yprs_trans, cprs_yprs, sstas_yprs/
+] -L0 -T0 -T1 -HT2 #H #HW12 destruct /3 width=4/
+qed-.
+
+fact dxprs_ltpr_cprs_aux: ∀h,g,L0,T0.
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
+ ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
+ ∀U1. ⦃h, L1⦄ ⊢ T1 •*➡*[g] U1 →
+ ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡* T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*➡*[g] U2 & L2 ⊢ U1 ➡* U2.
+#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 * #W1 #HTW1 #HWU1 #L2 #HL12 #T2 #HT12
+elim (sstas_ltpr_cprs_aux … IH3 IH2 IH1 … H01 … HT12 … HTW1) // -L0 -T0 -T1 #W2 #HTW2 #HW12
+lapply (ltpr_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_dxprs_aux: ∀h,g,L0,T0.
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
+ ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
+ ∀l,U1. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l+1, U1⦄ → ∀T2. ⦃h, L1⦄ ⊢ T1 •*➡*[g] T2 →
+ ∃∃U,U2. ⦃h, L1⦄ ⊢ U1 •*[g] U & ⦃h, L1⦄ ⊢ T2 •*[g] U2 & L1 ⊢ U ⬌* U2.
+#h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #H01 #HT1 #l #U1 #HTU1 #T2 * #T #HT1T #HTT2
+elim (sstas_strip … HT1T … HTU1) #HU1T destruct [ -HT1T | -L0 -T0 -T1 ]
+[ elim (ssta_ltpr_cprs_aux … IH2 IH1 … HTU1 L1 … HTT2) // -L0 -T0 -T /3 width=5/
+| @(ex3_2_intro …T2 HU1T) // /2 width=1/
+]
+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/computation/dxprs_dxprs.ma".
+include "basic_2/dynamic/snv_cpcs.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties on context-free parallel reduction for local environments *****)
+
+fact snv_ltpr_tpr_aux: ∀h,g,L0,T0.
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
+ ∀L1,T1. L0 = L1 → T0 = T1 → IH_snv_ltpr_tpr h g L1 T1.
+#h #g #L0 #T0 #IH4 #IH3 #IH2 #IH1 #L1 * * [||||*]
+[ #k #HL0 #HT0 #H1 #L2 #_ #X #H2 destruct -IH4 -IH3 -IH2 -IH1 -L1
+ >(tpr_inv_atom1 … H2) -X //
+| #i #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct -IH4 -IH3 -IH2
+ elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
+ >(tpr_inv_atom1 … H2) -X
+ elim (ltpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
+ elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+ lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) -HLK1 #HLK1
+ lapply (IH1 … HK12 … HV12) -IH1 -HV12 -HK12 // -HV1 [ /2 width=1/ ] -HLK1 /2 width=5/
+| #p #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct -IH4 -IH3 -IH2 -IH1
+ elim (snv_inv_gref … H1)
+| #a #I #V1 #T1 #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct -IH4 -IH3 -IH2
+ elim (snv_inv_bind … H1) -H1 #HV1 #HT1
+ elim (tpr_inv_bind1 … H2) -H2 *
+ [ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct
+ lapply (tps_lsubr_trans … HT02 (L2.ⓑ{I}V2) ?) -HT02 /2 width=1/ #HT02
+ lapply (IH1 … HL12 … HV12) // [ /2 width=1/ ] #HV2
+ lapply (snv_ltpr_cpr_aux … HT1 (L2.ⓑ{I}V2) … T2 ?) -HT1
+ [ /3 width=5 by cpr_intro, tps_tpss/ | /2 width=1/ | /3 width=1/ ] -IH1 -T0 /2 width=1/
+ | #T2 #HT12 #HXT2 #H1 #H2 destruct
+ lapply (IH1 … HT1 (L2.ⓓV1) … HT12) -IH1 -HT1 -HT12 [1,2: /2 width=1/ ] -HL12 #HT2
+ lapply (snv_inv_lift … HT2 L2 … HXT2) -T2 // /2 width=1/
+ ]
+| #V1 #T1 #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct
+ elim (snv_inv_appl … H1) -H1 #a #W10 #W1 #U1 #l #HV1 #HT1 #HVW1 #HW10 #HTU1
+ elim (tpr_inv_appl1 … H2) -H2 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct -IH4
+ lapply (IH1 … HV1 … HL12 … HV12) [ /2 width=1/ ] #HV2
+ lapply (IH1 … HT1 … HL12 … HT12) [ /2 width=1/ ] #HT2
+ elim (IH3 … HVW1 … HL12 … HV12) -HVW1 -HV12 // -HV1 [2: /2 width=1/ ] #W2 #HVW2 #HW12
+ elim (dxprs_ltpr_cprs_aux … IH2 IH1 IH3 … HTU1 … HL12 T2) // [2: /3 width=1/ |3: /2 width=1/ ] -IH2 -IH1 -IH3 -HT1 -HT12 -HTU1 #X #HTU2 #H
+ elim (cprs_inv_abst1 Abst W1 … H) -H #W20 #U2 #HW120 #_ #H destruct
+ lapply (ltpr_cprs_conf … HL12 … HW10) -L1 #HW10
+ lapply (cpcs_cprs_strap1 … HW10 … HW120) -W1 #HW120
+ lapply (cpcs_canc_sn … HW12 HW120) -W10 #HW20
+ elim (cpcs_inv_cprs … HW20) -HW20 #W0 #HW20 #HW200
+ lapply (dxprs_cprs_trans … (ⓛ{a}W0.U2) HTU2 ?) [ /2 width=1/ ] -HW200 -HTU2 /2 width=8/
+ | #b #V2 #W20 #T20 #T2 #HV12 #HT202 #H1 #H2 destruct
+ elim (snv_inv_bind … HT1) -HT1 #HW20 #HT20
+ elim (dxprs_inv_abst1 … HTU1) -HTU1 #W30 #T30 #HW230 #_ #H destruct -T30
+ lapply (cprs_div … HW10 … HW230) -W30 #HW120
+ elim (snv_fwd_ssta … HW20) #l0 #U20 #HWU20
+ elim (ssta_fwd_correct … HVW1) <minus_plus_m_m #U10 #HWU10
+ elim (ssta_ltpr_cpcs_aux … IH1 IH3 … HW20 … HWU10 … HWU20) // -IH3 -HWU10
+ [2: /3 width=5/ |3: /2 width=1/
+ |4: /4 width=4 by ygt_yprs_trans, fw_ygt, sstas_yprs, ssta_sstas/
+ ] #H #_ -IH2 -U10 destruct
+ lapply (IH4 … HT20 (L1.ⓓV1) ?) [ /2 width=6/ | /2 width=1/ ] -U20 -W10 -l0 -IH4 -HT20 -HW20 #HT20
+ lapply (IH1 … HL12 … HV12) // [ /2 width=1/ ] #HV2
+ lapply (IH1 … HT20 … (L2.ⓓV2) … HT202) [1,2: /2 width=1/ ] -L1 -V1 -W20 -T20 /2 width=1/
+ | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct -IH4
+ elim (snv_inv_bind … HT1) -HT1 #HW0 #HT0
+ elim (dxprs_inv_abbr_abst … HTU1) -HTU1 #X #HTU0 #HX #H destruct
+ elim (lift_inv_bind1 … HX) -HX #W3 #U3 #HW13 #_ #H destruct
+ lapply (ltpr_cprs_conf … HL12 … HW10) -HW10 #HW10
+ elim (dxprs_ltpr_cprs_aux … IH2 IH1 IH3 … HTU0 (L2.ⓓW2) … T2 ?) // [2: /3 width=1/ |3,4: /2 width=1/ ] -IH2 -HTU0 #X #HTU2 #H
+ elim (cprs_inv_abst1 Abst W3 … H) -H #W #U2 #HW1 #_ #H destruct -U3
+ elim (IH3 … HVW1 … HL12 … HV10) // /2 width=1/ -IH3 -HVW1 #X #H1 #H2
+ lapply (cpcs_canc_sn … H2 HW10) -W10 #H2
+ elim (lift_total X 0 1) #W20 #H3
+ lapply (ssta_lift … H1 (L2.ⓓW2) … HV02 … H3) /2 width=1/ -H1 #HVW20
+ lapply (cpcs_lift (L2.ⓓW2) … H3 … HW13 H2) /2 width=1/ -HW13 -H3 -H2 #HW320
+ lapply (cpcs_cprs_strap1 … HW320 … HW1) -W3 #HW20
+ elim (cpcs_inv_cprs … HW20) -HW20 #W3 #HW203 #HW3
+ lapply (dxprs_cprs_trans … (ⓛ{a}W3.U2) HTU2 ?) [ /2 width=1/ ] -HW3 -HTU2 #HTU2
+ lapply (IH1 … HL12 … HW02) // [ /2 width=1/ ] -HW0 #HW2
+ lapply (IH1 … HL12 … HV10) // [ /2 width=1/ ] -HV1 -HV10 #HV0
+ lapply (IH1 … HT0 … (L2.ⓓW2) … HT02) [1,2: /2 width=1/ ] -L1 -HT02 -HW02 #HT2
+ lapply (snv_lift … HV0 (L2.ⓓW2) … HV02) /2 width=1/ -HV0 -HV02 /3 width=8/
+ ]
+| #W1 #T1 #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct -IH4 -IH2
+ elim (snv_inv_cast … H1) -H1 #U1 #l #HW1 #HT1 #HTU1 #HUW1
+ elim (tpr_inv_cast1 … H2) -H2
+ [ * #W2 #T2 #HW12 #HT12 #H destruct
+ lapply (cpcs_cprs_strap1 … HUW1 W2 ?) [ /3 width=1/ ] -HUW1 #H1
+ lapply (IH1 … HL12 … HW12) // /2 width=1/ -HW1 -HW12 #HW2
+ lapply (IH1 … HL12 … HT12) // /2 width=1/ -IH1 #HT2
+ elim (IH3 … HTU1 … HL12 … HT12) // /2 width=1/ -IH3 -HT1 -HT12 -HTU1 #U2 #HTU2 #HU12
+ lapply (ltpr_cpcs_conf … HL12 … H1) -L1 #H1
+ lapply (cpcs_canc_sn … HU12 H1) -U1 /2 width=4/
+ | #H -IH3 -HW1 -HTU1 -HUW1
+ lapply (IH1 … HL12 … H) // /2 width=1/
+ ]
+]
+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/equivalence/lsubss_ssta.ma".
+include "basic_2/dynamic/snv_cpcs.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties on context-free parallel reduction for local environments *****)
+
+fact ssta_ltpr_tpr_aux: ∀h,g,L0,T0.
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
+ (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
+ ∀L1,T1. L0 = L1 → T0 = T1 → IH_ssta_ltpr_tpr h g L1 T1.
+#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 * * [|||| *]
+[ #k #_ #_ #_ #X2 #l #H2 #L2 #HL12 #X3 #H3 -IH3 -IH2 -IH1
+ elim (ssta_inv_sort1 … H2) -H2 #Hkl #H destruct
+ >(tpr_inv_atom1 … H3) -X3 /4 width=6/
+| #i #HL0 #HT0 #H1 #X2 #l #H2 #L2 #HL12 #X3 #H3 destruct -IH3 -IH2
+ elim (snv_inv_lref … H1) -H1 #I0 #K0 #V0 #H #HV1
+ elim (ssta_inv_lref1 … H2) -H2 * #K1
+ >(tpr_inv_atom1 … H3) -X3
+ [ #V1 #W1 #HLK1 #HVW1 #HWU1
+ lapply (ldrop_mono … H … HLK1) -H #H destruct
+ lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
+ elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
+ elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+ elim (IH1 … HVW1 … HK12 … HV12) -IH1 -HVW1 -HV12 // [2: /2 width=1/ ] -V1 #W2 #HVW2 #HW12
+ lapply (ldrop_fwd_ldrop2 … HLK2) #H2
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (cpcs_lift … H2 … HWU1 … HWU2 HW12) -H2 -W1 /3 width=6/
+ | #V1 #W1 #l0 #HLK1 #HVW1 #HVU1 #H destruct
+ lapply (ldrop_mono … H … HLK1) -H #H destruct
+ lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
+ elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 #X #H #HLK2
+ elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+ elim (IH1 … HVW1 … HK12 … HV12) -IH1 -HVW1 -HK12 // [2: /2 width=1/ ] -HV1 -HKV1 #W2 #HVW2 #_ -W1
+ elim (lift_total V2 0 (i+1)) #U2 #HVU2
+ lapply (tpr_lift … HV12 … HVU1 … HVU2) -V1 /4 width=6/
+ ]
+| #p #_ #HT0 #H1 destruct -IH3 -IH2 -IH1
+ elim (snv_inv_gref … H1)
+| #a #I #V1 #T1 #HL0 #HT0 #H1 #X2 #l #H2 #L2 #HL12 #X3 #H3 destruct -IH3 -IH2
+ elim (snv_inv_bind … H1) -H1 #_ #HT1
+ elim (ssta_inv_bind1 … H2) -H2 #U1 #HTU1 #H destruct
+ elim (tpr_inv_bind1 … H3) -H3 *
+ [ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct
+ lapply (tps_lsubr_trans … HT02 (L2.ⓑ{I}V2) ?) -HT02 [ /2 width=1/ ] #HT02
+ elim (ssta_ltpr_cpr_aux … HT1 … HTU1 (L2.ⓑ{I}V2) … T2) -HT1 -HTU1
+ [2: /3 width=5 by cpr_intro, tps_tpss/ |3: /2 width=1/ |4: /3 width=1/ ] -IH1 -T0 -HL12 #U2 #HTU2 #HU12
+ lapply (cpcs_bind2 … V1 … HU12 … a) -HU12 [ /2 width=1/ ] -HV12 /3 width=3/
+ | #T2 #HT12 #HT2 #H1 #H2 destruct
+ elim (IH1 … HTU1 (L2.ⓓV1) … T2) -IH1 -HTU1 // [2,3: /2 width=1/ ] -T1 -HL12 #U2 #HTU2 #HU12
+ lapply (cpcs_bind1 … HU12 … V1 … true) // -HU12 #HU12
+ elim (ssta_inv_lift1 … HTU2 … HT2) -T2 [3: /2 width=1/ |2: skip ] #U #HXU #HU2
+ lapply (cpcs_cpr_strap1 … HU12 U ?) -HU12 [ /3 width=3/ ] -U2 /2 width=3/
+ ]
+| #V1 #T1 #HL0 #HT0 #H1 #X2 #l #H2 #L2 #HL12 #X3 #H3 destruct
+ elim (snv_inv_appl … H1) -H1 #a #W1 #W10 #U10 #l0 #HV1 #HT1 #HVW1 #HW10 #HTU10
+ elim (ssta_inv_appl1 … H2) -H2 #U1 #HTU1 #H destruct
+ elim (tpr_inv_appl1 … H3) -H3 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct -a -l0 -W1 -W10 -U10 -HV1 -IH3 -IH2
+ elim (IH1 … HTU1 … HL12 … HT12) -IH1 -HTU1 -HL12 // [2: /2 width=1/ ] -T1 #U2 #HTU2 #HU12
+ lapply (cpcs_flat … V1 V2 … HU12) -HU12 [ /2 width=1/ ] -HV12 /3 width=3/
+ | #b #V2 #W #T2 #T20 #HV12 #HT20 #H1 #H2 destruct
+ elim (snv_inv_bind … HT1) -HT1 #HW #HT2
+ elim (ssta_inv_bind1 … HTU1) -HTU1 #U2 #HTU2 #H destruct
+ elim (dxprs_inv_abst1 … HTU10) -HTU10 #W0 #U0 #HW0 #_ #H destruct
+ lapply (cprs_div … HW10 … HW0) -W0 #HW1W
+ elim (ssta_fwd_correct … HVW1) <minus_plus_m_m #X1 #HWX1
+ elim (snv_fwd_ssta … HW) #l1 #V #HWV
+ lapply (IH3 … HVW1) -IH3 // [ /2 width=1/ ] #HW1
+ elim (ssta_ltpr_cpcs_aux … IH2 IH1 … HWX1 … HWV …) -IH2 -HWX1 //
+ [2: /2 width=1/
+ |3: /4 width=4 by ygt_yprs_trans, fw_ygt, sstas_yprs, ssta_sstas/
+ ] #H #_ destruct -X1
+ elim (IH1 … HVW1 … HL12 … HV12) -HVW1 // -HV1 [2: /2 width=1/ ] #W2 #HVW2 #HW12
+ elim (IH1 … HWV … HL12 W) -HWV // -HW [2: /2 width=1/ ] #V0 #HWV0 #_
+ elim (IH1 … HTU2 (L2.ⓛW) … HT20) -IH1 -HTU2 -HT20 // [2,3: /2 width=1/ ] -HT2 #U20 #HTU20 #HU20
+ lapply (ltpr_cpcs_conf … HL12 … HW1W) -L1 #HW1W
+ lapply (cpcs_canc_sn … HW12 HW1W) -W1 #HW2
+ elim (lsubss_ssta_trans … HTU20 (L2.ⓓV2) ?) -HTU20
+ [ #U #HTU20 #HUU20 -HWV0 -W2
+ @(ex2_intro … (ⓓ{b}V2.U)) [ /2 width=1/ ] -h -l -l1 -V -V0 -T2 -T20 -U0
+ @(cpcs_cprs_strap2 … (ⓓ{b}V2.U2)) [ /3 width=1/ ] -V1
+ @(cpcs_canc_dx … (ⓓ{b}V2.U20)) /2 width=2/
+ | -b -l -V -V1 -T2 -T20 -U0 -U2 -U20 /2 width=6/
+ ]
+ | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct -a -l0 -W1 -W10 -HV1 -IH3 -IH2
+ elim (ssta_inv_bind1 … HTU1) -HTU1 #U0 #HTU0 #H destruct
+ elim (snv_inv_bind … HT1) -HT1 #_ #HT0
+ elim (IH1 … HTU0 (L2.ⓓW2) … HT02) -IH1 -HTU0 // [2,3: /2 width=1/ ] -T0 -HL12 #U2 #HTU2 #HU02
+ lapply (cpcs_bind2 … W0 … HU02 b) -HU02 [ /2 width=1/ ] #HU02
+ lapply (cpcs_flat … V1 V0 … HU02 Appl) [ /2 width=1/ ] -HV10 -HU02 #HU02
+ lapply (cpcs_cpr_strap1 … HU02 (ⓓ{b}W2.ⓐV2.U2) ?) -HU02 [ /3 width=3/ ] -V0 -HW02 /4 width=3/
+ ]
+| #U0 #T1 #HL0 #HT0 #H1 #X2 #l #H2 #L2 #HL12 #X3 #H3 destruct -IH3 -IH2
+ elim (snv_inv_cast … H1) -H1 #T0 #l0 #_ #HT1 #HT10 #_
+ lapply (ssta_inv_cast1 … H2) -H2 #HTU1
+ elim (ssta_mono … HT10 … HTU1) -HT10 #H1 #H2 destruct
+ elim (tpr_inv_cast1 … H3) -H3
+ [ * #U2 #T2 #_ #HT12 #H destruct
+ elim (IH1 … HTU1 … HL12 … HT12) -IH1 -HTU1 -HL12 // [2: /2 width=1/ ] -T1 -U0 /3 width=3/
+ | #HT1X3
+ elim (IH1 … HTU1 … HL12 … HT1X3) -IH1 -HTU1 -HL12 // [2: /2 width=1/ ] -U0 -T1 /2 width=3/
+ ]
+]
+qed-.
}
]
[ { "stratified native validity" * } {
- [ "snv ( ⦃?,?⦄ ⊢ ? ¡[?] )" "snv_lift" + "snv_lpss" + "snv_aaa" + "snv_ssta" + "snv_sstas" + "snv_ssta_ltpr" + "snv_ltpr" + "snv_cpcs" * ]
+ [ "snv ( ⦃?,?⦄ ⊢ ? ¡[?] )" "snv_lift" + "snv_lpss" + "snv_aaa" + "snv_ssta" + "snv_sstas" + "snv_ssta_lpr" + "snv_lpr" + "snv_cpcs" * ]
}
]
}
projects / helm.git / commitdiff