(* *)
(**************************************************************************)
-include "basic_2/relocation/ldrop_ldrop.ma".
-include "basic_2/substitution/fqus_alt.ma".
-include "basic_2/static/ssta.ma".
+include "basic_2/substitution/ldrop_ldrop.ma".
+include "basic_2/multiple/fqus_alt.ma".
+include "basic_2/static/sta.ma".
+include "basic_2/static/da.ma".
include "basic_2/reduction/cpx.ma".
(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
(* Advanced properties ******************************************************)
-lemma ssta_cpx: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •[h, g] T2 →
- ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
+lemma sta_cpx: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •[h] T2 →
+ ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
#h #g #G #L #T1 #T2 #l #H elim H -G -L -T1 -T2
-[ /3 width=4 by cpx_sort, da_inv_sort/
-| #G #L #K #V #U #W #i #HLK #_ #HWU #IHVW #H
+[ /3 width=4 by cpx_st, da_inv_sort/
+| #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #IHV12 #H
elim (da_inv_lref … H) -H * #K0 #V0 [| #l0 ] #HLK0
lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct /3 width=7 by cpx_delta/
-| #G #L #K #W #U #l0 #i #HLK #_ #HWU #H
+| #G #L #K #W1 #W2 #V1 #i #HLK #_ #HWV1 #IHW12 #H
elim (da_inv_lref … H) -H * #K0 #W0 [| #l1 ] #HLK0
- lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct /2 width=7 by cpx_delta/
+ lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct /3 width=7 by cpx_delta/
| /4 width=2 by cpx_bind, da_inv_bind/
| /4 width=3 by cpx_flat, da_inv_flat/
-| /4 width=3 by cpx_tau, da_inv_flat/
+| /4 width=3 by cpx_eps, da_inv_flat/
]
qed.
>(lift_mono … H1 … H2) -H1 -H2 //
| #G #K #k #l #Hkl #L #s #d #e #_ #U1 #H1 #U2 #H2
>(lift_inv_sort1 … H1) -U1
- >(lift_inv_sort1 … H2) -U2 /2 width=2 by cpx_sort/
+ >(lift_inv_sort1 … H2) -U2 /2 width=2 by cpx_st/
| #I #G #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #s #d #e #HLK #U1 #H #U2 #HWU2
elim (lift_inv_lref1 … H) * #Hid #H destruct
[ elim (lift_trans_ge … HVW2 … HWU2) -W2 // <minus_plus #W2 #HVW2 #HWU2
elim (lift_inv_bind1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
elim (lift_conf_O1 … HTU2 … HT2) -T2 /4 width=6 by cpx_zeta, ldrop_skip/
| #G #K #V #T1 #T2 #_ #IHT12 #L #s #d #e #HLK #U1 #H #U2 #HTU2
- elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=6 by cpx_tau/
+ elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=6 by cpx_eps/
| #G #K #V1 #V2 #T #_ #IHV12 #L #s #d #e #HLK #U1 #H #U2 #HVU2
- elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=6 by cpx_ti/
+ elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=6 by cpx_ct/
| #a #G #K #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L #s #d #e #HLK #X1 #HX1 #X2 #HX2
elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
| lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by cpx_atom, lift_gref, ex2_intro/
]
| #G #L #k #l #Hkl #K #s #d #e #_ #T1 #H
- lapply (lift_inv_sort2 … H) -H #H destruct /3 width=3 by cpx_sort, lift_sort, ex2_intro/
+ lapply (lift_inv_sort2 … H) -H #H destruct /3 width=3 by cpx_st, lift_sort, ex2_intro/
| #I #G #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #s #d #e #HLK #T1 #H
elim (lift_inv_lref2 … H) -H * #Hid #H destruct
[ elim (ldrop_conf_lt … HLK … HLV) -L // #L #U #HKL #HLV #HUV
elim (lift_div_le … HU2 … HTU) -U /3 width=5 by cpx_zeta, ex2_intro/
| #G #L #V #U1 #U2 #_ #IHU12 #K #s #d #e #HLK #X #H
elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHU12 … HLK … HTU1) -L -U1 /3 width=3 by cpx_tau, ex2_intro/
+ elim (IHU12 … HLK … HTU1) -L -U1 /3 width=3 by cpx_eps, ex2_intro/
| #G #L #V1 #V2 #U1 #_ #IHV12 #K #s #d #e #HLK #X #H
elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1) -L -V1 /3 width=3 by cpx_ti, ex2_intro/
+ elim (IHV12 … HLK … HWV1) -L -V1 /3 width=3 by cpx_ct, ex2_intro/
| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #K #s #d #e #HLK #X #HX
elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
]
qed-.
-lemma fqu_ssta_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 •[h, g] U2 →
- ∀l. ⦃G2, L2⦄ ⊢ T2 ▪[h, g] l+1 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
-/3 width=5 by fqu_cpx_trans, ssta_cpx/ qed-.
+lemma fqu_sta_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 •[h] U2 →
+ ∀l. ⦃G2, L2⦄ ⊢ T2 ▪[h, g] l+1 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+/3 width=5 by fqu_cpx_trans, sta_cpx/ qed-.
lemma fquq_cpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
]
qed-.
-lemma fquq_ssta_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 •[h, g] U2 →
- ∀l. ⦃G2, L2⦄ ⊢ T2 ▪[h, g] l+1 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
-/3 width=5 by fquq_cpx_trans, ssta_cpx/ qed-.
+lemma fquq_sta_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 •[h] U2 →
+ ∀l. ⦃G2, L2⦄ ⊢ T2 ▪[h, g] l+1 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+/3 width=5 by fquq_cpx_trans, sta_cpx/ qed-.
lemma fqup_cpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
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