(**************************************************************************)
include "basic_2/reduction/cnx_lift.ma".
-include "basic_2/computation/acp.ma".
+include "basic_2/computation/gcp.ma".
include "basic_2/computation/csx.ma".
(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
(* Relocation properties ****************************************************)
(* Basic_1: was just: sn3_lift *)
-lemma csx_lift: ∀h,g,G,L2,L1,T1,d,e. ⦃G, L1⦄ ⊢ ⬊*[h, g] T1 →
- â\88\80T2. â\87©[d, e] L2 â\89¡ L1 â\86\92 â\87§[d, e] T1 ≡ T2 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T2.
-#h #g #G #L2 #L1 #T1 #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL21 #HT12
+lemma csx_lift: ∀h,g,G,L2,L1,T1,s,d,e. ⦃G, L1⦄ ⊢ ⬊*[h, g] T1 →
+ â\88\80T2. â¬\87[s, d, e] L2 â\89¡ L1 â\86\92 â¬\86[d, e] T1 ≡ T2 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T2.
+#h #g #G #L2 #L1 #T1 #s #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL21 #HT12
@csx_intro #T #HLT2 #HT2
elim (cpx_inv_lift1 … HLT2 … HL21 … HT12) -HLT2 #T0 #HT0 #HLT10
@(IHT1 … HLT10) // -L1 -L2 #H destruct
->(lift_mono … HT0 … HT12) in HT2; -T1 /2 width=1/
+>(lift_mono … HT0 … HT12) in HT2; -T1 /2 width=1 by/
qed.
(* Basic_1: was just: sn3_gen_lift *)
-lemma csx_inv_lift: ∀h,g,G,L2,L1,T1,d,e. ⦃G, L1⦄ ⊢ ⬊*[h, g] T1 →
- â\88\80T2. â\87©[d, e] L1 â\89¡ L2 â\86\92 â\87§[d, e] T2 ≡ T1 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T2.
-#h #g #G #L2 #L1 #T1 #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL12 #HT21
+lemma csx_inv_lift: ∀h,g,G,L2,L1,T1,s,d,e. ⦃G, L1⦄ ⊢ ⬊*[h, g] T1 →
+ â\88\80T2. â¬\87[s, d, e] L1 â\89¡ L2 â\86\92 â¬\86[d, e] T2 ≡ T1 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T2.
+#h #g #G #L2 #L1 #T1 #s #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL12 #HT21
@csx_intro #T #HLT2 #HT2
elim (lift_total T d e) #T0 #HT0
lapply (cpx_lift … HLT2 … HL12 … HT21 … HT0) -HLT2 #HLT10
@(IHT1 … HLT10) // -L1 -L2 #H destruct
->(lift_inj … HT0 … HT21) in HT2; -T1 /2 width=1/
+>(lift_inj … HT0 … HT21) in HT2; -T1 /2 width=1 by/
qed.
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was: sn3_gen_def *)
+lemma csx_inv_lref_bind: ∀h,g,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V →
+ ⦃G, L⦄ ⊢ ⬊*[h, g] #i → ⦃G, K⦄ ⊢ ⬊*[h, g] V.
+#h #g #I #G #L #K #V #i #HLK #Hi
+elim (lift_total V 0 (i+1))
+/4 width=9 by csx_inv_lift, csx_cpx_trans, cpx_delta, drop_fwd_drop2/
+qed-.
+
(* Advanced properties ******************************************************)
(* Basic_1: was just: sn3_abbr *)
-lemma csx_lref_bind: â\88\80h,g,I,G,L,K,V,i. â\87©[0, i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ ⬊*[h, g] V → ⦃G, L⦄ ⊢ ⬊*[h, g] #i.
+lemma csx_lref_bind: â\88\80h,g,I,G,L,K,V,i. â¬\87[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ ⬊*[h, g] V → ⦃G, L⦄ ⊢ ⬊*[h, g] #i.
#h #g #I #G #L #K #V #i #HLK #HV
@csx_intro #X #H #Hi
elim (cpx_inv_lref1 … H) -H
[ #H destruct elim Hi //
| -Hi * #I0 #K0 #V0 #V1 #HLK0 #HV01 #HV1
- lapply (ldrop_mono … HLK0 … HLK) -HLK #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #HLK
- @(csx_lift … HLK HV1) -HLK -HV1
- @(csx_cpx_trans … HV) -HV //
+ lapply (drop_mono … HLK0 … HLK) -HLK #H destruct
+ /3 width=8 by csx_lift, csx_cpx_trans, drop_fwd_drop2/
]
qed.
elim (cpx_inv_appl1_simple … H1) // -H1
#V0 #T0 #HLV0 #HLT10 #H destruct
elim (eq_false_inv_tpair_dx … H2) -H2
-[ -IHV -HT1 #HT10
- @(csx_cpx_trans … (ⓐV.T0)) /2 width=1/ -HLV0
- @IHT1 -IHT1 // /2 width=1/
+[ -IHV -HT1 /4 width=3 by csx_cpx_trans, cpx_pair_sn/
| -HLT10 * #H #HV0 destruct
- @IHV -IHV // -HT1 /2 width=1/ -HV0
- #T2 #HLT02 #HT02
- @(csx_cpx_trans … (ⓐV.T2)) /2 width=1/ -HLV0
- @IHT1 -IHT1 // -HLT02 /2 width=1/
+ @IHV /4 width=3 by csx_cpx_trans, cpx_pair_sn/ (**) (* full auto 17s *)
]
qed.
-(* Advanced inversion lemmas ************************************************)
+lemma csx_fqu_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, g] T2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/2 width=8 by csx_inv_lref_bind, csx_inv_lift, csx_fwd_flat_dx, csx_fwd_bind_dx, csx_fwd_pair_sn/
+qed-.
-(* Basic_1: was: sn3_gen_def *)
-lemma csx_inv_lref_bind: ∀h,g,I,G,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V →
- ⦃G, L⦄ ⊢ ⬊*[h, g] #i → ⦃G, K⦄ ⊢ ⬊*[h, g] V.
-#h #g #I #G #L #K #V #i #HLK #Hi
-elim (lift_total V 0 (i+1)) #V0 #HV0
-lapply (ldrop_fwd_ldrop2 … HLK) #H0LK
-@(csx_inv_lift … H0LK … HV0) -H0LK
-@(csx_cpx_trans … Hi) -Hi /2 width=7/
+lemma csx_fquq_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, g] T2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #H elim (fquq_inv_gen … H12) -H12
+[ /2 width=5 by csx_fqu_conf/
+| * #HG #HL #HT destruct //
+]
+qed-.
+
+lemma csx_fqup_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+ ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, g] T2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+/3 width=5 by csx_fqu_conf/
+qed-.
+
+lemma csx_fqus_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, g] T2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #H elim (fqus_inv_gen … H12) -H12
+[ /2 width=5 by csx_fqup_conf/
+| * #HG #HL #HT destruct //
+]
qed-.
(* Main properties **********************************************************)
-theorem csx_acp: ∀h,g. acp (cpx h g) (eq …) (csx h g).
-#h #g @mk_acp
-[ #G #L elim (deg_total h g 0)
- #l #Hl lapply (cnx_sort_iter … L … Hl) /2 width=2/
-| @cnx_lift
+theorem csx_gcp: ∀h,g. gcp (cpx h g) (eq …) (csx h g).
+#h #g @mk_gcp
+[ normalize /3 width=13 by cnx_lift/
+| #G #L elim (deg_total h g 0) /3 width=8 by cnx_sort_iter, ex_intro/
+| /2 width=8 by csx_lift/
| /2 width=3 by csx_fwd_flat_dx/
-| /2 width=1/
]
qed.
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