-lemma cpx_teqx_conf_rex: ∀h,G. R_confluent2_rex … (cpx h G) cdeq (cpx h G) cdeq.
-#h #G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1 /2 width=3 by ex2_intro/
-[ #G #L0 #s0 #X0 #H0 #L1 #HL01 #L2 #HL02
- elim (teqx_inv_sort1 … H0) -H0 #s1 #H destruct
- /3 width=3 by teqx_sort, ex2_intro/
-| #I #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
- >(teqx_inv_lref1 … H0) -H0
- elim (rpx_inv_zero_pair_sn … H1) -H1 #K1 #X1 #HK01 #HX1 #H destruct
- elim (reqx_inv_zero_pair_sn … H2) -H2 #K2 #X2 #HK02 #HX2 #H destruct
- elim (IH X2 … HK01 … HK02) // -K0 -V0 #V #HV1 #HV2
- elim (teqx_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_delta, ex2_intro/
-| #I0 #G #K0 #V1 #W1 #i #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
- >(teqx_inv_lref1 … H0) -H0
- elim (rpx_inv_lref_bind_sn … H1) -H1 #I1 #K1 #HK01 #H destruct
- elim (reqx_inv_lref_bind_sn … H2) -H2 #I2 #K2 #HK02 #H destruct
- elim (IH … HK01 … HK02) [|*: //] -K0 #V #HV1 #HV2
- elim (teqx_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_lref, ex2_intro/
-| #p #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
- elim (teqx_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
- elim (rpx_inv_bind … H1) -H1 #HL01 #H1
- elim (reqx_inv_bind … H2) -H2 #HL02 #H2
- lapply (reqx_bind_repl_dx … H2 (BPair I V2) ?) -H2 /2 width=1 by ext2_pair/ #H2
- elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
- elim (IHT … HT02 … H1 … H2) -L0 -T0
- /3 width=5 by cpx_bind, teqx_pair, ex2_intro/
-| #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
- elim (teqx_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
- elim (rpx_inv_flat … H1) -H1 #HL01 #H1
- elim (reqx_inv_flat … H2) -H2 #HL02 #H2
- elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
- elim (IHT … HT02 … H1 … H2) -L0 -V0 -T0
- /3 width=5 by cpx_flat, teqx_pair, ex2_intro/
-| #G #L0 #V0 #U0 #T0 #T1 #HTU0 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
- elim (teqx_inv_pair1 … H0) -H0 #V2 #U2 #HV02 #HU02 #H destruct
- elim (rpx_inv_bind … H1) -H1 #HL01 #H1
- elim (reqx_inv_bind … H2) -H2 #HL02 #H2
- lapply (rpx_inv_lifts_bi … H1 (Ⓣ) … HTU0) -H1 [6:|*: /3 width=2 by drops_refl, drops_drop/ ] #H1
- lapply (reqx_inv_lifts_bi … H2 (Ⓣ) … HTU0) -H2 [6:|*: /3 width=2 by drops_refl, drops_drop/ ] #H2
+lemma cpx_teqx_repl_reqx (G) (L0) (T0):
+ ∀T1. ❪G,L0❫ ⊢ T0 ⬈ T1 → ∀T2. T0 ≛ T2 → ∀T3. T1 ≛ T3 →
+ ∀L2. L0 ≛[T0] L2 → ❪G,L2❫ ⊢ T2 ⬈ T3.
+#G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1
+[ * #x0 #G #L0 #X2 #HX2 #X3 #HX3 #L2 #_
+ [ elim (teqx_inv_sort1 … HX2) -HX2 #x2 #H destruct
+ elim (teqx_inv_sort1 … HX3) -HX3 #x3 #H destruct //
+ | lapply (teqx_inv_lref1 … HX2) -HX2 #H destruct
+ lapply (teqx_inv_lref1 … HX3) -HX3 #H destruct //
+ | lapply (teqx_inv_gref1 … HX2) -HX2 #H destruct
+ lapply (teqx_inv_gref1 … HX3) -HX3 #H destruct //
+ ]
+| #G #L0 #s0 #s1 #X2 #HX2 #X3 #HX3 #L2 #HL02
+ elim (teqx_inv_sort1 … HX2) -HX2 #s2 #H destruct
+ elim (teqx_inv_sort1 … HX3) -HX3 #s3 #H destruct //
+| #I #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #X2 #HX2 #X3 #HX3 #L2 #HL2
+ lapply (teqx_inv_lref1 … HX2) -HX2 #H destruct
+ elim (reqx_inv_zero_pair_sn … HL2) -HL2 #K2 #V2 #HK02 #HV02 #H destruct
+ elim (teqx_inv_lifts_sn … HX3 … HVW1) -W1 #V3 #HVX3 #HV13
+ /3 width=3 by cpx_delta/
+| #I0 #G #K0 #V1 #W1 #i #_ #IH #HVW1 #X2 #HX2 #X3 #HX3 #L2 #HL2
+ lapply (teqx_inv_lref1 … HX2) -HX2 #H destruct
+ elim (reqx_inv_lref_bind_sn … HL2) -HL2 #I2 #K2 #HK02 #H destruct
+ elim (teqx_inv_lifts_sn … HX3 … HVW1) -W1 #V3 #HVX3 #HV13
+ /3 width=3 by cpx_lref/
+| #p #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X2 #HX2 #X3 #HX3 #L2 #HL02
+ elim (teqx_inv_pair1 … HX2) -HX2 #V2 #T2 #HV02 #HT02 #H destruct
+ elim (teqx_inv_pair1 … HX3) -HX3 #V3 #T3 #HV13 #HT13 #H destruct
+ elim (reqx_inv_bind … HL02) -HL02 #HV0 #HT0
+ lapply (reqx_bind_repl_dx … HT0 (BPair I V2) ?) -HT0
+ /2 width=1 by ext2_pair/ #HT0
+ /3 width=1 by cpx_bind/
+| #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X2 #HX2 #X3 #HX3 #L2 #HL02
+ elim (teqx_inv_pair1 … HX2) -HX2 #V2 #T2 #HV02 #HT02 #H destruct
+ elim (teqx_inv_pair1 … HX3) -HX3 #V3 #T3 #HV13 #HT13 #H destruct
+ elim (reqx_inv_flat … HL02) -HL02 #HV0 #HT0
+ /3 width=1 by cpx_flat/
+| #G #L0 #V0 #U0 #T0 #T1 #HTU0 #_ #IH #X2 #HX2 #X3 #HX3 #L2 #HL02
+ elim (teqx_inv_pair1 … HX2) -HX2 #V2 #U2 #HV02 #HU02 #H destruct
+ elim (reqx_inv_bind … HL02) -HL02 #HV0 #HU0
+ lapply (reqx_inv_lifts_bi … HU0 (Ⓣ) … HTU0) -HU0
+ [6:|*: /3 width=2 by drops_refl, drops_drop/ ] #HT0