- | ∃∃T2. ❪G,L❫ ⊢ T1 ➡[h,0] T2 & (T1 ≛ T2 → ⊥).
-#h #G #L #T1
-@(fqup_wf_ind_eq (Ⓣ) … G L T1) -G -L -T1 #G0 #L0 #T0 #IH #G #L * *
-[ #s #HG #HL #HT destruct -IH
- /3 width=4 by cnr_sort, or_introl/
-| #i #HG #HL #HT destruct -IH
- elim (drops_F_uni L i)
- [ /3 width=6 by cnr_lref_atom, or_introl/
- | * * [ #I | * #V ] #K #HLK
- [ /3 width=7 by cnr_lref_unit, or_introl/
- | elim (lifts_total V 𝐔❨↑i❩) #W #HVW
- @or_intror @(ex2_intro … W) [ /2 width=6 by cpm_delta_drops/ ] #H
- lapply (teqx_inv_lref1 … H) -H #H destruct
- /2 width=5 by lifts_inv_lref2_uni_lt/
- | /3 width=7 by cnr_lref_abst, or_introl/
- ]
- ]
-| #l #HG #HL #HT destruct -IH
- /3 width=4 by cnr_gref, or_introl/
-| #p * [ cases p ] #V1 #T1 #HG #HL #HT destruct
- [ elim (cpr_subst h G (L.ⓓV1) T1 0 L V1) [| /2 width=1 by drops_refl/ ] #T2 #X2 #HT12 #HXT2 -IH
- elim (teqx_dec T1 T2) [ -HT12 #HT12 | #HnT12 ]
- [ elim (teqx_inv_lifts_dx … HT12 … HXT2) -T2 #X1 #HXT1 #_ -X2
- @or_intror @(ex2_intro … X1) [ /2 width=3 by cpm_zeta/ ] #H
- /2 width=7 by teqx_lifts_inv_pair_sn/
- | @or_intror @(ex2_intro … (+ⓓV1.T2)) [ /2 width=1 by cpm_bind/ ] #H
- elim (teqx_inv_pair … H) -H /2 width=1 by/
- ]
- | elim (IH G L V1) [ elim (IH G (L.ⓓV1) T1) [| * | // ] | * | // ] -IH
- [ #HT1 #HV1 /3 width=6 by cnr_abbr_neg, or_introl/
- | #T2 #HT12 #HnT12 #_
- @or_intror @(ex2_intro … (-ⓓV1.T2)) [ /2 width=1 by cpm_bind/ ] #H
- elim (teqx_inv_pair … H) -H /2 width=1 by/
- | #V2 #HV12 #HnV12
- @or_intror @(ex2_intro … (-ⓓV2.T1)) [ /2 width=1 by cpr_pair_sn/ ] #H
- elim (teqx_inv_pair … H) -H /2 width=1 by/
- ]
- | elim (IH G L V1) [ elim (IH G (L.ⓛV1) T1) [| * | // ] | * | // ] -IH
- [ #HT1 #HV1 /3 width=6 by cnr_abst, or_introl/
- | #T2 #HT12 #HnT12 #_
- @or_intror @(ex2_intro … (ⓛ[p]V1.T2)) [ /2 width=1 by cpm_bind/ ] #H
- elim (teqx_inv_pair … H) -H /2 width=1 by/
- | #V2 #HV12 #HnV12
- @or_intror @(ex2_intro … (ⓛ[p]V2.T1)) [ /2 width=1 by cpr_pair_sn/ ] #H
- elim (teqx_inv_pair … H) -H /2 width=1 by/
- ]
- ]
-| * #V1 #T1 #HG #HL #HT destruct [| -IH ]
- [ elim (IH G L V1) [ elim (IH G L T1) [| * | // ] | * | // ] -IH
- [ #HT1 #HV1
- elim (simple_dec_ex T1) [| * #p * #W1 #U1 #H destruct ]
- [ /3 width=6 by cnr_appl_simple, or_introl/
- | elim (lifts_total V1 𝐔❨1❩) #X1 #HVX1
- @or_intror @(ex2_intro … (ⓓ[p]W1.ⓐX1.U1)) [ /2 width=3 by cpm_theta/ ] #H
- elim (teqx_inv_pair … H) -H #H destruct
- | @or_intror @(ex2_intro … (ⓓ[p]ⓝW1.V1.U1)) [ /2 width=1 by cpm_beta/ ] #H
- elim (teqx_inv_pair … H) -H #H destruct
- ]
- | #T2 #HT12 #HnT12 #_
- @or_intror @(ex2_intro … (ⓐV1.T2)) [ /2 width=1 by cpm_appl/ ] #H
- elim (teqx_inv_pair … H) -H /2 width=1 by/
- | #V2 #HV12 #HnV12
- @or_intror @(ex2_intro … (ⓐV2.T1)) [ /2 width=1 by cpr_pair_sn/ ] #H
- elim (teqx_inv_pair … H) -H /2 width=1 by/
- ]
- | @or_intror @(ex2_intro … T1) [ /2 width=1 by cpm_eps/ ] #H
- /2 width=4 by teqx_inv_pair_xy_y/
- ]
-]
-qed-.