qed.
theorem cpxs_theta_rc: ∀h,g,a,G,L,V1,V,V2,W1,W2,T1,T2.
- â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, g] V â\86\92 â\87§[0, 1] V ≡ V2 →
+ â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, g] V â\86\92 â¬\86[0, 1] V ≡ V2 →
⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 →
⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, g] ⓓ{a}W2.ⓐV2.T2.
#h #g #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H @(cpxs_ind … H) -W2
qed.
theorem cpxs_theta: ∀h,g,a,G,L,V1,V,V2,W1,W2,T1,T2.
- â\87§[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 →
+ â¬\86[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 →
⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ V1 ➡*[h, g] V →
⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, g] ⓓ{a}W2.ⓐV2.T2.
#h #g #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1
∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 &
U2 = ⓐV2. T2
| ∃∃a,W,T. ⦃G, L⦄ ⊢ T1 ➡*[h, g] ⓛ{a}W.T & ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ➡*[h, g] U2
- | â\88\83â\88\83a,V0,V2,V,T. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡*[h, g] V0 & â\87§[0,1] V0 ≡ V2 &
+ | â\88\83â\88\83a,V0,V2,V,T. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡*[h, g] V0 & â¬\86[0,1] V0 ≡ V2 &
⦃G, L⦄ ⊢ T1 ➡*[h, g] ⓓ{a}V.T & ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡*[h, g] U2.
#h #g #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 [ /3 width=5 by or3_intro0, ex3_2_intro/ ]
#U #U2 #_ #HU2 * *
[ #I #G #L #V1 #V2 #HV12 #_ elim (lift_total V2 0 1)
#U2 #HVU2 @(ex3_intro … U2)
[1,3: /3 width=7 by fqu_drop, cpxs_delta, drop_pair, drop_drop/
- | #H destruct /2 width=7 by lift_inv_lref2_be/
+ | #H destruct
+ lapply (lift_inv_lref2_be … HVU2 ? ?) -HVU2 //
]
| #I #G #L #V1 #T #V2 #HV12 #H @(ex3_intro … (②{I}V2.T))
[1,3: /2 width=4 by fqu_pair_sn, cpxs_pair_sn/
[1,3: /2 width=4 by fqu_flat_dx, cpxs_flat/
| #H0 destruct /2 width=1 by/
]
-| #G #L #K #T1 #U1 #e #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (e+1))
+| #G #L #K #T1 #U1 #m #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (m+1))
#U2 #HTU2 @(ex3_intro … U2)
[1,3: /2 width=10 by cpxs_lift, fqu_drop/
| #H0 destruct /3 width=5 by lift_inj/