qed.
theorem cprs_theta_rc: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
- â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡ V â\86\92 â\87§[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
+ â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡ V â\86\92 â¬\86[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H @(cprs_ind … H) -W2
/3 width=5 by cprs_trans, cprs_theta_dx, cprs_bind_dx/
qed.
theorem cprs_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
- â\87§[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
+ â¬\86[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
⦃G, L⦄ ⊢ V1 ➡* V → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(cprs_ind_dx … H) -V1
/3 width=3 by cprs_trans, cprs_theta_rc, cprs_flat_dx/
U2 = ⓐV2. T2
| ∃∃a,W,T. ⦃G, L⦄ ⊢ T1 ➡* ⓛ{a}W.T &
⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ➡* U2
- | â\88\83â\88\83a,V0,V2,V,T. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡* 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¡* V0 & â¬\86[0,1] V0 ≡ V2 &
⦃G, L⦄ ⊢ T1 ➡* ⓓ{a}V.T &
⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡* U2.
#G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
| #a #V2 #W #W2 #T #T2 #HV02 #HW2 #HT2 #H1 #H2 destruct
lapply (cprs_strap1 … HV10 … HV02) -V0 #HV12
lapply (lsubr_cpr_trans … HT2 (L.ⓓⓝW.V1) ?) -HT2
- /5 width=5 by cprs_bind, cprs_flat_dx, cpr_cprs, lsubr_abst, ex2_3_intro, or3_intro1/
+ /5 width=5 by cprs_bind, cprs_flat_dx, cpr_cprs, lsubr_beta, ex2_3_intro, or3_intro1/
| #a #V #V2 #W0 #W2 #T #T2 #HV0 #HV2 #HW02 #HT2 #H1 #H2 destruct
/5 width=10 by cprs_flat_sn, cprs_bind_dx, cprs_strap1, ex4_5_intro, or3_intro2/
]