(* Advanced properties ******************************************************)
-lemma csn_lpx_conf: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡[g] L2 →
- ∀T. ⦃h, L1⦄ ⊢ ⬊*[g] T → ⦃h, L2⦄ ⊢ ⬊*[g] T.
+lemma csn_lpx_conf: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡[h, g] L2 →
+ ∀T. ⦃h, L1⦄ ⊢ ⬊*[h, g] T → ⦃h, L2⦄ ⊢ ⬊*[h, g] T.
#h #g #L1 #L2 #HL12 #T #H @(csn_ind_alt … H) -T #T #_ #IHT
@csn_intro #T0 #HLT0 #HT0
@IHT /2 width=2/ -IHT -HT0 /2 width=3 by lpx_cpx_trans/
qed.
-lemma csn_abst: ∀h,g,a,L,W. ⦃h, L⦄ ⊢ ⬊*[g] W →
- ∀T. ⦃h, L.ⓛW⦄ ⊢ ⬊*[g] T → ⦃h, L⦄ ⊢ ⬊*[g] ⓛ{a}W.T.
+lemma csn_abst: ∀h,g,a,L,W. ⦃G, L⦄ ⊢ ⬊*[h, g] W →
+ ∀T. ⦃h, L.ⓛW⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓛ{a}W.T.
#h #g #a #L #W #HW @(csn_ind … HW) -W #W #_ #IHW #T #HT @(csn_ind … HT) -T #T #HT #IHT
@csn_intro #X #H1 #H2
elim (cpx_inv_abst1 … H1) -H1
]
qed.
-lemma csn_abbr: ∀h,g,a,L,V. ⦃h, L⦄ ⊢ ⬊*[g] V →
- ∀T. ⦃h, L.ⓓV⦄ ⊢ ⬊*[g] T → ⦃h, L⦄ ⊢ ⬊*[g] ⓓ{a}V. T.
+lemma csn_abbr: ∀h,g,a,L,V. ⦃G, L⦄ ⊢ ⬊*[h, g] V →
+ ∀T. ⦃h, L.ⓓV⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V. T.
#h #g #a #L #V #HV elim HV -V #V #_ #IHV #T #HT @(csn_ind_alt … HT) -T #T #HT #IHT
@csn_intro #X #H1 #H2
elim (cpx_inv_abbr1 … H1) -H1 *
]
qed.
-fact csn_appl_beta_aux: ∀h,g,a,L,U1. ⦃h, L⦄ ⊢ ⬊*[g] U1 →
- ∀V,W,T1. U1 = ⓓ{a}ⓝW.V.T1 → ⦃h, L⦄ ⊢ ⬊*[g] ⓐV.ⓛ{a}W.T1.
+fact csn_appl_beta_aux: ∀h,g,a,L,U1. ⦃G, L⦄ ⊢ ⬊*[h, g] U1 →
+ ∀V,W,T1. U1 = ⓓ{a}ⓝW.V.T1 → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.ⓛ{a}W.T1.
#h #g #a #L #X #H @(csn_ind … H) -X
#X #HT1 #IHT1 #V #W #T1 #H1 destruct
@csn_intro #X #H1 #H2
qed-.
(* Basic_1: was just: sn3_beta *)
-lemma csn_appl_beta: ∀h,g,a,L,V,W,T. ⦃h, L⦄ ⊢ ⬊*[g] ⓓ{a}ⓝW.V.T → ⦃h, L⦄ ⊢ ⬊*[g] ⓐV.ⓛ{a}W.T.
+lemma csn_appl_beta: ∀h,g,a,L,V,W,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}ⓝW.V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.ⓛ{a}W.T.
/2 width=3 by csn_appl_beta_aux/ qed.
-fact csn_appl_theta_aux: ∀h,g,a,L,U. ⦃h, L⦄ ⊢ ⬊*[g] U → ∀V1,V2. ⇧[0, 1] V1 ≡ V2 →
- ∀V,T. U = ⓓ{a}V.ⓐV2.T → ⦃h, L⦄ ⊢ ⬊*[g] ⓐV1.ⓓ{a}V.T.
+fact csn_appl_theta_aux: ∀h,g,a,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U → ∀V1,V2. ⇧[0, 1] V1 ≡ V2 →
+ ∀V,T. U = ⓓ{a}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV1.ⓓ{a}V.T.
#h #g #a #L #X #H @(csn_ind_alt … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct
lapply (csn_fwd_pair_sn … HVT) #HV
lapply (csn_fwd_bind_dx … HVT) -HVT #HVT
qed-.
lemma csn_appl_theta: ∀h,g,a,V1,V2. ⇧[0, 1] V1 ≡ V2 →
- ∀L,V,T. ⦃h, L⦄ ⊢ ⬊*[g] ⓓ{a}V.ⓐV2.T → ⦃h, L⦄ ⊢ ⬊*[g] ⓐV1.ⓓ{a}V.T.
+ ∀L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV1.ⓓ{a}V.T.
/2 width=5 by csn_appl_theta_aux/ qed.
(* Basic_1: was just: sn3_appl_appl *)
-lemma csn_appl_simple_tstc: ∀h,g,L,V. ⦃h, L⦄ ⊢ ⬊*[g] V → ∀T1. ⦃h, L⦄ ⊢ ⬊*[g] T1 →
- (∀T2. ⦃h, L⦄ ⊢ T1 ➡*[g] T2 → (T1 ≃ T2 → ⊥) → ⦃h, L⦄ ⊢ ⬊*[g] ⓐV.T2) →
- 𝐒⦃T1⦄ → ⦃h, L⦄ ⊢ ⬊*[g] ⓐV.T1.
+lemma csn_appl_simple_tstc: ∀h,g,L,V. ⦃G, L⦄ ⊢ ⬊*[h, g] V → ∀T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
+ (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 ≃ T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T2) →
+ 𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T1.
#h #g #L #V #H @(csn_ind … H) -V #V #_ #IHV #T1 #H @(csn_ind … H) -T1 #T1 #H1T1 #IHT1 #H2T1 #H3T1
@csn_intro #X #HL #H
elim (cpx_inv_appl1_simple … HL) -HL //