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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 include "basic_2/rt_computation/cpxs_toeq_vector.ma".
16 include "basic_2/rt_computation/csx_simple_toeq.ma".
17 include "basic_2/rt_computation/csx_lsubr.ma".
18 include "basic_2/rt_computation/csx_lpx.ma".
19 include "basic_2/rt_computation/csx_vector.ma".
21 (* STRONGLY NORMALIZING TERM VECTORS FOR UNBOUND PARALLEL RT-TRANSITION *****)
23 (* Advanced properties ************************************* ****************)
25 (* Basic_1: was just: sn3_appls_beta *)
26 lemma csx_applv_beta (h) (G):
27 ∀p,L,Vs,V,W,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓓ{p}ⓝW.V.T⦄ →
28 ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓐV.ⓛ{p}W.T⦄.
29 #h #G #p #L #Vs elim Vs -Vs /2 width=1 by csx_appl_beta/
30 #V0 #Vs #IHV #V #W #T #H1T
31 lapply (csx_fwd_pair_sn … H1T) #HV0
32 lapply (csx_fwd_flat_dx … H1T) #H2T
33 @csx_appl_simple_toeq /2 width=1 by applv_simple, simple_flat/ -IHV -HV0 -H2T
35 elim (cpxs_fwd_beta_vector … H) -H #H
37 | -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
41 lemma csx_applv_delta_drops (h) (G):
42 ∀I,L,K,V1,i. ⬇*[i] L ≘ K.ⓑ{I}V1 →
44 ∀Vs. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.V2⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.#i⦄.
45 #h #G #I #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
46 [ /4 width=11 by csx_inv_lifts, csx_lref_pair_drops, drops_isuni_fwd_drop2/
48 lapply (csx_fwd_pair_sn … H1T) #HV
49 lapply (csx_fwd_flat_dx … H1T) #H2T
50 @csx_appl_simple_toeq /2 width=1 by applv_simple, simple_atom/ -IHV -HV -H2T
52 elim (cpxs_fwd_delta_drops_vector … HLK … HV12 … H) -HLK -HV12 -H #H
54 | -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
59 (* Basic_1: was just: sn3_appls_abbr *)
60 lemma csx_applv_theta (h) (G):
61 ∀p,L,V1b,V2b. ⬆*[1] V1b ≘ V2b →
62 ∀V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.ⒶV2b.T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶV1b.ⓓ{p}V.T⦄.
63 #h #G #p #L #V1b #V2b * -V1b -V2b /2 width=1 by/
64 #V1b #V2b #V1 #V2 #HV12 #H
65 generalize in match HV12; -HV12 generalize in match V2; -V2 generalize in match V1; -V1
66 elim H -V1b -V2b /2 width=3 by csx_appl_theta/
67 #V1b #V2b #V1 #V2 #HV12 #HV12b #IHV12b #W1 #W2 #HW12 #V #T #H
68 lapply (csx_appl_theta … H … HW12) -H -HW12 #H
69 lapply (csx_fwd_pair_sn … H) #HW1
70 lapply (csx_fwd_flat_dx … H) #H1
71 @csx_appl_simple_toeq /2 width=3 by simple_flat/ -IHV12b -HW1 -H1 #X #H1 #H2
72 elim (cpxs_fwd_theta_vector … (V2⨮V2b) … H1) -H1 /2 width=1 by liftsv_cons/ -HV12b -HV12
73 [ -H #H elim H2 -H2 //
74 | -H2 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
78 (* Basic_1: was just: sn3_appls_cast *)
79 lemma csx_applv_cast (h) (G):
80 ∀L,Vs,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.U⦄ →
81 ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓝU.T⦄.
82 #h #G #L #Vs elim Vs -Vs /2 width=1 by csx_cast/
83 #V #Vs #IHV #U #H1U #T #H1T
84 lapply (csx_fwd_pair_sn … H1U) #HV
85 lapply (csx_fwd_flat_dx … H1U) #H2U
86 lapply (csx_fwd_flat_dx … H1T) #H2T
87 @csx_appl_simple_toeq /2 width=1 by applv_simple, simple_flat/ -IHV -HV -H2U -H2T
89 elim (cpxs_fwd_cast_vector … H) -H #H
90 [ -H1U -H1T elim H0 -H0 //
91 | -H1U -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
92 | -H1T -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/