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