1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/computation/acp_cr.ma".
16 include "basic_2/computation/cpxs_tstc_vector.ma".
17 include "basic_2/computation/csx_lpx.ma".
18 include "basic_2/computation/csx_vector.ma".
20 (* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERM VECTORS *************)
22 (* Advanced properties ******************************************************)
24 (* Basic_1: was just: sn3_appls_lref *)
25 lemma csx_applv_cnx: ∀h,g,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ →
26 ∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] Vs → ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.T.
27 #h #g #G #L #T #H1T #H2T #Vs elim Vs -Vs [ #_ @(cnx_csx … H2T) ] (**) (* /2 width=1/ does not work *)
29 elim (csxv_inv_cons … H) -H #HV #HVs
30 @csx_appl_simple_tstc // -HV /2 width=1/ -IHV -HVs
32 lapply (cpxs_fwd_cnx_vector … H) -H // -H1T -H2T #H
36 lemma csx_applv_sort: ∀h,g,G,L,k,Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] Vs → ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.⋆k.
37 #h #g #G #L #k elim (deg_total h g k)
38 #l generalize in match k; -k @(nat_ind_plus … l) -l [ /3 width=1/ ]
39 #l #IHl #k #Hkl lapply (deg_next_SO … Hkl) -Hkl
40 #Hkl #Vs elim Vs -Vs /2 width=1/
42 elim (csxv_inv_cons … HVVs) #HV #HVs
43 @csx_appl_simple_tstc // -HV /2 width=1/ -IHVs -HVs
45 elim (cpxs_fwd_sort_vector … H) -H #H
47 | -H0 @(csx_cpxs_trans … (Ⓐ(V@Vs).⋆(next h k))) /2 width=1/
51 (* Basic_1: was just: sn3_appls_beta *)
52 lemma csx_applv_beta: ∀h,g,a,G,L,Vs,V,W,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.ⓓ{a}ⓝW.V.T →
53 ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs. ⓐV.ⓛ{a}W.T.
54 #h #g #a #G #L #Vs elim Vs -Vs /2 width=1/
55 #V0 #Vs #IHV #V #W #T #H1T
56 lapply (csx_fwd_pair_sn … H1T) #HV0
57 lapply (csx_fwd_flat_dx … H1T) #H2T
58 @csx_appl_simple_tstc // -HV0 /2 width=1/ -IHV -H2T
60 elim (cpxs_fwd_beta_vector … H) -H #H
62 | -H0 @(csx_cpxs_trans … H1T) -H1T /2 width=1/
66 lemma csx_applv_delta: ∀h,g,I,G,L,K,V1,i. ⇩[0, i] L ≡ K.ⓑ{I}V1 →
67 ∀V2. ⇧[0, i + 1] V1 ≡ V2 →
68 ∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] (ⒶVs.V2) → ⦃G, L⦄ ⊢ ⬊*[h, g] (ⒶVs.#i).
69 #h #g #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
71 lapply (ldrop_fwd_ldrop2 … HLK) #HLK0
72 lapply (csx_inv_lift … H … HLK0 HV12) -V2 -HLK0 /2 width=5/
74 lapply (csx_fwd_pair_sn … H1T) #HV
75 lapply (csx_fwd_flat_dx … H1T) #H2T
76 @csx_appl_simple_tstc // -HV /2 width=1/ -IHV -H2T
78 elim (cpxs_fwd_delta_vector … HLK … HV12 … H) -HLK -HV12 -H #H
80 | -H0 @(csx_cpxs_trans … H1T) -H1T /2 width=1/
85 (* Basic_1: was just: sn3_appls_abbr *)
86 lemma csx_applv_theta: ∀h,g,a,G,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
87 ∀V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V.ⒶV2s.T →
88 ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶV1s.ⓓ{a}V.T.
89 #h #g #a #G #L #V1s #V2s * -V1s -V2s /2 width=1/
90 #V1s #V2s #V1 #V2 #HV12 #H
91 generalize in match HV12; -HV12 generalize in match V2; -V2 generalize in match V1; -V1
92 elim H -V1s -V2s /2 width=3/
93 #V1s #V2s #V1 #V2 #HV12 #HV12s #IHV12s #W1 #W2 #HW12 #V #T #H
94 lapply (csx_appl_theta … HW12 … H) -H -HW12 #H
95 lapply (csx_fwd_pair_sn … H) #HW1
96 lapply (csx_fwd_flat_dx … H) #H1
97 @csx_appl_simple_tstc // -HW1 /2 width=3/ -IHV12s -H1 #X #H1 #H2
98 elim (cpxs_fwd_theta_vector … (V2@V2s) … H1) -H1 /2 width=1/ -HV12s -HV12
99 [ -H #H elim H2 -H2 //
100 | -H2 #H1 @(csx_cpxs_trans … H) -H /2 width=1/
104 (* Basic_1: was just: sn3_appls_cast *)
105 lemma csx_applv_cast: ∀h,g,G,L,Vs,W,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.W → ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.T →
106 ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.ⓝW.T.
107 #h #g #G #L #Vs elim Vs -Vs /2 width=1/
108 #V #Vs #IHV #W #T #H1W #H1T
109 lapply (csx_fwd_pair_sn … H1W) #HV
110 lapply (csx_fwd_flat_dx … H1W) #H2W
111 lapply (csx_fwd_flat_dx … H1T) #H2T
112 @csx_appl_simple_tstc // -HV /2 width=1/ -IHV -H2W -H2T
114 elim (cpxs_fwd_cast_vector … H) -H #H
115 [ -H1W -H1T elim H0 -H0 //
116 | -H1W -H0 @(csx_cpxs_trans … H1T) -H1T /2 width=1/
117 | -H1T -H0 @(csx_cpxs_trans … H1W) -H1W /2 width=1/
121 theorem csx_acr: ∀h,g. acr (cpx h g) (eq …) (csx h g) (λG,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T).
126 | #G #L #V1s #V2s #HV12s #a #V #T #H #HV
127 @(csx_applv_theta … HV12s) -HV12s