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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "basic_2/rt_computation/cpxs_lpx.ma".
16 include "basic_2/rt_computation/csx_cpxs.ma".
18 (* STRONGLY NORMALIZING TERMS FOR EXTENDED PARALLEL RT-TRANSITION ***********)
20 (* Properties with extended parallel rt-transition on all entries ***********)
22 lemma csx_lpx_conf (G) (L1):
24 ∀L2. ❨G,L1❩ ⊢ ⬈ L2 → ❨G,L2❩ ⊢ ⬈*𝐒 T.
25 #G #L1 #T #H @(csx_ind_cpxs … H) -T
26 /4 width=3 by csx_intro, lpx_cpx_trans/
29 (* Advanced properties ******************************************************)
31 lemma csx_abst (G) (L):
33 ∀T. ❨G,L.ⓛW❩ ⊢ ⬈*𝐒 T → ❨G,L❩ ⊢ ⬈*𝐒 ⓛ[p]W.T.
35 @(csx_ind … HW) -W #W #_ #IHW #T #HT
36 @(csx_ind … HT) -T #T #HT #IHT
38 elim (cpx_inv_abst1 … H1) -H1 #W0 #T0 #HLW0 #HLT0 #H destruct
39 elim (tneqx_inv_pair … H2) -H2
41 | -IHT #H lapply (csx_cpx_trans … HLT0) // -HT #HT0
42 /4 width=5 by csx_lpx_conf, lpx_pair/
43 | -IHW -HT /4 width=3 by csx_cpx_trans, cpx_pair_sn/
47 lemma csx_abbr (G) (L):
49 ∀T. ❨G,L.ⓓV❩ ⊢ ⬈*𝐒 T → ❨G,L❩ ⊢ ⬈*𝐒 ⓓ[p]V.T.
51 @(csx_ind … HV) -V #V #_ #IHV #T #HT
52 @(csx_ind_cpxs … HT) -T #T #HT #IHT
54 elim (cpx_inv_abbr1 … H1) -H1 *
55 [ #V1 #T1 #HLV1 #HLT1 #H destruct
56 elim (tneqx_inv_pair … H2) -H2
58 | /4 width=5 by csx_cpx_trans, csx_lpx_conf, lpx_pair/
59 | -IHV /4 width=3 by csx_cpx_trans, cpx_cpxs, cpx_pair_sn/
62 /5 width=7 by csx_cpx_trans, csx_inv_lifts, drops_drop, drops_refl/
66 lemma csx_bind (G) (L):
67 ∀p,I,V. ❨G,L❩ ⊢ ⬈*𝐒 V →
68 ∀T. ❨G,L.ⓑ[I]V❩ ⊢ ⬈*𝐒 T → ❨G,L❩ ⊢ ⬈*𝐒 ⓑ[p,I]V.T.
69 #G #L #p * #V #HV #T #HT
70 /2 width=1 by csx_abbr, csx_abst/
73 fact csx_appl_theta_aux (G) (L):
74 ∀p,U. ❨G,L❩ ⊢ ⬈*𝐒 U → ∀V1,V2. ⇧[1] V1 ≘ V2 →
75 ∀V,T. U = ⓓ[p]V.ⓐV2.T → ❨G,L❩ ⊢ ⬈*𝐒 ⓐV1.ⓓ[p]V.T.
77 @(csx_ind_cpxs … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct
78 lapply (csx_fwd_pair_sn … HVT) #HV
79 lapply (csx_fwd_bind_dx … HVT) -HVT #HVT
81 elim (cpx_inv_appl1 … HL) -HL *
82 [ -HV #V0 #Y #HLV10 #HL #H0 destruct
83 elim (cpx_inv_abbr1 … HL) -HL *
84 [ #V3 #T3 #HV3 #HLT3 #H0 destruct
85 elim (cpx_lifts_sn … HLV10 (Ⓣ) … (L.ⓓV) … HV12) -HLV10 /3 width=1 by drops_refl, drops_drop/ #V4 #HV04 #HV24
86 elim (teqx_dec (ⓓ[p]V.ⓐV2.T) (ⓓ[p]V3.ⓐV4.T3)) #H0
87 [ -IHVT -HV3 -HV24 -HLT3
88 elim (teqx_inv_pair … H0) -H0 #_ #HV3 #H0
89 elim (teqx_inv_pair … H0) -H0 #_ #HV24 #HT3
90 elim (tneqx_inv_pair … H) -H #H elim H -H -G -L
91 /3 width=6 by teqx_inv_lifts_bi, teqx_pair/
92 | -V1 @(IHVT … H0 … HV04) -V0 /4 width=1 by cpx_cpxs, cpx_flat, cpx_bind/
94 | #T0 #HT0 #HLT0 #H0 destruct -H -IHVT
95 lapply (csx_inv_lifts … HVT (Ⓣ) … L ???) -HVT
96 [5:|*: /3 width=4 by drops_refl, drops_drop, lifts_flat/ ] -V2 -T #HVT
97 /3 width=5 by csx_cpx_trans, cpx_flat/
99 | -HV -HV12 -HVT -IHVT -H #b #V0 #W0 #W1 #T0 #T1 #_ #_ #_ #H destruct
100 | -IHVT -H #b #V0 #V3 #W0 #W1 #T0 #T1 #HLV10 #HV03 #HLW01 #HLT01 #H1 #H2 destruct
101 lapply (cpx_lifts_bi … HLV10 (Ⓣ) … (L.ⓓW0) … HV12 … HV03) -HLV10 -HV12 -HV03 /3 width=1 by drops_refl, drops_drop/ #HLV23
102 @csx_abbr /2 width=3 by csx_cpx_trans/ -HV
103 @(csx_lpx_conf … (L.ⓓW0)) /2 width=1 by lpx_pair/ -W1
104 /4 width=5 by csx_cpxs_trans, cpx_cpxs, cpx_flat/
108 lemma csx_appl_theta (G) (L):
109 ∀p,V,V2,T. ❨G,L❩ ⊢ ⬈*𝐒 ⓓ[p]V.ⓐV2.T →
110 ∀V1. ⇧[1] V1 ≘ V2 → ❨G,L❩ ⊢ ⬈*𝐒 ⓐV1.ⓓ[p]V.T.
111 /2 width=5 by csx_appl_theta_aux/ qed.