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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_lfpx.ma".
16 include "basic_2/rt_computation/csx_drops.ma".
17 include "basic_2/rt_computation/csx_cpxs.ma".
19 (* STRONGLY NORMALIZING TERMS FOR UNCOUNTED PARALLEL RT-TRANSITION **********)
21 (* Advanced properties ******************************************************)
23 (* Basic_2A1: was just: csx_lpx_conf *)
24 lemma csx_lfpx_conf: ∀h,o,G,L1,T. ⦃G, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ →
25 ∀L2. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → ⦃G, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
26 #h #o #G #L1 #T #H @(csx_ind_cpxs … H) -T
27 /5 width=3 by csx_intro, lfpx_cpx_trans, lfpx_cpxs_conf/
30 lemma csx_abst: ∀h,o,p,G,L,W. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃W⦄ →
31 ∀T. ⦃G, L.ⓛW⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓛ{p}W.T⦄.
32 #h #o #p #G #L #W #HW @(csx_ind … HW) -W
33 #W #_ #IHW #T #HT @(csx_ind … HT) -T #T #HT #IHT
35 elim (cpx_inv_abst1 … H1) -H1
36 #W0 #T0 #HLW0 #HLT0 #H destruct
37 elim (tdneq_inv_pair … H2) -H2
39 | -IHT #H lapply (csx_cpx_trans … o … HLT0) // -HT
40 #HT0 lapply (csx_lfpx_conf … HT0 … (L.ⓛW0)) -HT0 /4 width=1 by lfpx_pair/
41 | -IHW -HT /4 width=3 by csx_cpx_trans, cpx_pair_sn/
45 lemma csx_abbr: ∀h,o,p,G,L,V. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ →
46 ∀T. ⦃G, L.ⓓV⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓓ{p}V.T⦄.
47 #h #o #p #G #L #V #HV @(csx_ind … HV) -V
48 #V #_ #IHV #T #HT @(csx_ind_cpxs … HT) -T #T #HT #IHT
50 elim (cpx_inv_abbr1 … H1) -H1 *
51 [ #V1 #T1 #HLV1 #HLT1 #H destruct
52 elim (tdneq_inv_pair … H2) -H2
54 | /4 width=3 by csx_cpx_trans, csx_lfpx_conf, lfpx_pair/
55 | -IHV /4 width=3 by csx_cpx_trans, cpx_cpxs, cpx_pair_sn/
58 /4 width=7 by csx_cpx_trans, csx_inv_lifts, drops_drop, drops_refl/
62 fact csx_appl_theta_aux: ∀h,o,p,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃U⦄ → ∀V1,V2. ⬆*[1] V1 ≡ V2 →
63 ∀V,T. U = ⓓ{p}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄.
64 #h #o #p #G #L #X #H @(csx_ind_cpxs … H) -X
65 #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct
66 lapply (csx_fwd_pair_sn … HVT) #HV
67 lapply (csx_fwd_bind_dx … HVT) -HVT #HVT
69 elim (cpx_inv_appl1 … HL) -HL *
70 [ -HV #V0 #Y #HLV10 #HL #H0 destruct
71 elim (cpx_inv_abbr1 … HL) -HL *
72 [ -HVT #V3 #T3 #HV3 #HLT3 #H0 destruct
73 elim (cpx_lifts_sn … HLV10 (Ⓣ) … (L.ⓓV) … HV12) -HLV10 /3 width=1 by drops_refl, drops_drop/ #V4 #HV04 #HV24
74 elim (tdeq_dec h o (ⓓ{p}V.ⓐV2.T) (ⓓ{p}V3.ⓐV4.T3)) #H0
75 [ -IHVT -HV3 -HV24 -HLT3
76 elim (tdeq_inv_pair … H0) -H0 #_ #HV3 #H0
77 elim (tdeq_inv_pair … H0) -H0 #_ #HV24 #HT3
78 elim (tdneq_inv_pair … H) -H #H elim H -H -G -L
79 /3 width=6 by tdeq_inv_lifts_bi, tdeq_pair/
80 | -V1 @(IHVT … H0 … HV04) -o -V0 /4 width=1 by cpx_cpxs, cpx_flat, cpx_bind/
82 | -H -IHVT #T0 #HLT0 #HT0 #H0 destruct
83 lapply (csx_cpx_trans … HVT (ⓐV2.T0) ?) /2 width=1 by cpx_flat/ -T #HVT0
84 lapply (csx_inv_lifts … HVT0 (Ⓣ) … L ???) -HVT0
85 /3 width=5 by csx_cpx_trans, cpx_pair_sn, drops_refl, drops_drop, lifts_flat/
87 | -HV -HV12 -HVT -IHVT -H #b #V0 #W0 #W1 #T0 #T1 #_ #_ #_ #H destruct
88 | -IHVT -H #b #V0 #V3 #W0 #W1 #T0 #T1 #HLV10 #HV03 #HLW01 #HLT01 #H1 #H2 destruct
89 lapply (cpx_lifts_bi … HLV10 (Ⓣ) … (L.ⓓW0) … HV12 … HV03) -HLV10 -HV12 -HV03 /3 width=1 by drops_refl, drops_drop/ #HLV23
90 @csx_abbr /2 width=3 by csx_cpx_trans/ -HV
91 @(csx_lfpx_conf … (L.ⓓW0)) /2 width=1 by lfpx_pair/ -W1
92 /4 width=5 by csx_cpxs_trans, cpx_cpxs, cpx_flat/
96 lemma csx_appl_theta: ∀h,o,p,G,L,V,V2,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓓ{p}V.ⓐV2.T⦄ →
97 ∀V1. ⬆*[1] V1 ≡ V2 → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄.
98 /2 width=5 by csx_appl_theta_aux/ qed.