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 *)
13 (**************************************************************************)
15 include "basic_2/rt_computation/cpxs_lfpx.ma".
16 include "basic_2/rt_computation/csx_cpxs.ma".
18 (* STRONGLY NORMALIZING TERMS FOR UNCOUNTED PARALLEL RT-TRANSITION **********)
20 (* Properties with uncounted parallel rt-transition on referred entries *****)
22 (* Basic_2A1: was just: csx_lpx_conf *)
23 lemma csx_lfpx_conf: ∀h,o,G,L1,T. ⦃G, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ →
24 ∀L2. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → ⦃G, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
25 #h #o #G #L1 #T #H @(csx_ind_cpxs … H) -T
26 /5 width=3 by csx_intro, lfpx_cpx_trans, lfpx_cpxs_conf/
29 (* Advanced properties ******************************************************)
31 lemma csx_abst: ∀h,o,p,G,L,W. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃W⦄ →
32 ∀T. ⦃G, L.ⓛW⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓛ{p}W.T⦄.
33 #h #o #p #G #L #W #HW @(csx_ind … HW) -W
34 #W #_ #IHW #T #HT @(csx_ind … HT) -T #T #HT #IHT
36 elim (cpx_inv_abst1 … H1) -H1
37 #W0 #T0 #HLW0 #HLT0 #H destruct
38 elim (tdneq_inv_pair … H2) -H2
40 | -IHT #H lapply (csx_cpx_trans … o … HLT0) // -HT
41 #HT0 lapply (csx_lfpx_conf … HT0 … (L.ⓛW0)) -HT0 /4 width=1 by lfpx_pair_refl/
42 | -IHW -HT /4 width=3 by csx_cpx_trans, cpx_pair_sn/
46 lemma csx_abbr: ∀h,o,p,G,L,V. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ →
47 ∀T. ⦃G, L.ⓓV⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓓ{p}V.T⦄.
48 #h #o #p #G #L #V #HV @(csx_ind … HV) -V
49 #V #_ #IHV #T #HT @(csx_ind_cpxs … HT) -T #T #HT #IHT
51 elim (cpx_inv_abbr1 … H1) -H1 *
52 [ #V1 #T1 #HLV1 #HLT1 #H destruct
53 elim (tdneq_inv_pair … H2) -H2
55 | /4 width=3 by csx_cpx_trans, csx_lfpx_conf, lfpx_pair_refl/
56 | -IHV /4 width=3 by csx_cpx_trans, cpx_cpxs, cpx_pair_sn/
59 /4 width=7 by csx_cpx_trans, csx_inv_lifts, drops_drop, drops_refl/
63 fact csx_appl_theta_aux: ∀h,o,p,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃U⦄ → ∀V1,V2. ⬆*[1] V1 ≡ V2 →
64 ∀V,T. U = ⓓ{p}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄.
65 #h #o #p #G #L #X #H @(csx_ind_cpxs … H) -X
66 #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct
67 lapply (csx_fwd_pair_sn … HVT) #HV
68 lapply (csx_fwd_bind_dx … HVT) -HVT #HVT
70 elim (cpx_inv_appl1 … HL) -HL *
71 [ -HV #V0 #Y #HLV10 #HL #H0 destruct
72 elim (cpx_inv_abbr1 … HL) -HL *
73 [ -HVT #V3 #T3 #HV3 #HLT3 #H0 destruct
74 elim (cpx_lifts_sn … HLV10 (Ⓣ) … (L.ⓓV) … HV12) -HLV10 /3 width=1 by drops_refl, drops_drop/ #V4 #HV04 #HV24
75 elim (tdeq_dec h o (ⓓ{p}V.ⓐV2.T) (ⓓ{p}V3.ⓐV4.T3)) #H0
76 [ -IHVT -HV3 -HV24 -HLT3
77 elim (tdeq_inv_pair … H0) -H0 #_ #HV3 #H0
78 elim (tdeq_inv_pair … H0) -H0 #_ #HV24 #HT3
79 elim (tdneq_inv_pair … H) -H #H elim H -H -G -L
80 /3 width=6 by tdeq_inv_lifts_bi, tdeq_pair/
81 | -V1 @(IHVT … H0 … HV04) -o -V0 /4 width=1 by cpx_cpxs, cpx_flat, cpx_bind/
83 | -H -IHVT #T0 #HLT0 #HT0 #H0 destruct
84 lapply (csx_cpx_trans … HVT (ⓐV2.T0) ?) /2 width=1 by cpx_flat/ -T #HVT0
85 lapply (csx_inv_lifts … HVT0 (Ⓣ) … L ???) -HVT0
86 /3 width=5 by csx_cpx_trans, cpx_pair_sn, drops_refl, drops_drop, lifts_flat/
88 | -HV -HV12 -HVT -IHVT -H #b #V0 #W0 #W1 #T0 #T1 #_ #_ #_ #H destruct
89 | -IHVT -H #b #V0 #V3 #W0 #W1 #T0 #T1 #HLV10 #HV03 #HLW01 #HLT01 #H1 #H2 destruct
90 lapply (cpx_lifts_bi … HLV10 (Ⓣ) … (L.ⓓW0) … HV12 … HV03) -HLV10 -HV12 -HV03 /3 width=1 by drops_refl, drops_drop/ #HLV23
91 @csx_abbr /2 width=3 by csx_cpx_trans/ -HV
92 @(csx_lfpx_conf … (L.ⓓW0)) /2 width=1 by lfpx_pair_refl/ -W1
93 /4 width=5 by csx_cpxs_trans, cpx_cpxs, cpx_flat/
97 lemma csx_appl_theta: ∀h,o,p,G,L,V,V2,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓓ{p}V.ⓐV2.T⦄ →
98 ∀V1. ⬆*[1] V1 ≡ V2 → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄.
99 /2 width=5 by csx_appl_theta_aux/ qed.