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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_transition/lpx_reqg.ma".
16 include "basic_2/rt_computation/csx_drops.ma".
18 (* STRONGLY NORMALIZING TERMS FOR EXTENDED PARALLEL RT-TRANSITION ***********)
20 (* Advanced properties ******************************************************)
22 lemma csx_teqg_trans (S) (G) (L):
23 reflexive … S → symmetric … S →
25 ∀T2. T1 ≛[S] T2 → ❪G,L❫ ⊢ ⬈*𝐒 T2.
26 #S #G #L #H1S #H2S #T1 #H @(csx_ind … H) -T1 #T #_ #IH #T2 #HT2
27 @csx_intro #T1 #HT21 #HnT21
28 lapply (teqg_cpx_trans … HT2 … HT21) // -HT21 #HT1
29 /5 width=4 by teqg_teqx, teqg_canc_sn, teqg_refl/
32 lemma csx_cpx_trans (G) (L):
34 ∀T2. ❪G,L❫ ⊢ T1 ⬈ T2 → ❪G,L❫ ⊢ ⬈*𝐒 T2.
35 #G #L #T1 #H @(csx_ind … H) -T1 #T1 #HT1 #IHT1 #T2 #HLT12
36 elim (teqx_dec T1 T2) /3 width=6 by csx_teqg_trans/
39 (* Basic_1: was just: sn3_cast *)
40 lemma csx_cast (G) (L):
42 ∀T. ❪G,L❫ ⊢ ⬈*𝐒 T → ❪G,L❫ ⊢ ⬈*𝐒 ⓝW.T.
43 #G #L #W #HW @(csx_ind … HW) -W
44 #W #HW #IHW #T #HT @(csx_ind … HT) -T
45 #T #HT #IHT @csx_intro
46 #X #H1 #H2 elim (cpx_inv_cast1 … H1) -H1
47 [ * #W0 #T0 #HLW0 #HLT0 #H destruct
48 elim (tneqx_inv_pair … H2) -H2
49 [ -W -T #H elim H -H //
50 | -HW -IHT /3 width=3 by csx_cpx_trans/
51 | -HW -HT -IHW /4 width=3 by csx_cpx_trans, cpx_pair_sn/
53 |*: /3 width=3 by csx_cpx_trans/
57 (* Basic_1: was just: sn3_abbr *)
58 (* Basic_2A1: was: csx_lref_bind *)
59 lemma csx_lref_pair_drops (G) (L):
60 ∀I,K,V,i. ⇩[i] L ≘ K.ⓑ[I]V →
61 ❪G,K❫ ⊢ ⬈*𝐒 V → ❪G,L❫ ⊢ ⬈*𝐒 #i.
62 #G #L #I #K #V #i #HLK #HV
63 @csx_intro #X #H #Hi elim (cpx_inv_lref1_drops … H) -H
64 [ #H destruct elim Hi //
65 | -Hi * #I0 #K0 #V0 #V1 #HLK0 #HV01 #HV1
66 lapply (drops_mono … HLK0 … HLK) -HLK #H destruct
67 /3 width=8 by csx_lifts, csx_cpx_trans, drops_isuni_fwd_drop2/
71 (* Advanced inversion lemmas ************************************************)
73 (* Basic_1: was: sn3_gen_def *)
74 (* Basic_2A1: was: csx_inv_lref_bind *)
75 lemma csx_inv_lref_pair_drops (G) (L):
76 ∀I,K,V,i. ⇩[i] L ≘ K.ⓑ[I]V →
77 ❪G,L❫ ⊢ ⬈*𝐒 #i → ❪G,K❫ ⊢ ⬈*𝐒 V.
78 #G #L #I #K #V #i #HLK #Hi
79 elim (lifts_total V (𝐔❨↑i❩))
80 /4 width=9 by csx_inv_lifts, csx_cpx_trans, cpx_delta_drops, drops_isuni_fwd_drop2/
83 lemma csx_inv_lref_drops (G) (L):
86 | ∃∃I,K. ⇩[i] L ≘ K.ⓤ[I]
87 | ∃∃I,K,V. ⇩[i] L ≘ K.ⓑ[I]V & ❪G,K❫ ⊢ ⬈*𝐒 V.
88 #G #L #i #H elim (drops_F_uni L i) /2 width=1 by or3_intro0/
89 * * /4 width=9 by csx_inv_lref_pair_drops, ex2_3_intro, ex1_2_intro, or3_intro2, or3_intro1/