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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 include "basic_2/rt_transition/lpx_reqx.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_teqx_trans (G) (L):
24 ∀T2. T1 ≛ T2 → ❪G,L❫ ⊢ ⬈*𝐒 T2.
25 #G #L #T1 #H @(csx_ind … H) -T1 #T #_ #IH #T2 #HT2
26 @csx_intro #T1 #HT21 #HnT21
27 lapply (teqx_cpx_trans … HT2 … HT21) -HT21 #HT1
28 /4 width=5 by teqx_repl/
31 lemma csx_cpx_trans (G) (L):
33 ∀T2. ❪G,L❫ ⊢ T1 ⬈ T2 → ❪G,L❫ ⊢ ⬈*𝐒 T2.
34 #G #L #T1 #H @(csx_ind … H) -T1 #T1 #HT1 #IHT1 #T2 #HLT12
35 elim (teqx_dec T1 T2) /3 width=4 by csx_teqx_trans/
38 (* Basic_1: was just: sn3_cast *)
39 lemma csx_cast (G) (L):
41 ∀T. ❪G,L❫ ⊢ ⬈*𝐒 T → ❪G,L❫ ⊢ ⬈*𝐒 ⓝW.T.
42 #G #L #W #HW @(csx_ind … HW) -W
43 #W #HW #IHW #T #HT @(csx_ind … HT) -T
44 #T #HT #IHT @csx_intro
45 #X #H1 #H2 elim (cpx_inv_cast1 … H1) -H1
46 [ * #W0 #T0 #HLW0 #HLT0 #H destruct
47 elim (tneqx_inv_pair … H2) -H2
48 [ -W -T #H elim H -H //
49 | -HW -IHT /3 width=3 by csx_cpx_trans/
50 | -HW -HT -IHW /4 width=3 by csx_cpx_trans, cpx_pair_sn/
52 |*: /3 width=3 by csx_cpx_trans/
56 (* Basic_1: was just: sn3_abbr *)
57 (* Basic_2A1: was: csx_lref_bind *)
58 lemma csx_lref_pair_drops (G) (L):
59 ∀I,K,V,i. ⇩[i] L ≘ K.ⓑ[I]V →
60 ❪G,K❫ ⊢ ⬈*𝐒 V → ❪G,L❫ ⊢ ⬈*𝐒 #i.
61 #G #L #I #K #V #i #HLK #HV
62 @csx_intro #X #H #Hi elim (cpx_inv_lref1_drops … H) -H
63 [ #H destruct elim Hi //
64 | -Hi * #I0 #K0 #V0 #V1 #HLK0 #HV01 #HV1
65 lapply (drops_mono … HLK0 … HLK) -HLK #H destruct
66 /3 width=8 by csx_lifts, csx_cpx_trans, drops_isuni_fwd_drop2/
70 (* Advanced inversion lemmas ************************************************)
72 (* Basic_1: was: sn3_gen_def *)
73 (* Basic_2A1: was: csx_inv_lref_bind *)
74 lemma csx_inv_lref_pair_drops (G) (L):
75 ∀I,K,V,i. ⇩[i] L ≘ K.ⓑ[I]V →
76 ❪G,L❫ ⊢ ⬈*𝐒 #i → ❪G,K❫ ⊢ ⬈*𝐒 V.
77 #G #L #I #K #V #i #HLK #Hi
78 elim (lifts_total V (𝐔❨↑i❩))
79 /4 width=9 by csx_inv_lifts, csx_cpx_trans, cpx_delta_drops, drops_isuni_fwd_drop2/
82 lemma csx_inv_lref_drops (G) (L):
85 | ∃∃I,K. ⇩[i] L ≘ K.ⓤ[I]
86 | ∃∃I,K,V. ⇩[i] L ≘ K.ⓑ[I]V & ❪G,K❫ ⊢ ⬈*𝐒 V.
87 #G #L #i #H elim (drops_F_uni L i) /2 width=1 by or3_intro0/
88 * * /4 width=9 by csx_inv_lref_pair_drops, ex2_3_intro, ex1_2_intro, or3_intro2, or3_intro1/