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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 UNBOUND PARALLEL RT-TRANSITION ************)
20 (* Advanced properties ******************************************************)
22 lemma csx_teqx_trans (h) (G) (L):
23 ∀T1. ❪G,L❫ ⊢ ⬈*𝐒[h] T1 →
24 ∀T2. T1 ≛ T2 → ❪G,L❫ ⊢ ⬈*𝐒[h] T2.
25 #h #G #L #T1 #H @(csx_ind … H) -T1 #T #_ #IH #T2 #HT2
26 @csx_intro #T1 #HT21 #HnT21 elim (teqx_cpx_trans … HT2 … HT21) -HT21
27 /4 width=5 by teqx_repl/
30 lemma csx_cpx_trans (h) (G) (L):
31 ∀T1. ❪G,L❫ ⊢ ⬈*𝐒[h] T1 →
32 ∀T2. ❪G,L❫ ⊢ T1 ⬈[h] T2 → ❪G,L❫ ⊢ ⬈*𝐒[h] T2.
33 #h #G #L #T1 #H @(csx_ind … H) -T1 #T1 #HT1 #IHT1 #T2 #HLT12
34 elim (teqx_dec T1 T2) /3 width=4 by csx_teqx_trans/
37 (* Basic_1: was just: sn3_cast *)
38 lemma csx_cast (h) (G) (L):
39 ∀W. ❪G,L❫ ⊢ ⬈*𝐒[h] W →
40 ∀T. ❪G,L❫ ⊢ ⬈*𝐒[h] T → ❪G,L❫ ⊢ ⬈*𝐒[h] ⓝW.T.
41 #h #G #L #W #HW @(csx_ind … HW) -W
42 #W #HW #IHW #T #HT @(csx_ind … HT) -T
43 #T #HT #IHT @csx_intro
44 #X #H1 #H2 elim (cpx_inv_cast1 … H1) -H1
45 [ * #W0 #T0 #HLW0 #HLT0 #H destruct
46 elim (tneqx_inv_pair … H2) -H2
47 [ -W -T #H elim H -H //
48 | -HW -IHT /3 width=3 by csx_cpx_trans/
49 | -HW -HT -IHW /4 width=3 by csx_cpx_trans, cpx_pair_sn/
51 |*: /3 width=3 by csx_cpx_trans/
55 (* Basic_1: was just: sn3_abbr *)
56 (* Basic_2A1: was: csx_lref_bind *)
57 lemma csx_lref_pair_drops (h) (G) (L):
58 ∀I,K,V,i. ⇩[i] L ≘ K.ⓑ[I]V →
59 ❪G,K❫ ⊢ ⬈*𝐒[h] V → ❪G,L❫ ⊢ ⬈*𝐒[h] #i.
60 #h #G #L #I #K #V #i #HLK #HV
61 @csx_intro #X #H #Hi elim (cpx_inv_lref1_drops … H) -H
62 [ #H destruct elim Hi //
63 | -Hi * #I0 #K0 #V0 #V1 #HLK0 #HV01 #HV1
64 lapply (drops_mono … HLK0 … HLK) -HLK #H destruct
65 /3 width=8 by csx_lifts, csx_cpx_trans, drops_isuni_fwd_drop2/
69 (* Advanced inversion lemmas ************************************************)
71 (* Basic_1: was: sn3_gen_def *)
72 (* Basic_2A1: was: csx_inv_lref_bind *)
73 lemma csx_inv_lref_pair_drops (h) (G) (L):
74 ∀I,K,V,i. ⇩[i] L ≘ K.ⓑ[I]V →
75 ❪G,L❫ ⊢ ⬈*𝐒[h] #i → ❪G,K❫ ⊢ ⬈*𝐒[h] V.
76 #h #G #L #I #K #V #i #HLK #Hi
77 elim (lifts_total V (𝐔❨↑i❩))
78 /4 width=9 by csx_inv_lifts, csx_cpx_trans, cpx_delta_drops, drops_isuni_fwd_drop2/
81 lemma csx_inv_lref_drops (h) (G) (L):
82 ∀i. ❪G,L❫ ⊢ ⬈*𝐒[h] #i →
84 | ∃∃I,K. ⇩[i] L ≘ K.ⓤ[I]
85 | ∃∃I,K,V. ⇩[i] L ≘ K.ⓑ[I]V & ❪G,K❫ ⊢ ⬈*𝐒[h] V.
86 #h #G #L #i #H elim (drops_F_uni L i) /2 width=1 by or3_intro0/
87 * * /4 width=9 by csx_inv_lref_pair_drops, ex2_3_intro, ex1_2_intro, or3_intro2, or3_intro1/