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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/reduction/cnx_lift.ma".
16 include "basic_2/computation/acp.ma".
17 include "basic_2/computation/csx.ma".
19 (* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
21 (* Relocation properties ****************************************************)
23 (* Basic_1: was just: sn3_lift *)
24 lemma csx_lift: ∀h,g,G,L2,L1,T1,s,d,e. ⦃G, L1⦄ ⊢ ⬊*[h, g] T1 →
25 ∀T2. ⇩[s, d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T2.
26 #h #g #G #L2 #L1 #T1 #s #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL21 #HT12
27 @csx_intro #T #HLT2 #HT2
28 elim (cpx_inv_lift1 … HLT2 … HL21 … HT12) -HLT2 #T0 #HT0 #HLT10
29 @(IHT1 … HLT10) // -L1 -L2 #H destruct
30 >(lift_mono … HT0 … HT12) in HT2; -T1 /2 width=1 by/
33 (* Basic_1: was just: sn3_gen_lift *)
34 lemma csx_inv_lift: ∀h,g,G,L2,L1,T1,s,d,e. ⦃G, L1⦄ ⊢ ⬊*[h, g] T1 →
35 ∀T2. ⇩[s, d, e] L1 ≡ L2 → ⇧[d, e] T2 ≡ T1 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T2.
36 #h #g #G #L2 #L1 #T1 #s #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL12 #HT21
37 @csx_intro #T #HLT2 #HT2
38 elim (lift_total T d e) #T0 #HT0
39 lapply (cpx_lift … HLT2 … HL12 … HT21 … HT0) -HLT2 #HLT10
40 @(IHT1 … HLT10) // -L1 -L2 #H destruct
41 >(lift_inj … HT0 … HT21) in HT2; -T1 /2 width=1 by/
44 (* Advanced inversion lemmas ************************************************)
46 (* Basic_1: was: sn3_gen_def *)
47 lemma csx_inv_lref_bind: ∀h,g,I,G,L,K,V,i. ⇩[i] L ≡ K.ⓑ{I}V →
48 ⦃G, L⦄ ⊢ ⬊*[h, g] #i → ⦃G, K⦄ ⊢ ⬊*[h, g] V.
49 #h #g #I #G #L #K #V #i #HLK #Hi
50 elim (lift_total V 0 (i+1))
51 /4 width=9 by csx_inv_lift, csx_cpx_trans, cpx_delta, ldrop_fwd_drop2/
54 (* Advanced properties ******************************************************)
56 (* Basic_1: was just: sn3_abbr *)
57 lemma csx_lref_bind: ∀h,g,I,G,L,K,V,i. ⇩[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ ⬊*[h, g] V → ⦃G, L⦄ ⊢ ⬊*[h, g] #i.
58 #h #g #I #G #L #K #V #i #HLK #HV
60 elim (cpx_inv_lref1 … H) -H
61 [ #H destruct elim Hi //
62 | -Hi * #I0 #K0 #V0 #V1 #HLK0 #HV01 #HV1
63 lapply (ldrop_mono … HLK0 … HLK) -HLK #H destruct
64 /3 width=8 by csx_lift, csx_cpx_trans, ldrop_fwd_drop2/
68 lemma csx_appl_simple: ∀h,g,G,L,V. ⦃G, L⦄ ⊢ ⬊*[h, g] V → ∀T1.
69 (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T2) →
70 𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T1.
71 #h #g #G #L #V #H @(csx_ind … H) -V #V #_ #IHV #T1 #IHT1 #HT1
73 elim (cpx_inv_appl1_simple … H1) // -H1
74 #V0 #T0 #HLV0 #HLT10 #H destruct
75 elim (eq_false_inv_tpair_dx … H2) -H2
76 [ -IHV -HT1 /4 width=3 by csx_cpx_trans, cpx_pair_sn/
77 | -HLT10 * #H #HV0 destruct
78 @IHV /4 width=3 by csx_cpx_trans, cpx_pair_sn/ (**) (* full auto 17s *)
82 lemma csx_fqu_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
83 ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, g] T2.
84 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
85 /2 width=8 by csx_inv_lref_bind, csx_inv_lift, csx_fwd_flat_dx, csx_fwd_bind_dx, csx_fwd_pair_sn/
88 lemma csx_fquq_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
89 ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, g] T2.
90 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #H elim (fquq_inv_gen … H12) -H12
91 [ /2 width=5 by csx_fqu_conf/
92 | * #HG #HL #HT destruct //
96 lemma csx_fqup_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
97 ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, g] T2.
98 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
99 /3 width=5 by csx_fqu_conf/
102 lemma csx_fqus_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
103 ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, g] T2.
104 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #H elim (fqus_inv_gen … H12) -H12
105 [ /2 width=5 by csx_fqup_conf/
106 | * #HG #HL #HT destruct //
110 (* Main properties **********************************************************)
112 theorem csx_acp: ∀h,g. acp (cpx h g) (eq …) (csx h g).
114 [ #G #L elim (deg_total h g 0) /3 width=8 by cnx_sort_iter, ex_intro/
115 | /3 width=13 by cnx_lift/
116 | /2 width=3 by csx_fwd_flat_dx/
117 | /2 width=1 by csx_cast/