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 *)
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15 include "basic_2/rt_transition/lpr_drops.ma".
16 include "basic_2/rt_computation/cpms_lpr.ma".
17 include "basic_2/rt_computation/fpbg_fqup.ma".
18 include "basic_2/rt_conversion/cpce_drops.ma".
19 include "basic_2/rt_conversion/lpce_drops.ma".
20 include "basic_2/dynamic/cnv_drops.ma".
22 (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
24 definition IH (h) (a): relation3 genv lenv term ≝
25 λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![h,a] →
26 ∀n,T1. ⦃G,L0⦄ ⊢ T0 ➡[n,h] T1 → ∀T2. ⦃G,L0⦄ ⊢ T0 ⬌η[h] T2 →
27 ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ⬌η[h] L2 →
28 ∃∃T. ⦃G,L1⦄ ⊢ T1 ⬌η[h] T & ⦃G,L2⦄ ⊢ T2 ➡[n,h] T.
30 (* Properties with eta-conversion for full local environments ***************)
32 lemma pippo_aux (h) (a) (G0) (L0) (T0):
33 (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH h a G L T) →
36 [ #s #_ #_ #n #X1 #HX1 #X2 #HX2 #L1 #HL01 #L2 #HL02
37 elim (cpm_inv_sort1 … HX1) -HX1 #H #Hn destruct
38 lapply (cpce_inv_sort_sn … HX2) -HX2 #H destruct
39 /3 width=3 by cpce_sort, cpm_sort, ex2_intro/
40 | #i #IH #Hi #n #X1 #HX1 #X2 #HX2 #L1 #HL01 #L2 #HL02
41 elim (cnv_inv_lref_drops … Hi) -Hi #I #K0 #W0 #HLK0 #HW0
42 elim (lpr_drops_conf … HLK0 … HL01) [| // ] #Y1 #H1 #HLK1
43 elim (lpr_inv_pair_sn … H1) -H1 #K1 #W1 #HK01 #HW01 #H destruct
44 elim (lpce_drops_conf … HLK0 … HL02) [| // ] #Y2 #H2 #HLK2
45 elim (lpce_inv_pair_sn … H2) -H2 #K2 #W2 #HK02 #HW02 #H destruct
46 elim (cpm_inv_lref1_drops … HX1) -HX1 *
48 elim (cpce_inv_lref_sn_drops_pair … HX2 … HLK0) -HX2 *
49 [ #H1 #H2 destruct -L0 -K0 -W0
50 /3 width=3 by cpce_ldef_drops, ex2_intro/
51 | #H1 #HW #H2 destruct -L0 -W2 -HW0 -HK02
52 @(ex2_intro … (#i)) [| // ]
53 @(cpce_ldec_drops … HLK1) -HLK1 #n #p #V0 #U0 #HWU0
54 /4 width=10 by lpr_cpms_trans, cpms_step_sn/
55 | #n #p #W01 #W02 #V0 #V01 #V02 #U0 #H1 #HWU0 #HW001 #HW012 #HV001 #HV012 #H2 destruct
57 | lapply (drops_isuni_fwd_drop2 … HLK1) [ // ] -W1 #HLK1
58 #Y0 #X0 #W1 #HLY0 #HW01 #HWX1 -HL01 -HL02
59 lapply (drops_mono … HLY0 … HLK0) -HLY0 #H destruct
60 lapply (cpce_inv_lref_sn_drops_ldef … HX2 … HLK0) -HX2 #H destruct
61 elim (IH … HW0 … HW01 … HW02 … HK01 … HK02)
62 [| /3 width=2 by fqup_fpbg, fqup_lref/ ] -L0 -K0 #W #HW1 #HW2
63 elim (lifts_total W (𝐔❴↑i❵)) #V #HWV
64 /3 width=9 by cpce_lifts_bi, cpm_delta_drops, ex2_intro/
65 | lapply (drops_isuni_fwd_drop2 … HLK1) [ // ] -W1 #HLK1
66 #m #Y0 #X0 #W1 #HLY0 #HW01 #HWX1 #H destruct -HL01 -HL02
67 lapply (drops_mono … HLY0 … HLK0) -HLY0 #H destruct
68 elim (cpce_inv_lref_sn_drops_ldec … HX2 … HLK0) -HX2 *
70 elim (IH … HW0 … HW01 … HW02 … HK01 … HK02)
71 [| /3 width=2 by fqup_fpbg, fqup_lref/ ] -L0 -K0 #W #HW1 #HW2
72 elim (lifts_total W (𝐔❴↑i❵)) #V #HWV
73 /3 width=9 by cpce_lifts_bi, cpm_ell_drops, ex2_intro/
74 | lapply (drops_isuni_fwd_drop2 … HLK2) [ // ] -W2 #HLK2
75 #n #p #W01 #W02 #V0 #V01 #V02 #U0 #_ #HW001 #HW012 #_ #_ #H destruct -V0 -V01 -U0
76 elim (IH … HW0 … HW01 … HW001 … HK01 … HK02)
77 [| /3 width=2 by fqup_fpbg, fqup_lref/ ] -L0 -K0 #W #HW1 #HW2
78 elim (lifts_total W (𝐔❴↑i❵)) #V #HWV
79 /4 width=11 by cpce_lifts_bi, cpm_lifts_bi, cpm_ee, ex2_intro/
82 | #l #_ #_ #n #X1 #HX1 #X2 #HX2 #L1 #HL01 #L2 #HL02
83 elim (cpm_inv_gref1 … HX1) -HX1 #H1 #H2 destruct
84 lapply (cpce_inv_gref_sn … HX2) -HX2 #H destruct
85 /3 width=3 by cpce_gref, cpr_refl, ex2_intro/
88 lemma cpce_inv_eta_drops (h) (n) (G) (L) (i):
89 ∀X. ⦃G,L⦄ ⊢ #i ⬌η[h] X →
90 ∀K,W. ⇩*[i] L ≘ K.ⓛW →
91 ∀p,V1,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V1.U →
92 ∀V2. ⦃G,K⦄ ⊢ V1 ⬌η[h] V2 →
93 ∀W2. ⇧*[↑i] V2 ≘ W2 → X = +ⓛW2.ⓐ#0.#↑i.
95 theorem cpce_mono_cnv (h) (a) (G) (L):
96 ∀T. ⦃G,L⦄ ⊢ T ![h,a] →
97 ∀T1. ⦃G,L⦄ ⊢ T ⬌η[h] T1 → ∀T2. ⦃G,L⦄ ⊢ T ⬌η[h] T2 → T1 = T2.