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4 (* ||A|| A project by Andrea Asperti *)
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15 include "basic_2/rt_computation/cpms_drops.ma".
16 include "basic_2/dynamic/cnv.ma".
18 (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
20 (* Advanced dproperties *****************************************************)
22 (* Basic_2A1: uses: snv_lref *)
23 lemma cnv_lref_drops (h) (a) (G):
24 ∀I,K,V,i,L. ❨G,K❩ ⊢ V ![h,a] →
25 ⇩[i] L ≘ K.ⓑ[I]V → ❨G,L❩ ⊢ #i ![h,a].
26 #h #a #G #I #K #V #i elim i -i
28 lapply (drops_fwd_isid … H ?) -H // #H destruct
29 /2 width=1 by cnv_zero/
31 elim (drops_inv_succ … H) -H #J0 #K0 #HK0 #H destruct
32 /3 width=1 by cnv_lref/
36 (* Advanced inversion lemmas ************************************************)
38 (* Basic_2A1: uses: snv_inv_lref *)
39 lemma cnv_inv_lref_drops (h) (a) (G):
40 ∀i,L. ❨G,L❩ ⊢ #i ![h,a] →
41 ∃∃I,K,V. ⇩[i] L ≘ K.ⓑ[I]V & ❨G,K❩ ⊢ V ![h,a].
44 elim (cnv_inv_zero … H) -H #I #K #V #HV #H destruct
45 /3 width=5 by drops_refl, ex2_3_intro/
47 elim (cnv_inv_lref … H) -H #I #L #HL #H destruct
48 elim (IH … HL) -IH -HL #J #K #V #HLK #HV
49 /3 width=5 by drops_drop, ex2_3_intro/
53 lemma cnv_inv_lref_pair (h) (a) (G):
54 ∀i,L. ❨G,L❩ ⊢ #i ![h,a] →
55 ∀I,K,V. ⇩[i] L ≘ K.ⓑ[I]V → ❨G,K❩ ⊢ V ![h,a].
56 #h #a #G #i #L #H #I #K #V #HLK
57 elim (cnv_inv_lref_drops … H) -H #Z #Y #X #HLY #HX
58 lapply (drops_mono … HLY … HLK) -L #H destruct //
61 lemma cnv_inv_lref_atom (h) (a) (b) (G):
62 ∀i,L. ❨G,L❩ ⊢ #i ![h,a] → ⇩*[b,𝐔❨i❩] L ≘ ⋆ → ⊥.
63 #h #a #b #G #i #L #H #Hi
64 elim (cnv_inv_lref_drops … H) -H #Z #Y #X #HLY #_
65 lapply (drops_gen b … HLY) -HLY #HLY
66 lapply (drops_mono … HLY … Hi) -L #H destruct
69 lemma cnv_inv_lref_unit (h) (a) (G):
70 ∀i,L. ❨G,L❩ ⊢ #i ![h,a] →
71 ∀I,K. ⇩[i] L ≘ K.ⓤ[I] → ⊥.
72 #h #a #G #i #L #H #I #K #HLK
73 elim (cnv_inv_lref_drops … H) -H #Z #Y #X #HLY #_
74 lapply (drops_mono … HLY … HLK) -L #H destruct
77 (* Properties with generic slicing for local environments *******************)
79 (* Basic_2A1: uses: snv_lift *)
80 lemma cnv_lifts (h) (a): ∀G. d_liftable1 (cnv h a G).
82 @(fqup_wf_ind_eq (Ⓣ) … G K T) -G -K -T #G0 #K0 #T0 #IH #G #K * * [|||| * ]
83 [ #s #HG #HK #HT #_ #b #f #L #_ #X #H2 destruct
84 >(lifts_inv_sort1 … H2) -X -K -f //
85 | #i #HG #HK #HT #H1 #b #f #L #HLK #X #H2 destruct
86 elim (cnv_inv_lref_drops … H1) -H1 #I0 #K0 #V #HK0 #HV
87 elim (lifts_inv_lref1 … H2) -H2 #j #Hf #H destruct
88 (**) (* this should be a lemma *)
89 lapply (drops_trans … HLK … HK0 ??) -HLK [3,6: |*: // ] #H
90 elim (drops_split_trans … H) -H [1,6: |*: /2 width=6 by after_uni_dx/ ] #Y #HL #HY
91 lapply (drops_tls_at … Hf … HY) -HY #HY
92 elim (drops_inv_skip2 … HY) -HY #Z #L0 #HLK0 #HZ #H destruct
93 elim (liftsb_inv_pair_sn … HZ) -HZ #W #HVW #H destruct
94 (**) (* end of the lemma *)
95 /4 width=8 by cnv_lref_drops, fqup_lref, drops_inv_gen/
96 | #l #HG #HK #HT #H1 #b #f #L #_ #X #_ destruct
97 elim (cnv_inv_gref … H1)
98 | #p #I #V #T #HG #HK #HT #H1 #b #f #L #HLK #X #H2 destruct
99 elim (cnv_inv_bind … H1) -H1 #HV #HT
100 elim (lifts_inv_bind1 … H2) -H2 #W #U #HVW #HTU #H destruct
101 /5 width=8 by cnv_bind, drops_skip, ext2_pair/
102 | #V #T #HG #HK #HT #H1 #b #f #L #HLK #X #H2 destruct
103 elim (cnv_inv_appl … H1) #n #p #W0 #U0 #Ha #HV #HT #HVW0 #HTW0
104 elim (lifts_inv_flat1 … H2) -H2 #W #U #HVW #HTU #H destruct
105 elim (lifts_total W0 f)
106 elim (lifts_total U0 (⫯f))
107 /4 width=17 by cnv_appl, cpms_lifts_bi, lifts_bind/
108 | #V #T #HG #HK #HT #H1 #b #f #L #HLK #X #H2 destruct
109 elim (cnv_inv_cast … H1) #U0 #HV #HT #HVU0 #HTU0
110 elim (lifts_inv_flat1 … H2) -H2 #W #U #HVW #HTU #H destruct
111 elim (lifts_total U0 f)
112 /3 width=12 by cnv_cast, cpms_lifts_bi/
116 (* Inversion lemmas with generic slicing for local environments *************)
118 (* Basic_2A1: uses: snv_inv_lift *)
119 lemma cnv_inv_lifts (h) (a): ∀G. d_deliftable1 (cnv h a G).
121 @(fqup_wf_ind_eq (Ⓣ) … G L U) -G -L -U #G0 #L0 #U0 #IH #G #L * * [|||| * ]
122 [ #s #HG #HL #HU #H1 #b #f #K #HLK #X #H2 destruct
123 >(lifts_inv_sort2 … H2) -X -L -f //
124 | #j #HG #HL #HU #H1 #b #f #K #HLK #X #H2 destruct
125 elim (cnv_inv_lref_drops … H1) -H1 #I0 #L0 #W #HL0 #HW
126 elim (lifts_inv_lref2 … H2) -H2 #i #Hf #H destruct
127 (**) (* this should be a lemma *)
128 lapply (drops_split_div … HLK (𝐔❨i❩) ???) -HLK [4,8: * |*: // ] #Y0 #HK #HLY0
129 lapply (drops_conf … HL0 … HLY0 ??) -HLY0 [3,6: |*: /2 width=6 by after_uni_dx/ ] #HLY0
130 lapply (drops_tls_at … Hf … HLY0) -HLY0 #HLY0
131 elim (drops_inv_skip1 … HLY0) -HLY0 #Z #K0 #HLK0 #HZ #H destruct
132 elim (liftsb_inv_pair_dx … HZ) -HZ #V #HVW #H destruct
133 (**) (* end of the lemma *)
134 /4 width=8 by cnv_lref_drops, fqup_lref, drops_inv_F/
135 | #l #HG #HL #HU #H1 #b #f #K #_ #X #_ destruct
136 elim (cnv_inv_gref … H1)
137 | #p #I #W #U #HG #HL #HU #H1 #b #f #K #HLK #X #H2 destruct
138 elim (cnv_inv_bind … H1) -H1 #HW #HU
139 elim (lifts_inv_bind2 … H2) -H2 #V #T #HVW #HTU #H destruct
140 /5 width=8 by cnv_bind, drops_skip, ext2_pair/
141 | #W #U #HG #HL #HU #H1 #b #f #K #HLK #X #H2 destruct
142 elim (cnv_inv_appl … H1) #n #p #W0 #U0 #Ha #HW #HU #HW0 #HU0
143 elim (lifts_inv_flat2 … H2) -H2 #V #T #HVW #HTU #H destruct
144 elim (cpms_inv_lifts_sn … HW0 … HLK … HVW) -HW0 #V0 #HVW0 #HV0
145 elim (cpms_inv_lifts_sn … HU0 … HLK … HTU) -HU0 #X0 #H #HT0
146 elim (lifts_inv_bind2 … H) -H #X #T0 #HX #HTU0 #H destruct
147 lapply (lifts_inj … HX … HVW0) -HX #H destruct
148 /3 width=8 by cnv_appl/
149 | #W #U #HG #HL #HU #H1 #b #f #K #HLK #X #H2 destruct
150 elim (cnv_inv_cast … H1) #U0 #HW #HU #HWU0 #HU0
151 elim (lifts_inv_flat2 … H2) -H2 #V #T #HVW #HTU #H destruct
152 elim (cpms_inv_lifts_sn … HWU0 … HLK … HVW) -HWU0 #V0 #HVU0 #HV0
153 elim (cpms_inv_lifts_sn … HU0 … HLK … HTU) -HU0 #X #HX #HTV0
154 lapply (lifts_inj … HX … HVU0) -HX #H destruct
155 /3 width=8 by cnv_cast/