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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 "static_2/relocation/drops_drops.ma".
16 include "static_2/s_computation/fqup_weight.ma".
17 include "static_2/s_computation/fqup_drops.ma".
18 include "basic_2/rt_transition/cpg.ma".
20 (* BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *****************)
22 (* Advanced properties ******************************************************)
24 lemma cpg_delta_drops: ∀Rt,c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≘ K.ⓓV → ⦃G, K⦄ ⊢ V ⬈[Rt, c, h] V2 →
25 ⬆*[↑i] V2 ≘ T2 → ⦃G, L⦄ ⊢ #i ⬈[Rt, c, h] T2.
26 #Rt #c #h #G #K #V #V2 #i elim i -i
27 [ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_delta/
28 | #i #IH #L0 #T0 #H0 #HV2 #HVT2
29 elim (drops_inv_succ … H0) -H0 #I #L #HLK #H destruct
30 elim (lifts_split_trans … HVT2 (𝐔❴↑i❵) (𝐔❴1❵) ?) -HVT2 /3 width=3 by cpg_lref/
34 lemma cpg_ell_drops: ∀Rt,c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≘ K.ⓛV → ⦃G, K⦄ ⊢ V ⬈[Rt,c, h] V2 →
35 ⬆*[↑i] V2 ≘ T2 → ⦃G, L⦄ ⊢ #i ⬈[Rt, c+𝟘𝟙, h] T2.
36 #Rt #c #h #G #K #V #V2 #i elim i -i
37 [ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_ell/
38 | #i #IH #L0 #T0 #H0 #HV2 #HVT2
39 elim (drops_inv_succ … H0) -H0 #I #L #HLK #H destruct
40 elim (lifts_split_trans … HVT2 (𝐔❴↑i❵) (𝐔❴1❵) ?) -HVT2 /3 width=3 by cpg_lref/
44 (* Advanced inversion lemmas ************************************************)
46 lemma cpg_inv_lref1_drops: ∀Rt,c,h,G,i,L,T2. ⦃G, L⦄ ⊢ #i ⬈[Rt,c, h] T2 →
48 | ∃∃cV,K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 &
49 ⬆*[↑i] V2 ≘ T2 & c = cV
50 | ∃∃cV,K,V,V2. ⬇*[i] L ≘ K.ⓛV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 &
51 ⬆*[↑i] V2 ≘ T2 & c = cV + 𝟘𝟙.
52 #Rt #c #h #G #i elim i -i
53 [ #L #T2 #H elim (cpg_inv_zero1 … H) -H * /3 width=1 by or3_intro0, conj/
54 /4 width=8 by drops_refl, ex4_4_intro, or3_intro2, or3_intro1/
55 | #i #IH #L #T2 #H elim (cpg_inv_lref1 … H) -H * /3 width=1 by or3_intro0, conj/
56 #I #K #V2 #H #HVT2 #H0 destruct elim (IH … H) -IH -H
57 [ * #H1 #H2 destruct lapply (lifts_inv_lref1_uni … HVT2) -HVT2 #H destruct /3 width=1 by or3_intro0, conj/ ] *
58 #cV #L #W #W2 #HKL #HW2 #HWV2 #H destruct
59 lapply (lifts_trans … HWV2 … HVT2 ??) -V2
60 /4 width=8 by drops_drop, ex4_4_intro, or3_intro2, or3_intro1/
64 lemma cpg_inv_atom1_drops: ∀Rt,c,h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ⬈[Rt, c, h] T2 →
66 | ∃∃s. T2 = ⋆(next h s) & I = Sort s & c = 𝟘𝟙
67 | ∃∃cV,i,K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 &
68 ⬆*[↑i] V2 ≘ T2 & I = LRef i & c = cV
69 | ∃∃cV,i,K,V,V2. ⬇*[i] L ≘ K.ⓛV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 &
70 ⬆*[↑i] V2 ≘ T2 & I = LRef i & c = cV + 𝟘𝟙.
71 #Rt #c #h * #n #G #L #T2 #H
72 [ elim (cpg_inv_sort1 … H) -H *
73 /3 width=3 by or4_intro0, or4_intro1, ex3_intro, conj/
74 | elim (cpg_inv_lref1_drops … H) -H *
75 /3 width=10 by or4_intro0, or4_intro2, or4_intro3, ex5_5_intro, conj/
76 | elim (cpg_inv_gref1 … H) -H
77 /3 width=1 by or4_intro0, conj/
81 (* Properties with generic slicing for local environments *******************)
83 (* Note: it should use drops_split_trans_pair2 *)
84 lemma cpg_lifts_sn: ∀Rt. reflexive … Rt →
85 ∀c,h,G. d_liftable2_sn … lifts (cpg Rt h c G).
86 #Rt #HRt #c #h #G #K #T generalize in match c; -c
87 @(fqup_wf_ind_eq (Ⓣ) … G K T) -G -K -T #G0 #K0 #T0 #IH #G #K * *
88 [ #s #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct -IH
89 lapply (lifts_inv_sort1 … H1) -H1 #H destruct
90 elim (cpg_inv_sort1 … H2) -H2 * #H1 #H2 destruct
91 /2 width=3 by cpg_atom, cpg_ess, lifts_sort, ex2_intro/
92 | #i1 #HG #HK #HT #c #T2 #H2 #b #f #L #HLK #X1 #H1 destruct
93 elim (cpg_inv_lref1_drops … H2) -H2 *
94 [ #H1 #H2 destruct /3 width=3 by cpg_refl, ex2_intro/ ]
95 #cV #K0 #V #V2 #HK0 #HV2 #HVT2 #H destruct
96 elim (lifts_inv_lref1 … H1) -H1 #i2 #Hf #H destruct
97 lapply (drops_trans … HLK … HK0 ??) -HLK [3,6: |*: // ] #H
98 elim (drops_split_trans … H) -H [1,6: |*: /2 width=6 by after_uni_dx/ ] #Y #HL0 #HY
99 lapply (drops_tls_at … Hf … HY) -HY #HY
100 elim (drops_inv_skip2 … HY) -HY #Z #L0 #HLK0 #HZ #H destruct
101 elim (liftsb_inv_pair_sn … HZ) -HZ #W #HVW #H destruct
102 elim (IH … HV2 … HLK0 … HVW) -IH /2 width=2 by fqup_lref/ -K -K0 -V #W2 #HVW2 #HW2
103 elim (lifts_total W2 (𝐔❴↑i2❵)) #U2 #HWU2
104 lapply (lifts_trans … HVW2 … HWU2 ??) -HVW2 [3,6: |*: // ] #HVU2
105 lapply (lifts_conf … HVT2 … HVU2 f ?) -V2 [1,3: /2 width=3 by after_uni_succ_sn/ ]
106 /4 width=8 by cpg_ell_drops, cpg_delta_drops, drops_inv_gen, ex2_intro/
107 | #l #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct -IH
108 lapply (lifts_inv_gref1 … H1) -H1 #H destruct
109 elim (cpg_inv_gref1 … H2) -H2 #H1 #H2 destruct
110 /2 width=3 by cpg_atom, lifts_gref, ex2_intro/
111 | #p #I #V1 #T1 #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct
112 elim (lifts_inv_bind1 … H1) -H1 #W1 #U1 #HVW1 #HTU1 #H destruct
113 elim (cpg_inv_bind1 … H2) -H2 *
114 [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
115 elim (IH … HV12 … HLK … HVW1) -HV12 //
116 elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
117 /3 width=5 by cpg_bind, lifts_bind, ex2_intro/
118 | #cT #T2 #HT21 #HTX2 #H1 #H2 #H3 destruct
119 elim (lifts_trans4_one … HT21 … HTU1) -HTU1 #U2 #HTU2 #HU21
120 elim (IH … HTX2 … HLK … HTU2) [| /3 width=1 by fqup_zeta/ ] -K -V1 -T1 -T2
121 /3 width=5 by cpg_zeta, ex2_intro/
123 | * #V1 #T1 #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct
124 elim (lifts_inv_flat1 … H1) -H1 #W1 #U1 #HVW1 #HTU1 #H destruct
125 [ elim (cpg_inv_appl1 … H2) -H2 *
126 [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
127 elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
128 elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
129 /3 width=5 by cpg_appl, lifts_flat, ex2_intro/
130 | #cV #cY #cT #a #V2 #Y1 #Y2 #T0 #T2 #HV12 #HY12 #HT12 #H1 #H2 #H3 destruct
131 elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct
132 elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
133 elim (IH … HY12 … HLK … HYZ1) -HY12 //
134 elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
135 /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/
136 | #cV #cY #cT #a #V2 #V20 #Y1 #Y2 #T0 #T2 #HV12 #HV20 #HY12 #HT12 #H1 #H2 #H3 destruct
137 elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct
138 elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 // #W2 #HVW2 #HW12
139 elim (IH … HY12 … HLK … HYZ1) -HY12 //
140 elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
141 elim (lifts_total W2 (𝐔❴1❵)) #W20 #HW20
142 lapply (lifts_trans … HVW2 … HW20 ??) -HVW2 [3: |*: // ] #H
143 lapply (lifts_conf … HV20 … H (⫯f) ?) -V2 /2 width=3 by after_uni_one_sn/
144 /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/
146 | elim (cpg_inv_cast1 … H2) -H2 *
147 [ #cV #cT #V2 #T2 #HV12 #HT12 #HcVT #H1 #H2 destruct
148 elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
149 elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
150 /3 width=5 by cpg_cast, lifts_flat, ex2_intro/
151 | #cT #HT12 #H destruct
152 elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
153 /3 width=3 by cpg_eps, ex2_intro/
154 | #cV #HV12 #H destruct
155 elim (IH … HV12 … HLK … HVW1) -IH -HV12 -HLK -HVW1 //
156 /3 width=3 by cpg_ee, ex2_intro/
162 lemma cpg_lifts_bi: ∀Rt. reflexive … Rt →
163 ∀c,h,G. d_liftable2_bi … lifts (cpg Rt h c G).
164 /3 width=12 by cpg_lifts_sn, d_liftable2_sn_bi, lifts_mono/ qed-.
166 (* Inversion lemmas with generic slicing for local environments *************)
168 lemma cpg_inv_lifts_sn: ∀Rt. reflexive … Rt →
169 ∀c,h,G. d_deliftable2_sn … lifts (cpg Rt h c G).
170 #Rt #HRt #c #h #G #L #U generalize in match c; -c
171 @(fqup_wf_ind_eq (Ⓣ) … G L U) -G -L -U #G0 #L0 #U0 #IH #G #L * *
172 [ #s #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct -IH
173 lapply (lifts_inv_sort2 … H1) -H1 #H destruct
174 elim (cpg_inv_sort1 … H2) -H2 * #H1 #H2 destruct
175 /2 width=3 by cpg_atom, cpg_ess, lifts_sort, ex2_intro/
176 | #i2 #HG #HL #HU #c #U2 #H2 #b #f #K #HLK #X1 #H1 destruct
177 elim (cpg_inv_lref1_drops … H2) -H2 *
178 [ #H1 #H2 destruct /3 width=3 by cpg_refl, ex2_intro/ ]
179 #cW #L0 #W #W2 #HL0 #HW2 #HWU2 #H destruct
180 elim (lifts_inv_lref2 … H1) -H1 #i1 #Hf #H destruct
181 lapply (drops_split_div … HLK (𝐔❴i1❵) ???) -HLK [4,8: * |*: // ] #Y0 #HK0 #HLY0
182 lapply (drops_conf … HL0 … HLY0 ??) -HLY0 [3,6: |*: /2 width=6 by after_uni_dx/ ] #HLY0
183 lapply (drops_tls_at … Hf … HLY0) -HLY0 #HLY0
184 elim (drops_inv_skip1 … HLY0) -HLY0 #Z #K0 #HLK0 #HZ #H destruct
185 elim (liftsb_inv_pair_dx … HZ) -HZ #V #HVW #H destruct
186 elim (IH … HW2 … HLK0 … HVW) -IH /2 width=2 by fqup_lref/ -L -L0 -W #V2 #HVW2 #HV2
187 lapply (lifts_trans … HVW2 … HWU2 ??) -W2 [3,6: |*: // ] #HVU2
188 elim (lifts_split_trans … HVU2 ? f) -HVU2 [1,4: |*: /2 width=4 by after_uni_succ_sn/ ]
189 /4 width=8 by cpg_ell_drops, cpg_delta_drops, drops_inv_F, ex2_intro/
190 | #l #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct -IH
191 lapply (lifts_inv_gref2 … H1) -H1 #H destruct
192 elim (cpg_inv_gref1 … H2) -H2 #H1 #H2 destruct
193 /2 width=3 by cpg_atom, lifts_gref, ex2_intro/
194 | #p #I #W1 #U1 #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct
195 elim (lifts_inv_bind2 … H1) -H1 #V1 #T1 #HVW1 #HTU1 #H destruct
196 elim (cpg_inv_bind1 … H2) -H2 *
197 [ #cW #cU #W2 #U2 #HW12 #HU12 #H1 #H2 destruct
198 elim (IH … HW12 … HLK … HVW1) -HW12 //
199 elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
200 /3 width=5 by cpg_bind, lifts_bind, ex2_intro/
201 | #cU #U2 #HU21 #HUX2 #H1 #H2 #H3 destruct
202 elim (lifts_div4_one … HTU1 … HU21) -HTU1 #T2 #HT21 #HTU2
203 elim (IH … HUX2 … HLK … HTU2) [| /3 width=1 by fqup_zeta/ ] -L -W1 -U1 -U2
204 /3 width=5 by cpg_zeta, ex2_intro/
206 | * #W1 #U1 #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct
207 elim (lifts_inv_flat2 … H1) -H1 #V1 #T1 #HVW1 #HTU1 #H destruct
208 [ elim (cpg_inv_appl1 … H2) -H2 *
209 [ #cW #cU #W2 #U2 #HW12 #HU12 #H1 #H2 destruct
210 elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
211 elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
212 /3 width=5 by cpg_appl, lifts_flat, ex2_intro/
213 | #cW #cZ #cU #a #W2 #Z1 #Z2 #U0 #U2 #HW12 #HZ12 #HU12 #H1 #H2 #H3 destruct
214 elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct
215 elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
216 elim (IH … HZ12 … HLK … HYZ1) -HZ12 //
217 elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
218 /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/
219 | #cW #cZ #cU #a #W2 #W20 #Z1 #Z2 #U0 #U2 #HW12 #HW20 #HZ12 #HU12 #H1 #H2 #H3 destruct
220 elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct
221 elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 // #V2 #HVW2 #HV12
222 elim (IH … HZ12 … HLK … HYZ1) -HZ12 //
223 elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
224 lapply (lifts_trans … HVW2 … HW20 ??) -W2 [3: |*: // ] #H
225 elim (lifts_split_trans … H ? (⫯f)) -H [ |*: /2 width=3 by after_uni_one_sn/ ]
226 /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/
228 | elim (cpg_inv_cast1 … H2) -H2 *
229 [ #cW #cU #W2 #U2 #HW12 #HU12 #HcWU #H1 #H2 destruct
230 elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
231 elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
232 /3 width=5 by cpg_cast, lifts_flat, ex2_intro/
233 | #cU #HU12 #H destruct
234 elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
235 /3 width=3 by cpg_eps, ex2_intro/
236 | #cW #HW12 #H destruct
237 elim (IH … HW12 … HLK … HVW1) -IH -HW12 -HLK -HVW1 //
238 /3 width=3 by cpg_ee, ex2_intro/
244 lemma cpg_inv_lifts_bi: ∀Rt. reflexive … Rt →
245 ∀c,h,G. d_deliftable2_bi … lifts (cpg Rt h c G).
246 /3 width=12 by cpg_inv_lifts_sn, d_deliftable2_sn_bi, lifts_inj/ qed-.