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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/relocation/drops_drops.ma".
16 include "basic_2/s_computation/fqup_weight.ma".
17 include "basic_2/s_computation/fqup_drops.ma".
18 include "basic_2/rt_transition/cpg.ma".
20 (* CONTEXT-SENSITIVE GENERIC PARALLEL RT-TRANSITION FOR TERMS ***************)
22 (* Advanced properties ******************************************************)
24 lemma cpg_delta_drops: ∀c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≡ K.ⓓV → ⦃G, K⦄ ⊢ V ➡[c, h] V2 →
25 ⬆*[⫯i] V2 ≡ T2 → ⦃G, L⦄ ⊢ #i ➡[c, h] T2.
26 #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 #V0 #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: ∀c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≡ K.ⓛV → ⦃G, K⦄ ⊢ V ➡[c, h] V2 →
35 ⬆*[⫯i] V2 ≡ T2 → ⦃G, L⦄ ⊢ #i ➡[(↓c)+𝟘𝟙, h] T2.
36 #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 #V0 #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: ∀c,h,G,i,L,T2. ⦃G, L⦄ ⊢ #i ➡[c, h] T2 →
48 | ∃∃cV,K,V,V2. ⬇*[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ➡[cV, h] V2 &
49 ⬆*[⫯i] V2 ≡ T2 & c = cV
50 | ∃∃cV,K,V,V2. ⬇*[i] L ≡ K.ⓛV & ⦃G, K⦄ ⊢ V ➡[cV, h] V2 &
51 ⬆*[⫯i] V2 ≡ T2 & c = (↓cV) + 𝟘𝟙.
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 #V #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 (* Properties with generic slicing for local environments *******************)
66 lemma cpg_lifts: ∀c,h,G. d_liftable2 (cpg h c G).
67 #c #h #G #K #T generalize in match c; -c
68 @(fqup_wf_ind_eq … G K T) -G -K -T #G0 #K0 #T0 #IH #G #K * *
69 [ #s #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct -IH
70 lapply (lifts_inv_sort1 … H1) -H1 #H destruct
71 elim (cpg_inv_sort1 … H2) -H2 * #H1 #H2 destruct
72 /2 width=3 by cpg_atom, cpg_ess, lifts_sort, ex2_intro/
73 | #i1 #HG #HK #HT #c #T2 #H2 #b #f #L #HLK #X1 #H1 destruct
74 elim (cpg_inv_lref1_drops … H2) -H2 *
75 [ #H1 #H2 destruct /2 width=3 by ex2_intro/ ]
76 #cV #K0 #V #V2 #HK0 #HV2 #HVT2 #H destruct
77 elim (lifts_inv_lref1 … H1) -H1 #i2 #Hf #H destruct
78 lapply (drops_trans … HLK … HK0 ??) -HLK [3,6: |*: // ] #H
79 elim (drops_split_trans … H) -H [1,6: |*: /2 width=6 by after_uni_dx/ ] #Y #HL0 #HY
80 lapply (drops_inv_tls_at … Hf … HY) -HY #HY
81 elim (drops_inv_skip2 … HY) -HY #L0 #W #HLK0 #HVW #H destruct
82 elim (IH … HV2 … HLK0 … HVW) -IH /2 width=2 by fqup_lref/ -K -K0 -V #W2 #HVW2 #HW2
83 elim (lifts_total W2 (𝐔❴⫯i2❵)) #U2 #HWU2
84 lapply (lifts_trans … HVW2 … HWU2 ??) -HVW2 [3,6: |*: // ] #HVU2
85 lapply (lifts_conf … HVT2 … HVU2 f ?) -V2 [1,3: /2 width=3 by after_uni_succ_sn/ ]
86 /4 width=8 by cpg_ell_drops, cpg_delta_drops, drops_inv_gen, ex2_intro/
87 | #l #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct -IH
88 lapply (lifts_inv_gref1 … H1) -H1 #H destruct
89 elim (cpg_inv_gref1 … H2) -H2 #H1 #H2 destruct
90 /2 width=3 by cpg_atom, lifts_gref, ex2_intro/
91 | #p #I #V1 #T1 #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct
92 elim (lifts_inv_bind1 … H1) -H1 #W1 #U1 #HVW1 #HTU1 #H destruct
93 elim (cpg_inv_bind1 … H2) -H2 *
94 [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
95 elim (IH … HV12 … HLK … HVW1) -HV12 //
96 elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
97 /3 width=5 by cpg_bind, lifts_bind, ex2_intro/
98 | #cT #T2 #HT12 #HXT2 #H1 #H2 #H3 destruct
99 elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip/ ] #U2 #HTU2 #HU12
100 lapply (lifts_trans … HXT2 … HTU2 ??) -T2 [3: |*: // ] #HXU2
101 elim (lifts_split_trans … HXU2 f (𝐔❴⫯O❵)) [2: /2 width=1 by after_uni_one_dx/ ]
102 /3 width=5 by cpg_zeta, ex2_intro/
104 | #I #V1 #T1 #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct
105 elim (lifts_inv_flat1 … H1) -H1 #W1 #U1 #HVW1 #HTU1 #H destruct
106 elim (cpg_inv_flat1 … H2) -H2 *
107 [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
108 elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
109 elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
110 /3 width=5 by cpg_flat, lifts_flat, ex2_intro/
111 | #cT #HT12 #H1 #H2 destruct
112 elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
113 /3 width=3 by cpg_eps, ex2_intro/
114 | #cV #HV12 #H1 #H2 destruct
115 elim (IH … HV12 … HLK … HVW1) -IH -HV12 -HLK -HVW1 //
116 /3 width=3 by cpg_ee, ex2_intro/
117 | #cV #cY #cT #a #V2 #Y1 #Y2 #T0 #T2 #HV12 #HY12 #HT12 #H1 #H2 #H3 #H4 destruct
118 elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct
119 elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
120 elim (IH … HY12 … HLK … HYZ1) -HY12 //
121 elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
122 /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/
123 | #cV #cY #cT #a #V2 #V20 #Y1 #Y2 #T0 #T2 #HV12 #HV20 #HY12 #HT12 #H1 #H2 #H3 #H4 destruct
124 elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct
125 elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 // #W2 #HVW2 #HW12
126 elim (IH … HY12 … HLK … HYZ1) -HY12 //
127 elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
128 elim (lifts_total W2 (𝐔❴1❵)) #W20 #HW20
129 lapply (lifts_trans … HVW2 … HW20 ??) -HVW2 [3: |*: // ] #H
130 lapply (lifts_conf … HV20 … H (↑f) ?) -V2 /2 width=3 by after_uni_one_sn/
131 /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/
136 (* Inversion lemmas with generic slicing for local environments *************)
138 lemma cpg_inv_lift1: ∀c,h,G. d_deliftable2_sn (cpg h c G).
139 #c #h #G #L #U generalize in match c; -c
140 @(fqup_wf_ind_eq … G L U) -G -L -U #G0 #L0 #U0 #IH #G #L * *
141 [ #s #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct -IH
142 lapply (lifts_inv_sort2 … H1) -H1 #H destruct
143 elim (cpg_inv_sort1 … H2) -H2 * #H1 #H2 destruct
144 /2 width=3 by cpg_atom, cpg_ess, lifts_sort, ex2_intro/
145 | #i2 #HG #HL #HU #c #U2 #H2 #b #f #K #HLK #X1 #H1 destruct
146 elim (cpg_inv_lref1_drops … H2) -H2 *
147 [ #H1 #H2 destruct /2 width=3 by ex2_intro/ ]
148 #cW #L0 #W #W2 #HL0 #HW2 #HWU2 #H destruct
149 elim (lifts_inv_lref2 … H1) -H1 #i1 #Hf #H destruct
150 lapply (drops_split_div … HLK (𝐔❴i1❵) ???) -HLK [4,8: * |*: // ] #Y0 #HK0 #HLY0
151 lapply (drops_conf … HL0 … HLY0 ??) -HLY0 [3,6: |*: /2 width=6 by after_uni_dx/ ] #HLY0
152 lapply (drops_inv_tls_at … Hf … HLY0) -HLY0 #HLY0
153 elim (drops_inv_skip1 … HLY0) -HLY0 #K0 #V #HLK0 #HVW #H destruct
154 elim (IH … HW2 … HLK0 … HVW) -IH /2 width=2 by fqup_lref/ -L -L0 -W #V2 #HVW2 #HV2
155 lapply (lifts_trans … HVW2 … HWU2 ??) -W2 [3,6: |*: // ] #HVU2
156 elim (lifts_split_trans … HVU2 ? f) -HVU2 [1,4: |*: /2 width=4 by after_uni_succ_sn/ ]
157 /4 width=8 by cpg_ell_drops, cpg_delta_drops, drops_inv_F, ex2_intro/
158 | #l #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct -IH
159 lapply (lifts_inv_gref2 … H1) -H1 #H destruct
160 elim (cpg_inv_gref1 … H2) -H2 #H1 #H2 destruct
161 /2 width=3 by cpg_atom, lifts_gref, ex2_intro/
162 | #p #I #W1 #U1 #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct
163 elim (lifts_inv_bind2 … H1) -H1 #V1 #T1 #HVW1 #HTU1 #H destruct
164 elim (cpg_inv_bind1 … H2) -H2 *
165 [ #cW #cU #W2 #U2 #HW12 #HU12 #H1 #H2 destruct
166 elim (IH … HW12 … HLK … HVW1) -HW12 //
167 elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
168 /3 width=5 by cpg_bind, lifts_bind, ex2_intro/
169 | #cU #U2 #HU12 #HXU2 #H1 #H2 #H3 destruct
170 elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip/ ] #T2 #HTU2 #HT12
171 elim (lifts_div4_one … HTU2 … HXU2) -U2 /3 width=5 by cpg_zeta, ex2_intro/
173 | #I #W1 #U1 #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct
174 elim (lifts_inv_flat2 … H1) -H1 #V1 #T1 #HVW1 #HTU1 #H destruct
175 elim (cpg_inv_flat1 … H2) -H2 *
176 [ #cW #cU #W2 #U2 #HW12 #HU12 #H1 #H2 destruct
177 elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
178 elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
179 /3 width=5 by cpg_flat, lifts_flat, ex2_intro/
180 | #cU #HU12 #H1 #H2 destruct
181 elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
182 /3 width=3 by cpg_eps, ex2_intro/
183 | #cW #HW12 #H1 #H2 destruct
184 elim (IH … HW12 … HLK … HVW1) -IH -HW12 -HLK -HVW1 //
185 /3 width=3 by cpg_ee, ex2_intro/
186 | #cW #cZ #cU #a #W2 #Z1 #Z2 #U0 #U2 #HW12 #HZ12 #HU12 #H1 #H2 #H3 #H4 destruct
187 elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct
188 elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
189 elim (IH … HZ12 … HLK … HYZ1) -HZ12 //
190 elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
191 /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/
192 | #cW #cZ #cU #a #W2 #W20 #Z1 #Z2 #U0 #U2 #HW12 #HW20 #HZ12 #HU12 #H1 #H2 #H3 #H4 destruct
193 elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct
194 elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 // #V2 #HVW2 #HV12
195 elim (IH … HZ12 … HLK … HYZ1) -HZ12 //
196 elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
197 lapply (lifts_trans … HVW2 … HW20 ??) -W2 [3: |*: // ] #H
198 elim (lifts_split_trans … H ? (↑f)) -H [ |*: /2 width=3 by after_uni_one_sn/ ]
199 /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/