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 *)
13 (**************************************************************************)
15 include "ground_2/xoa/ex_5_5.ma".
16 include "static_2/relocation/drops_drops.ma".
17 include "static_2/s_computation/fqup_weight.ma".
18 include "static_2/s_computation/fqup_drops.ma".
19 include "basic_2/rt_transition/cpg.ma".
21 (* BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *****************)
23 (* Advanced properties ******************************************************)
25 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 →
26 ⇧[↑i] V2 ≘ T2 → ❪G,L❫ ⊢ #i ⬈[Rt,c,h] T2.
27 #Rt #c #h #G #K #V #V2 #i elim i -i
28 [ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_delta/
29 | #i #IH #L0 #T0 #H0 #HV2 #HVT2
30 elim (drops_inv_succ … H0) -H0 #I #L #HLK #H destruct
31 elim (lifts_split_trans … HVT2 (𝐔❨↑i❩) (𝐔❨1❩) ?) -HVT2 /3 width=3 by cpg_lref/
35 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 →
36 ⇧[↑i] V2 ≘ T2 → ❪G,L❫ ⊢ #i ⬈[Rt,c+𝟘𝟙,h] T2.
37 #Rt #c #h #G #K #V #V2 #i elim i -i
38 [ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_ell/
39 | #i #IH #L0 #T0 #H0 #HV2 #HVT2
40 elim (drops_inv_succ … H0) -H0 #I #L #HLK #H destruct
41 elim (lifts_split_trans … HVT2 (𝐔❨↑i❩) (𝐔❨1❩) ?) -HVT2 /3 width=3 by cpg_lref/
45 (* Advanced inversion lemmas ************************************************)
47 lemma cpg_inv_lref1_drops: ∀Rt,c,h,G,i,L,T2. ❪G,L❫ ⊢ #i ⬈[Rt,c,h] T2 →
49 | ∃∃cV,K,V,V2. ⇩[i] L ≘ K.ⓓV & ❪G,K❫ ⊢ V ⬈[Rt,cV,h] V2 &
50 ⇧[↑i] V2 ≘ T2 & c = cV
51 | ∃∃cV,K,V,V2. ⇩[i] L ≘ K.ⓛV & ❪G,K❫ ⊢ V ⬈[Rt,cV,h] V2 &
52 ⇧[↑i] V2 ≘ T2 & c = cV + 𝟘𝟙.
53 #Rt #c #h #G #i elim i -i
54 [ #L #T2 #H elim (cpg_inv_zero1 … H) -H * /3 width=1 by or3_intro0, conj/
55 /4 width=8 by drops_refl, ex4_4_intro, or3_intro2, or3_intro1/
56 | #i #IH #L #T2 #H elim (cpg_inv_lref1 … H) -H * /3 width=1 by or3_intro0, conj/
57 #I #K #V2 #H #HVT2 #H0 destruct elim (IH … H) -IH -H
59 lapply (lifts_inv_lref1_uni … HVT2) -HVT2 #H destruct
60 /3 width=1 by or3_intro0, conj/
62 #cV #L #W #W2 #HKL #HW2 #HWV2 #H destruct
63 lapply (lifts_trans … HWV2 … HVT2 (𝐔❨↑↑i❩) ?) -V2 [1,3: // ] #H (**) (* explicit rtmap *)
64 /4 width=8 by drops_drop, ex4_4_intro, or3_intro2, or3_intro1/
68 lemma cpg_inv_atom1_drops: ∀Rt,c,h,I,G,L,T2. ❪G,L❫ ⊢ ⓪[I] ⬈[Rt,c,h] T2 →
70 | ∃∃s. T2 = ⋆(⫯[h]s) & I = Sort s & c = 𝟘𝟙
71 | ∃∃cV,i,K,V,V2. ⇩[i] L ≘ K.ⓓV & ❪G,K❫ ⊢ V ⬈[Rt,cV,h] V2 &
72 ⇧[↑i] V2 ≘ T2 & I = LRef i & c = cV
73 | ∃∃cV,i,K,V,V2. ⇩[i] L ≘ K.ⓛV & ❪G,K❫ ⊢ V ⬈[Rt,cV,h] V2 &
74 ⇧[↑i] V2 ≘ T2 & I = LRef i & c = cV + 𝟘𝟙.
75 #Rt #c #h * #n #G #L #T2 #H
76 [ elim (cpg_inv_sort1 … H) -H *
77 /3 width=3 by or4_intro0, or4_intro1, ex3_intro, conj/
78 | elim (cpg_inv_lref1_drops … H) -H *
79 /3 width=10 by or4_intro0, or4_intro2, or4_intro3, ex5_5_intro, conj/
80 | elim (cpg_inv_gref1 … H) -H
81 /3 width=1 by or4_intro0, conj/
85 (* Properties with generic slicing for local environments *******************)
87 (* Note: it should use drops_split_trans_pair2 *)
88 lemma cpg_lifts_sn: ∀Rt. reflexive … Rt →
89 ∀c,h,G. d_liftable2_sn … lifts (cpg Rt h c G).
90 #Rt #HRt #c #h #G #K #T generalize in match c; -c
91 @(fqup_wf_ind_eq (Ⓣ) … G K T) -G -K -T #G0 #K0 #T0 #IH #G #K * *
92 [ #s #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct -IH
93 lapply (lifts_inv_sort1 … H1) -H1 #H destruct
94 elim (cpg_inv_sort1 … H2) -H2 * #H1 #H2 destruct
95 /2 width=3 by cpg_atom, cpg_ess, lifts_sort, ex2_intro/
96 | #i1 #HG #HK #HT #c #T2 #H2 #b #f #L #HLK #X1 #H1 destruct
97 elim (cpg_inv_lref1_drops … H2) -H2 *
98 [ #H1 #H2 destruct /3 width=3 by cpg_refl, ex2_intro/ ]
99 #cV #K0 #V #V2 #HK0 #HV2 #HVT2 #H destruct
100 elim (lifts_inv_lref1 … H1) -H1 #i2 #Hf #H destruct
101 lapply (drops_trans … HLK … HK0 ??) -HLK [3,6: |*: // ] #H
102 elim (drops_split_trans … H) -H [1,6: |*: /2 width=6 by after_uni_dx/ ] #Y #HL0 #HY
103 lapply (drops_tls_at … Hf … HY) -HY #HY
104 elim (drops_inv_skip2 … HY) -HY #Z #L0 #HLK0 #HZ #H destruct
105 elim (liftsb_inv_pair_sn … HZ) -HZ #W #HVW #H destruct
106 elim (IH … HV2 … HLK0 … HVW) -IH /2 width=2 by fqup_lref/ -K -K0 -V #W2 #HVW2 #HW2
107 elim (lifts_total W2 (𝐔❨↑i2❩)) #U2 #HWU2
108 lapply (lifts_trans … HVW2 … HWU2 ??) -HVW2 [3,6: |*: // ] #HVU2
109 lapply (lifts_conf … HVT2 … HVU2 f ?) -V2 [1,3: /2 width=3 by after_uni_succ_sn/ ]
110 /4 width=8 by cpg_ell_drops, cpg_delta_drops, drops_inv_gen, ex2_intro/
111 | #l #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct -IH
112 lapply (lifts_inv_gref1 … H1) -H1 #H destruct
113 elim (cpg_inv_gref1 … H2) -H2 #H1 #H2 destruct
114 /2 width=3 by cpg_atom, lifts_gref, ex2_intro/
115 | #p #I #V1 #T1 #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct
116 elim (lifts_inv_bind1 … H1) -H1 #W1 #U1 #HVW1 #HTU1 #H destruct
117 elim (cpg_inv_bind1 … H2) -H2 *
118 [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
119 elim (IH … HV12 … HLK … HVW1) -HV12 //
120 elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
121 /3 width=5 by cpg_bind, lifts_bind, ex2_intro/
122 | #cT #T2 #HT21 #HTX2 #H1 #H2 #H3 destruct
123 elim (lifts_trans4_one … HT21 … HTU1) -HTU1 #U2 #HTU2 #HU21
124 elim (IH … HTX2 … HLK … HTU2) [| /3 width=1 by fqup_zeta/ ] -K -V1 -T1 -T2
125 /3 width=5 by cpg_zeta, ex2_intro/
127 | * #V1 #T1 #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct
128 elim (lifts_inv_flat1 … H1) -H1 #W1 #U1 #HVW1 #HTU1 #H destruct
129 [ elim (cpg_inv_appl1 … H2) -H2 *
130 [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
131 elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
132 elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
133 /3 width=5 by cpg_appl, lifts_flat, ex2_intro/
134 | #cV #cY #cT #a #V2 #Y1 #Y2 #T0 #T2 #HV12 #HY12 #HT12 #H1 #H2 #H3 destruct
135 elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct
136 elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
137 elim (IH … HY12 … HLK … HYZ1) -HY12 //
138 elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
139 /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/
140 | #cV #cY #cT #a #V2 #V20 #Y1 #Y2 #T0 #T2 #HV12 #HV20 #HY12 #HT12 #H1 #H2 #H3 destruct
141 elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct
142 elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 // #W2 #HVW2 #HW12
143 elim (IH … HY12 … HLK … HYZ1) -HY12 //
144 elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
145 elim (lifts_total W2 (𝐔❨1❩)) #W20 #HW20
146 lapply (lifts_trans … HVW2 … HW20 ??) -HVW2 [3: |*: // ] #H
147 lapply (lifts_conf … HV20 … H (⫯f) ?) -V2 /2 width=3 by after_uni_one_sn/
148 /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/
150 | elim (cpg_inv_cast1 … H2) -H2 *
151 [ #cV #cT #V2 #T2 #HV12 #HT12 #HcVT #H1 #H2 destruct
152 elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
153 elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
154 /3 width=5 by cpg_cast, lifts_flat, ex2_intro/
155 | #cT #HT12 #H destruct
156 elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
157 /3 width=3 by cpg_eps, ex2_intro/
158 | #cV #HV12 #H destruct
159 elim (IH … HV12 … HLK … HVW1) -IH -HV12 -HLK -HVW1 //
160 /3 width=3 by cpg_ee, ex2_intro/
166 lemma cpg_lifts_bi: ∀Rt. reflexive … Rt →
167 ∀c,h,G. d_liftable2_bi … lifts (cpg Rt h c G).
168 /3 width=12 by cpg_lifts_sn, d_liftable2_sn_bi, lifts_mono/ qed-.
170 (* Inversion lemmas with generic slicing for local environments *************)
172 lemma cpg_inv_lifts_sn: ∀Rt. reflexive … Rt →
173 ∀c,h,G. d_deliftable2_sn … lifts (cpg Rt h c G).
174 #Rt #HRt #c #h #G #L #U generalize in match c; -c
175 @(fqup_wf_ind_eq (Ⓣ) … G L U) -G -L -U #G0 #L0 #U0 #IH #G #L * *
176 [ #s #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct -IH
177 lapply (lifts_inv_sort2 … H1) -H1 #H destruct
178 elim (cpg_inv_sort1 … H2) -H2 * #H1 #H2 destruct
179 /2 width=3 by cpg_atom, cpg_ess, lifts_sort, ex2_intro/
180 | #i2 #HG #HL #HU #c #U2 #H2 #b #f #K #HLK #X1 #H1 destruct
181 elim (cpg_inv_lref1_drops … H2) -H2 *
182 [ #H1 #H2 destruct /3 width=3 by cpg_refl, ex2_intro/ ]
183 #cW #L0 #W #W2 #HL0 #HW2 #HWU2 #H destruct
184 elim (lifts_inv_lref2 … H1) -H1 #i1 #Hf #H destruct
185 lapply (drops_split_div … HLK (𝐔❨i1❩) ???) -HLK [4,8: * |*: // ] #Y0 #HK0 #HLY0
186 lapply (drops_conf … HL0 … HLY0 ??) -HLY0 [3,6: |*: /2 width=6 by after_uni_dx/ ] #HLY0
187 lapply (drops_tls_at … Hf … HLY0) -HLY0 #HLY0
188 elim (drops_inv_skip1 … HLY0) -HLY0 #Z #K0 #HLK0 #HZ #H destruct
189 elim (liftsb_inv_pair_dx … HZ) -HZ #V #HVW #H destruct
190 elim (IH … HW2 … HLK0 … HVW) -IH /2 width=2 by fqup_lref/ -L -L0 -W #V2 #HVW2 #HV2
191 lapply (lifts_trans … HVW2 … HWU2 ??) -W2 [3,6: |*: // ] #HVU2
192 elim (lifts_split_trans … HVU2 ? f) -HVU2 [1,4: |*: /2 width=4 by after_uni_succ_sn/ ]
193 /4 width=8 by cpg_ell_drops, cpg_delta_drops, drops_inv_F, ex2_intro/
194 | #l #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct -IH
195 lapply (lifts_inv_gref2 … H1) -H1 #H destruct
196 elim (cpg_inv_gref1 … H2) -H2 #H1 #H2 destruct
197 /2 width=3 by cpg_atom, lifts_gref, ex2_intro/
198 | #p #I #W1 #U1 #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct
199 elim (lifts_inv_bind2 … H1) -H1 #V1 #T1 #HVW1 #HTU1 #H destruct
200 elim (cpg_inv_bind1 … H2) -H2 *
201 [ #cW #cU #W2 #U2 #HW12 #HU12 #H1 #H2 destruct
202 elim (IH … HW12 … HLK … HVW1) -HW12 //
203 elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
204 /3 width=5 by cpg_bind, lifts_bind, ex2_intro/
205 | #cU #U2 #HU21 #HUX2 #H1 #H2 #H3 destruct
206 elim (lifts_div4_one … HTU1 … HU21) -HTU1 #T2 #HT21 #HTU2
207 elim (IH … HUX2 … HLK … HTU2) [| /3 width=1 by fqup_zeta/ ] -L -W1 -U1 -U2
208 /3 width=5 by cpg_zeta, ex2_intro/
210 | * #W1 #U1 #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct
211 elim (lifts_inv_flat2 … H1) -H1 #V1 #T1 #HVW1 #HTU1 #H destruct
212 [ elim (cpg_inv_appl1 … H2) -H2 *
213 [ #cW #cU #W2 #U2 #HW12 #HU12 #H1 #H2 destruct
214 elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
215 elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
216 /3 width=5 by cpg_appl, lifts_flat, ex2_intro/
217 | #cW #cZ #cU #a #W2 #Z1 #Z2 #U0 #U2 #HW12 #HZ12 #HU12 #H1 #H2 #H3 destruct
218 elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct
219 elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
220 elim (IH … HZ12 … HLK … HYZ1) -HZ12 //
221 elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
222 /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/
223 | #cW #cZ #cU #a #W2 #W20 #Z1 #Z2 #U0 #U2 #HW12 #HW20 #HZ12 #HU12 #H1 #H2 #H3 destruct
224 elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct
225 elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 // #V2 #HVW2 #HV12
226 elim (IH … HZ12 … HLK … HYZ1) -HZ12 //
227 elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
228 lapply (lifts_trans … HVW2 … HW20 ??) -W2 [3: |*: // ] #H
229 elim (lifts_split_trans … H ? (⫯f)) -H [ |*: /2 width=3 by after_uni_one_sn/ ]
230 /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/
232 | elim (cpg_inv_cast1 … H2) -H2 *
233 [ #cW #cU #W2 #U2 #HW12 #HU12 #HcWU #H1 #H2 destruct
234 elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
235 elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
236 /3 width=5 by cpg_cast, lifts_flat, ex2_intro/
237 | #cU #HU12 #H destruct
238 elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
239 /3 width=3 by cpg_eps, ex2_intro/
240 | #cW #HW12 #H destruct
241 elim (IH … HW12 … HLK … HVW1) -IH -HW12 -HLK -HVW1 //
242 /3 width=3 by cpg_ee, ex2_intro/
248 lemma cpg_inv_lifts_bi: ∀Rt. reflexive … Rt →
249 ∀c,h,G. d_deliftable2_bi … lifts (cpg Rt h c G).
250 /3 width=12 by cpg_inv_lifts_sn, d_deliftable2_sn_bi, lifts_inj/ qed-.