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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 *)
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15 include "ground_2/xoa/ex_5_1.ma".
16 include "ground_2/xoa/ex_9_3.ma".
17 include "basic_2/rt_transition/cpm_tdeq.ma".
18 include "basic_2/rt_computation/fpbg_fqup.ma".
19 include "basic_2/dynamic/cnv_fsb.ma".
21 (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
23 (* Inversion lemmas with restricted rt-transition for terms *****************)
25 lemma cnv_cpr_tdeq_fwd_refl (h) (a) (G) (L):
26 ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → T1 ≛ T2 → ⦃G,L⦄ ⊢ T1 ![h,a] → T1 = T2.
27 #h #a #G #L #T1 #T2 #H @(cpr_ind … H) -G -L -T1 -T2
29 | #G #K #V1 #V2 #X2 #_ #_ #_ #H1 #_ -a -G -K -V1 -V2
30 lapply (tdeq_inv_lref1 … H1) -H1 #H destruct //
31 | #I #G #K #T2 #X2 #i #_ #_ #_ #H1 #_ -a -I -G -K -T2
32 lapply (tdeq_inv_lref1 … H1) -H1 #H destruct //
33 | #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #H1 #H2
34 elim (tdeq_inv_pair1 … H1) -H1 #V0 #T0 #HV0 #HT0 #H destruct
35 elim (cnv_inv_bind … H2) -H2 #HV1 #HT1
37 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #H1 #H2
38 elim (tdeq_inv_pair1 … H1) -H1 #V0 #T0 #HV0 #HT0 #H destruct
39 elim (cnv_fwd_flat … H2) -H2 #HV1 #HT1
41 | #G #K #V #T1 #X1 #X2 #HXT1 #HX12 #_ #H1 #H2
42 elim (cnv_fpbg_refl_false … H2) -a
43 @(fpbg_tdeq_div … H1) -H1
44 /3 width=9 by cpm_tdneq_cpm_fpbg, cpm_zeta, tdeq_lifts_inv_pair_sn/
45 | #G #L #U #T1 #T2 #HT12 #_ #H1 #H2
46 elim (cnv_fpbg_refl_false … H2) -a
47 @(fpbg_tdeq_div … H1) -H1
48 /3 width=6 by cpm_tdneq_cpm_fpbg, cpm_eps, tdeq_inv_pair_xy_y/
49 | #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H1 #_
50 elim (tdeq_inv_pair … H1) -H1 #H #_ #_ destruct
51 | #p #G #L #V1 #V2 #X2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H1 #_
52 elim (tdeq_inv_pair … H1) -H1 #H #_ #_ destruct
56 lemma cpm_tdeq_inv_bind_sn (h) (a) (n) (p) (I) (G) (L):
57 ∀V,T1. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ![h,a] →
58 ∀X. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛ X →
59 ∃∃T2. ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2.
60 #h #a #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
61 elim (cpm_inv_bind1 … H1) -H1 *
62 [ #XV #T2 #HXV #HT12 #H destruct
63 elim (tdeq_inv_pair … H2) -H2 #_ #H2XV #H2T12
64 elim (cnv_inv_bind … H0) -H0 #HV #HT
65 lapply (cnv_cpr_tdeq_fwd_refl … HXV H2XV HV) #H destruct -HXV -H2XV
66 /2 width=4 by ex5_intro/
67 | #X1 #HXT1 #HX1 #H1 #H destruct
68 elim (cnv_fpbg_refl_false … H0) -a
69 @(fpbg_tdeq_div … H2) -H2
70 /3 width=9 by cpm_tdneq_cpm_fpbg, cpm_zeta, tdeq_lifts_inv_pair_sn/
74 lemma cpm_tdeq_inv_appl_sn (h) (a) (n) (G) (L):
75 ∀V,T1. ⦃G,L⦄ ⊢ ⓐV.T1 ![h,a] →
76 ∀X. ⦃G,L⦄ ⊢ ⓐV.T1 ➡[n,h] X → ⓐV.T1 ≛ X →
77 ∃∃m,q,W,U1,T2. ad a m & ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L⦄ ⊢ V ➡*[1,h] W & ⦃G,L⦄ ⊢ T1 ➡*[m,h] ⓛ{q}W.U1
78 & ⦃G,L⦄⊢ T1 ![h,a] & ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓐV.T2.
79 #h #a #n #G #L #V #T1 #H0 #X #H1 #H2
80 elim (cpm_inv_appl1 … H1) -H1 *
81 [ #XV #T2 #HXV #HT12 #H destruct
82 elim (tdeq_inv_pair … H2) -H2 #_ #H2XV #H2T12
83 elim (cnv_inv_appl … H0) -H0 #m #q #W #U1 #Hm #HV #HT #HVW #HTU1
84 lapply (cnv_cpr_tdeq_fwd_refl … HXV H2XV HV) #H destruct -HXV -H2XV
85 /3 width=7 by ex8_5_intro/
86 | #q #V2 #W1 #W2 #XT #T2 #_ #_ #_ #H1 #H destruct -H0
87 elim (tdeq_inv_pair … H2) -H2 #H #_ #_ destruct
88 | #q #V2 #XV #W1 #W2 #XT #T2 #_ #_ #_ #_ #H1 #H destruct -H0
89 elim (tdeq_inv_pair … H2) -H2 #H #_ #_ destruct
93 lemma cpm_tdeq_inv_cast_sn (h) (a) (n) (G) (L):
94 ∀U1,T1. ⦃G,L⦄ ⊢ ⓝU1.T1 ![h,a] →
95 ∀X. ⦃G,L⦄ ⊢ ⓝU1.T1 ➡[n,h] X → ⓝU1.T1 ≛ X →
96 ∃∃U0,U2,T2. ⦃G,L⦄ ⊢ U1 ➡*[h] U0 & ⦃G,L⦄ ⊢ T1 ➡*[1,h] U0
97 & ⦃G,L⦄ ⊢ U1 ![h,a] & ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 & U1 ≛ U2
98 & ⦃G,L⦄ ⊢ T1 ![h,a] & ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓝU2.T2.
99 #h #a #n #G #L #U1 #T1 #H0 #X #H1 #H2
100 elim (cpm_inv_cast1 … H1) -H1 [ * || * ]
101 [ #U2 #T2 #HU12 #HT12 #H destruct
102 elim (tdeq_inv_pair … H2) -H2 #_ #H2U12 #H2T12
103 elim (cnv_inv_cast … H0) -H0 #U0 #HU1 #HT1 #HU10 #HT1U0
104 /2 width=7 by ex9_3_intro/
106 elim (cnv_fpbg_refl_false … H0) -a
107 @(fpbg_tdeq_div … H2) -H2
108 /3 width=6 by cpm_tdneq_cpm_fpbg, cpm_eps, tdeq_inv_pair_xy_y/
109 | #m #HU1X #H destruct
110 elim (cnv_fpbg_refl_false … H0) -a
111 @(fpbg_tdeq_div … H2) -H2
112 /3 width=6 by cpm_tdneq_cpm_fpbg, cpm_ee, tdeq_inv_pair_xy_x/
116 lemma cpm_tdeq_inv_bind_dx (h) (a) (n) (p) (I) (G) (L):
117 ∀X. ⦃G,L⦄ ⊢ X ![h,a] →
118 ∀V,T2. ⦃G,L⦄ ⊢ X ➡[n,h] ⓑ{p,I}V.T2 → X ≛ ⓑ{p,I}V.T2 →
119 ∃∃T1. ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T1.
120 #h #a #n #p #I #G #L #X #H0 #V #T2 #H1 #H2
121 elim (tdeq_inv_pair2 … H2) #V0 #T1 #_ #_ #H destruct
122 elim (cpm_tdeq_inv_bind_sn … H0 … H1 H2) -H0 -H1 -H2 #T0 #HV #HT1 #H1T12 #H2T12 #H destruct
123 /2 width=5 by ex5_intro/
126 (* Eliminators with restricted rt-transition for terms **********************)
128 lemma cpm_tdeq_ind (h) (a) (n) (G) (Q:relation3 …):
129 (∀I,L. n = 0 → Q L (⓪{I}) (⓪{I})) →
130 (∀L,s. n = 1 → Q L (⋆s) (⋆(⫯[h]s))) →
131 (∀p,I,L,V,T1. ⦃G,L⦄⊢ V![h,a] → ⦃G,L.ⓑ{I}V⦄⊢T1![h,a] →
132 ∀T2. ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
133 Q (L.ⓑ{I}V) T1 T2 → Q L (ⓑ{p,I}V.T1) (ⓑ{p,I}V.T2)
136 ∀L,V. ⦃G,L⦄ ⊢ V ![h,a] → ∀W. ⦃G,L⦄ ⊢ V ➡*[1,h] W →
137 ∀p,T1,U1. ⦃G,L⦄ ⊢ T1 ➡*[m,h] ⓛ{p}W.U1 → ⦃G,L⦄⊢ T1 ![h,a] →
138 ∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
139 Q L T1 T2 → Q L (ⓐV.T1) (ⓐV.T2)
141 (∀L,U0,U1,T1. ⦃G,L⦄ ⊢ U1 ➡*[h] U0 → ⦃G,L⦄ ⊢ T1 ➡*[1,h] U0 →
142 ∀U2. ⦃G,L⦄ ⊢ U1 ![h,a] → ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 → U1 ≛ U2 →
143 ∀T2. ⦃G,L⦄ ⊢ T1 ![h,a] → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
144 Q L U1 U2 → Q L T1 T2 → Q L (ⓝU1.T1) (ⓝU2.T2)
146 ∀L,T1. ⦃G,L⦄ ⊢ T1 ![h,a] →
147 ∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 → Q L T1 T2.
148 #h #a #n #G #Q #IH1 #IH2 #IH3 #IH4 #IH5 #L #T1
149 @(insert_eq_0 … G) #F
150 @(fqup_wf_ind_eq (Ⓣ) … F L T1) -L -T1 -F
151 #G0 #L0 #T0 #IH #F #L * [| * [| * ]]
152 [ #I #_ #_ #_ #_ #HF #X #H1X #H2X destruct -G0 -L0 -T0
153 elim (cpm_tdeq_inv_atom_sn … H1X H2X) -H1X -H2X *
154 [ #H1 #H2 destruct /2 width=1 by/
155 | #s #H1 #H2 #H3 destruct /2 width=1 by/
157 | #p #I #V #T1 #HG #HL #HT #H0 #HF #X #H1X #H2X destruct
158 elim (cpm_tdeq_inv_bind_sn … H0 … H1X H2X) -H0 -H1X -H2X #T2 #HV #HT1 #H1T12 #H2T12 #H destruct
160 | #V #T1 #HG #HL #HT #H0 #HF #X #H1X #H2X destruct
161 elim (cpm_tdeq_inv_appl_sn … H0 … H1X H2X) -H0 -H1X -H2X #m #q #W #U1 #T2 #Hm #HV #HVW #HTU1 #HT1 #H1T12 #H2T12 #H destruct
163 | #U1 #T1 #HG #HL #HT #H0 #HF #X #H1X #H2X destruct
164 elim (cpm_tdeq_inv_cast_sn … H0 … H1X H2X) -H0 -H1X -H2X #U0 #U2 #T2 #HU10 #HT1U0 #HU1 #H1U12 #H2U12 #HT1 #H1T12 #H2T12 #H destruct
169 (* Advanced properties with restricted rt-transition for terms **************)
171 lemma cpm_tdeq_free (h) (a) (n) (G) (L):
172 ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] →
173 ∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
174 ∀F,K. ⦃F,K⦄ ⊢ T1 ➡[n,h] T2.
175 #h #a #n #G #L #T1 #H0 #T2 #H1 #H2
176 @(cpm_tdeq_ind … H0 … H1 H2) -L -T1 -T2
177 [ #I #L #H #F #K destruct //
178 | #L #s #H #F #K destruct //
179 | #p #I #L #V #T1 #_ #_ #T2 #_ #_ #IH #F #K
180 /2 width=1 by cpm_bind/
181 | #m #_ #L #V #_ #W #_ #q #T1 #U1 #_ #_ #T2 #_ #_ #IH #F #K
182 /2 width=1 by cpm_appl/
183 | #L #U0 #U1 #T1 #_ #_ #U2 #_ #_ #_ #T2 #_ #_ #_ #IHU #IHT #F #K
184 /2 width=1 by cpm_cast/
188 (* Advanced inversion lemmas with restricted rt-transition for terms ********)
190 lemma cpm_tdeq_inv_bind_sn_void (h) (a) (n) (p) (I) (G) (L):
191 ∀V,T1. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ![h,a] →
192 ∀X. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛ X →
193 ∃∃T2. ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![h,a] & ⦃G,L.ⓧ⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2.
194 #h #a #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
195 elim (cpm_tdeq_inv_bind_sn … H0 … H1 H2) -H0 -H1 -H2 #T2 #HV #HT1 #H1T12 #H2T12 #H
196 /3 width=5 by ex5_intro, cpm_tdeq_free/