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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 "ground/xoa/ex_5_1.ma".
16 include "ground/xoa/ex_8_5.ma".
17 include "ground/xoa/ex_9_3.ma".
18 include "basic_2/rt_transition/cpm_teqx.ma".
19 include "basic_2/rt_computation/fpbg_cpm.ma".
20 include "basic_2/dynamic/cnv_fsb.ma".
22 (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
24 (* Inversion lemmas with restricted rt-transition for terms *****************)
26 lemma cnv_cpr_teqx_fwd_refl (h) (a) (G) (L):
27 ∀T1,T2. ❨G,L❩ ⊢ T1 ➡[h,0] T2 → T1 ≅ T2 → ❨G,L❩ ⊢ T1 ![h,a] → T1 = T2.
28 #h #a #G #L #T1 #T2 #H @(cpr_ind … H) -G -L -T1 -T2
30 | #G #K #V1 #V2 #X2 #_ #_ #_ #H1 #_ -a -G -K -V1 -V2
31 lapply (teqg_inv_lref1 … H1) -H1 #H destruct //
32 | #I #G #K #T2 #X2 #i #_ #_ #_ #H1 #_ -a -I -G -K -T2
33 lapply (teqg_inv_lref1 … H1) -H1 #H destruct //
34 | #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #H1 #H2
35 elim (teqx_inv_pair1 … H1) -H1 #V0 #T0 #HV0 #HT0 #H destruct
36 elim (cnv_inv_bind … H2) -H2 #HV1 #HT1
38 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #H1 #H2
39 elim (teqx_inv_pair1 … H1) -H1 #V0 #T0 #HV0 #HT0 #H destruct
40 elim (cnv_fwd_flat … H2) -H2 #HV1 #HT1
42 | #G #K #V #T1 #X1 #X2 #HXT1 #HX12 #_ #H1 #H2
43 elim (cnv_fpbg_refl_false … H2) -a
44 @(fpbg_teqg_div … H1) -H1
45 /3 width=9 by cpm_tneqx_cpm_fpbg, cpm_zeta, teqg_lifts_inv_pair_sn/
46 | #G #L #U #T1 #T2 #HT12 #_ #H1 #H2
47 elim (cnv_fpbg_refl_false … H2) -a
48 @(fpbg_teqg_div … H1) -H1
49 /3 width=7 by cpm_tneqx_cpm_fpbg, cpm_eps, teqg_inv_pair_xy_y/
50 | #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H1 #_
51 elim (teqx_inv_pair … H1) -H1 #H #_ #_ destruct
52 | #p #G #L #V1 #V2 #X2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H1 #_
53 elim (teqx_inv_pair … H1) -H1 #H #_ #_ destruct
57 lemma cpm_teqx_inv_bind_sn (h) (a) (n) (p) (I) (G) (L):
58 ∀V,T1. ❨G,L❩ ⊢ ⓑ[p,I]V.T1 ![h,a] →
59 ∀X. ❨G,L❩ ⊢ ⓑ[p,I]V.T1 ➡[h,n] X → ⓑ[p,I]V.T1 ≅ X →
60 ∃∃T2. ❨G,L❩ ⊢ V ![h,a] & ❨G,L.ⓑ[I]V❩ ⊢ T1 ![h,a] & ❨G,L.ⓑ[I]V❩ ⊢ T1 ➡[h,n] T2 & T1 ≅ T2 & X = ⓑ[p,I]V.T2.
61 #h #a #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
62 elim (cpm_inv_bind1 … H1) -H1 *
63 [ #XV #T2 #HXV #HT12 #H destruct
64 elim (teqx_inv_pair … H2) -H2 #_ #H2XV #H2T12
65 elim (cnv_inv_bind … H0) -H0 #HV #HT
66 lapply (cnv_cpr_teqx_fwd_refl … HXV H2XV HV) #H destruct -HXV -H2XV
67 /2 width=4 by ex5_intro/
68 | #X1 #HXT1 #HX1 #H1 #H destruct
69 elim (cnv_fpbg_refl_false … H0) -a
70 @(fpbg_teqg_div … H2) -H2
71 /3 width=9 by cpm_tneqx_cpm_fpbg, cpm_zeta, teqx_lifts_inv_pair_sn/
75 lemma cpm_teqx_inv_appl_sn (h) (a) (n) (G) (L):
76 ∀V,T1. ❨G,L❩ ⊢ ⓐV.T1 ![h,a] →
77 ∀X. ❨G,L❩ ⊢ ⓐV.T1 ➡[h,n] X → ⓐV.T1 ≅ X →
78 ∃∃m,q,W,U1,T2. ad a m & ❨G,L❩ ⊢ V ![h,a] & ❨G,L❩ ⊢ V ➡*[h,1] W & ❨G,L❩ ⊢ T1 ➡*[h,m] ⓛ[q]W.U1
79 & ❨G,L❩⊢ T1 ![h,a] & ❨G,L❩ ⊢ T1 ➡[h,n] T2 & T1 ≅ T2 & X = ⓐV.T2.
80 #h #a #n #G #L #V #T1 #H0 #X #H1 #H2
81 elim (cpm_inv_appl1 … H1) -H1 *
82 [ #XV #T2 #HXV #HT12 #H destruct
83 elim (teqx_inv_pair … H2) -H2 #_ #H2XV #H2T12
84 elim (cnv_inv_appl … H0) -H0 #m #q #W #U1 #Hm #HV #HT #HVW #HTU1
85 lapply (cnv_cpr_teqx_fwd_refl … HXV H2XV HV) #H destruct -HXV -H2XV
86 /3 width=7 by ex8_5_intro/
87 | #q #V2 #W1 #W2 #XT #T2 #_ #_ #_ #H1 #H destruct -H0
88 elim (teqx_inv_pair … H2) -H2 #H #_ #_ destruct
89 | #q #V2 #XV #W1 #W2 #XT #T2 #_ #_ #_ #_ #H1 #H destruct -H0
90 elim (teqx_inv_pair … H2) -H2 #H #_ #_ destruct
94 lemma cpm_teqx_inv_cast_sn (h) (a) (n) (G) (L):
95 ∀U1,T1. ❨G,L❩ ⊢ ⓝU1.T1 ![h,a] →
96 ∀X. ❨G,L❩ ⊢ ⓝU1.T1 ➡[h,n] X → ⓝU1.T1 ≅ X →
97 ∃∃U0,U2,T2. ❨G,L❩ ⊢ U1 ➡*[h,0] U0 & ❨G,L❩ ⊢ T1 ➡*[h,1] U0
98 & ❨G,L❩ ⊢ U1 ![h,a] & ❨G,L❩ ⊢ U1 ➡[h,n] U2 & U1 ≅ U2
99 & ❨G,L❩ ⊢ T1 ![h,a] & ❨G,L❩ ⊢ T1 ➡[h,n] T2 & T1 ≅ T2 & X = ⓝU2.T2.
100 #h #a #n #G #L #U1 #T1 #H0 #X #H1 #H2
101 elim (cpm_inv_cast1 … H1) -H1 [ * || * ]
102 [ #U2 #T2 #HU12 #HT12 #H destruct
103 elim (teqx_inv_pair … H2) -H2 #_ #H2U12 #H2T12
104 elim (cnv_inv_cast … H0) -H0 #U0 #HU1 #HT1 #HU10 #HT1U0
105 /2 width=7 by ex9_3_intro/
107 elim (cnv_fpbg_refl_false … H0) -a
108 @(fpbg_teqg_div … H2) -H2
109 /3 width=7 by cpm_tneqx_cpm_fpbg, cpm_eps, teqg_inv_pair_xy_y/
110 | #m #HU1X #H destruct
111 elim (cnv_fpbg_refl_false … H0) -a
112 @(fpbg_teqg_div … H2) -H2
113 /3 width=7 by cpm_tneqx_cpm_fpbg, cpm_ee, teqg_inv_pair_xy_x/
117 lemma cpm_teqx_inv_bind_dx (h) (a) (n) (p) (I) (G) (L):
118 ∀X. ❨G,L❩ ⊢ X ![h,a] →
119 ∀V,T2. ❨G,L❩ ⊢ X ➡[h,n] ⓑ[p,I]V.T2 → X ≅ ⓑ[p,I]V.T2 →
120 ∃∃T1. ❨G,L❩ ⊢ V ![h,a] & ❨G,L.ⓑ[I]V❩ ⊢ T1 ![h,a] & ❨G,L.ⓑ[I]V❩ ⊢ T1 ➡[h,n] T2 & T1 ≅ T2 & X = ⓑ[p,I]V.T1.
121 #h #a #n #p #I #G #L #X #H0 #V #T2 #H1 #H2
122 elim (teqx_inv_pair2 … H2) #V0 #T1 #_ #_ #H destruct
123 elim (cpm_teqx_inv_bind_sn … H0 … H1 H2) -H0 -H1 -H2 #T0 #HV #HT1 #H1T12 #H2T12 #H destruct
124 /2 width=5 by ex5_intro/
127 (* Eliminators with restricted rt-transition for terms **********************)
129 lemma cpm_teqx_ind (h) (a) (n) (G) (Q:relation3 …):
130 (∀I,L. n = 0 → Q L (⓪[I]) (⓪[I])) →
131 (∀L,s. n = 1 → Q L (⋆s) (⋆(⫯[h]s))) →
132 (∀p,I,L,V,T1. ❨G,L❩⊢ V![h,a] → ❨G,L.ⓑ[I]V❩⊢T1![h,a] →
133 ∀T2. ❨G,L.ⓑ[I]V❩ ⊢ T1 ➡[h,n] T2 → T1 ≅ T2 →
134 Q (L.ⓑ[I]V) T1 T2 → Q L (ⓑ[p,I]V.T1) (ⓑ[p,I]V.T2)
137 ∀L,V. ❨G,L❩ ⊢ V ![h,a] → ∀W. ❨G,L❩ ⊢ V ➡*[h,1] W →
138 ∀p,T1,U1. ❨G,L❩ ⊢ T1 ➡*[h,m] ⓛ[p]W.U1 → ❨G,L❩⊢ T1 ![h,a] →
139 ∀T2. ❨G,L❩ ⊢ T1 ➡[h,n] T2 → T1 ≅ T2 →
140 Q L T1 T2 → Q L (ⓐV.T1) (ⓐV.T2)
142 (∀L,U0,U1,T1. ❨G,L❩ ⊢ U1 ➡*[h,0] U0 → ❨G,L❩ ⊢ T1 ➡*[h,1] U0 →
143 ∀U2. ❨G,L❩ ⊢ U1 ![h,a] → ❨G,L❩ ⊢ U1 ➡[h,n] U2 → U1 ≅ U2 →
144 ∀T2. ❨G,L❩ ⊢ T1 ![h,a] → ❨G,L❩ ⊢ T1 ➡[h,n] T2 → T1 ≅ T2 →
145 Q L U1 U2 → Q L T1 T2 → Q L (ⓝU1.T1) (ⓝU2.T2)
147 ∀L,T1. ❨G,L❩ ⊢ T1 ![h,a] →
148 ∀T2. ❨G,L❩ ⊢ T1 ➡[h,n] T2 → T1 ≅ T2 → Q L T1 T2.
149 #h #a #n #G #Q #IH1 #IH2 #IH3 #IH4 #IH5 #L #T1
150 @(insert_eq_0 … G) #F
151 @(fqup_wf_ind_eq (Ⓣ) … F L T1) -L -T1 -F
152 #G0 #L0 #T0 #IH #F #L * [| * [| * ]]
153 [ #I #_ #_ #_ #_ #HF #X #H1X #H2X destruct -G0 -L0 -T0
154 elim (cpm_teqx_inv_atom_sn … H1X H2X) -H1X -H2X *
155 [ #H1 #H2 destruct /2 width=1 by/
156 | #s #H1 #H2 #H3 destruct /2 width=1 by/
158 | #p #I #V #T1 #HG #HL #HT #H0 #HF #X #H1X #H2X destruct
159 elim (cpm_teqx_inv_bind_sn … H0 … H1X H2X) -H0 -H1X -H2X #T2 #HV #HT1 #H1T12 #H2T12 #H destruct
161 | #V #T1 #HG #HL #HT #H0 #HF #X #H1X #H2X destruct
162 elim (cpm_teqx_inv_appl_sn … H0 … H1X H2X) -H0 -H1X -H2X #m #q #W #U1 #T2 #Hm #HV #HVW #HTU1 #HT1 #H1T12 #H2T12 #H destruct
164 | #U1 #T1 #HG #HL #HT #H0 #HF #X #H1X #H2X destruct
165 elim (cpm_teqx_inv_cast_sn … H0 … H1X H2X) -H0 -H1X -H2X #U0 #U2 #T2 #HU10 #HT1U0 #HU1 #H1U12 #H2U12 #HT1 #H1T12 #H2T12 #H destruct
170 (* Advanced properties with restricted rt-transition for terms **************)
172 lemma cpm_teqx_free (h) (a) (n) (G) (L):
173 ∀T1. ❨G,L❩ ⊢ T1 ![h,a] →
174 ∀T2. ❨G,L❩ ⊢ T1 ➡[h,n] T2 → T1 ≅ T2 →
175 ∀F,K. ❨F,K❩ ⊢ T1 ➡[h,n] T2.
176 #h #a #n #G #L #T1 #H0 #T2 #H1 #H2
177 @(cpm_teqx_ind … H0 … H1 H2) -L -T1 -T2
178 [ #I #L #H #F #K destruct //
179 | #L #s #H #F #K destruct //
180 | #p #I #L #V #T1 #_ #_ #T2 #_ #_ #IH #F #K
181 /2 width=1 by cpm_bind/
182 | #m #_ #L #V #_ #W #_ #q #T1 #U1 #_ #_ #T2 #_ #_ #IH #F #K
183 /2 width=1 by cpm_appl/
184 | #L #U0 #U1 #T1 #_ #_ #U2 #_ #_ #_ #T2 #_ #_ #_ #IHU #IHT #F #K
185 /2 width=1 by cpm_cast/
189 (* Advanced inversion lemmas with restricted rt-transition for terms ********)
191 lemma cpm_teqx_inv_bind_sn_void (h) (a) (n) (p) (I) (G) (L):
192 ∀V,T1. ❨G,L❩ ⊢ ⓑ[p,I]V.T1 ![h,a] →
193 ∀X. ❨G,L❩ ⊢ ⓑ[p,I]V.T1 ➡[h,n] X → ⓑ[p,I]V.T1 ≅ X →
194 ∃∃T2. ❨G,L❩ ⊢ V ![h,a] & ❨G,L.ⓑ[I]V❩ ⊢ T1 ![h,a] & ❨G,L.ⓧ❩ ⊢ T1 ➡[h,n] T2 & T1 ≅ T2 & X = ⓑ[p,I]V.T2.
195 #h #a #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
196 elim (cpm_teqx_inv_bind_sn … H0 … H1 H2) -H0 -H1 -H2 #T2 #HV #HT1 #H1T12 #H2T12 #H
197 /3 width=5 by ex5_intro, cpm_teqx_free/