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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 *)
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15 include "static_2/syntax/theq_simple_vector.ma".
16 include "static_2/relocation/lifts_vector.ma".
17 include "basic_2/rt_computation/cpxs_theq.ma".
19 (* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
21 (* Vector form of forward lemmas with head equivalence for terms ************)
23 lemma cpxs_fwd_sort_vector: ∀h,G,L,s,Vs,X2. ⦃G,L⦄ ⊢ ⒶVs.⋆s ⬈*[h] X2 → ⒶVs.⋆s ⩳ X2.
24 #h #G #L #s #Vs elim Vs -Vs /2 width=4 by cpxs_fwd_sort/
26 elim (cpxs_inv_appl1 … H) -H *
27 [ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by theq_pair/
29 lapply (IHVs … HT1) -IHVs -HT1 #HT1
30 elim (theq_inv_applv_bind_simple … HT1) //
31 | #p #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
32 lapply (IHVs … HT1) -IHVs -HT1 #HT1
33 elim (theq_inv_applv_bind_simple … HT1) //
37 (* Basic_2A1: was: cpxs_fwd_delta_vector *)
38 lemma cpxs_fwd_delta_drops_vector: ∀h,I,G,L,K,V1,i. ⬇*[i] L ≘ K.ⓑ{I}V1 →
40 ∀Vs,X2. ⦃G,L⦄ ⊢ ⒶVs.#i ⬈*[h] X2 →
41 ∨∨ ⒶVs.#i ⩳ X2 | ⦃G,L⦄ ⊢ ⒶVs.V2 ⬈*[h] X2.
42 #h #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs
43 elim Vs -Vs /2 width=5 by cpxs_fwd_delta_drops/
44 #V #Vs #IHVs #X2 #H -K -V1
45 elim (cpxs_inv_appl1 … H) -H *
46 [ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/
48 elim (IHVs … HT0) -IHVs -HT0 #HT0
49 [ elim (theq_inv_applv_bind_simple … HT0) //
50 | @or_intror -i (**) (* explicit constructor *)
51 @(cpxs_trans … HU) -X2
52 @(cpxs_strap1 … (ⓐV.ⓛ{q}W0.T0)) /3 width=1 by cpxs_flat_dx, cpx_beta/
54 | #q #V0 #V1 #V3 #T0 #HV0 #HV01 #HT0 #HU
55 elim (IHVs … HT0) -IHVs -HT0 #HT0
56 [ elim (theq_inv_applv_bind_simple … HT0) //
57 | @or_intror -i (**) (* explicit constructor *)
58 @(cpxs_trans … HU) -X2
59 @(cpxs_strap1 … (ⓐV0.ⓓ{q}V3.T0)) /3 width=3 by cpxs_flat, cpx_theta/
64 (* Basic_1: was just: pr3_iso_appls_beta *)
65 lemma cpxs_fwd_beta_vector: ∀h,p,G,L,Vs,V,W,T,X2. ⦃G,L⦄ ⊢ ⒶVs.ⓐV.ⓛ{p}W.T ⬈*[h] X2 →
66 ∨∨ ⒶVs.ⓐV.ⓛ{p}W. T ⩳ X2 | ⦃G,L⦄ ⊢ ⒶVs.ⓓ{p}ⓝW.V.T ⬈*[h] X2.
67 #h #p #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_beta/
68 #V0 #Vs #IHVs #V #W #T #X2 #H
69 elim (cpxs_inv_appl1 … H) -H *
70 [ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/
72 elim (IHVs … HT1) -IHVs -HT1 #HT1
73 [ elim (theq_inv_applv_bind_simple … HT1) //
74 | @or_intror (**) (* explicit constructor *)
75 @(cpxs_trans … HU) -X2
76 @(cpxs_strap1 … (ⓐV0.ⓛ{q}W1.T1)) /3 width=1 by cpxs_flat_dx, cpx_beta/
78 | #q #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
79 elim (IHVs … HT1) -IHVs -HT1 #HT1
80 [ elim (theq_inv_applv_bind_simple … HT1) //
81 | @or_intror (**) (* explicit constructor *)
82 @(cpxs_trans … HU) -X2
83 @(cpxs_strap1 … (ⓐV1.ⓓ{q}V3.T1)) /3 width=3 by cpxs_flat, cpx_theta/
88 (* Basic_1: was just: pr3_iso_appls_abbr *)
89 lemma cpxs_fwd_theta_vector: ∀h,G,L,V1b,V2b. ⬆*[1] V1b ≘ V2b →
90 ∀p,V,T,X2. ⦃G,L⦄ ⊢ ⒶV1b.ⓓ{p}V.T ⬈*[h] X2 →
91 ∨∨ ⒶV1b.ⓓ{p}V.T ⩳ X2 | ⦃G,L⦄ ⊢ ⓓ{p}V.ⒶV2b.T ⬈*[h] X2.
92 #h #G #L #V1b #V2b * -V1b -V2b /3 width=1 by or_intror/
93 #V1b #V2b #V1a #V2a #HV12a #HV12b #p
94 generalize in match HV12a; -HV12a
95 generalize in match V2a; -V2a
96 generalize in match V1a; -V1a
97 elim HV12b -V1b -V2b /2 width=1 by cpxs_fwd_theta/
98 #V1b #V2b #V1b #V2b #HV12b #_ #IHV12b #V1a #V2a #HV12a #V #T #X2 #H
99 elim (cpxs_inv_appl1 … H) -H *
100 [ -IHV12b -HV12a -HV12b #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/
101 | #q #W0 #T0 #HT0 #HU
102 elim (IHV12b … HV12b … HT0) -IHV12b -HT0 #HT0
104 elim (theq_inv_pair1 … HT0) #V1 #T1 #H destruct
105 | @or_intror -V1b (**) (* explicit constructor *)
106 @(cpxs_trans … HU) -X2
107 elim (cpxs_inv_abbr1_dx … HT0) -HT0 *
108 [ -HV12a #V1 #T1 #_ #_ #H destruct
109 | -V1b #X #HT1 #H #H0 destruct
110 elim (lifts_inv_bind1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
111 @(cpxs_trans … (+ⓓV.ⓐV2a.ⓛ{q}W1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2b
112 @(cpxs_strap2 … (ⓐV1a.ⓛ{q}W0.T0))
113 /4 width=7 by cpxs_beta_dx, cpx_zeta, lifts_bind, lifts_flat/
116 | #q #V0a #Va #V0 #T0 #HV10a #HV0a #HT0 #HU
117 elim (IHV12b … HV12b … HT0) -HV12b -IHV12b -HT0 #HT0
118 [ -HV12a -HV10a -HV0a -HU
119 elim (theq_inv_pair1 … HT0) #V1 #T1 #H destruct
120 | @or_intror -V1b -V1b (**) (* explicit constructor *)
121 @(cpxs_trans … HU) -X2
122 elim (cpxs_inv_abbr1_dx … HT0) -HT0 *
123 [ #V1 #T1 #HV1 #HT1 #H destruct
124 lapply (cpxs_lifts_bi … HV10a (Ⓣ) … (L.ⓓV) … HV12a … HV0a) -V1a -V0a /3 width=1 by drops_refl, drops_drop/ #HV2a
125 @(cpxs_trans … (ⓓ{p}V.ⓐV2a.T1)) /3 width=1 by cpxs_bind, cpxs_pair_sn, cpxs_flat_dx, cpxs_bind_dx/
126 | #X #HT1 #H #H0 destruct
127 elim (lifts_inv_bind1 … H) -H #V1 #T1 #HW01 #HT01 #H destruct
128 lapply (cpxs_lifts_bi … HV10a (Ⓣ) … (L.ⓓV0) … HV12a … HV0a) -V0a /3 width=1 by drops_refl, drops_drop/ #HV2a
129 @(cpxs_trans … (+ⓓV.ⓐV2a.ⓓ{q}V1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2b
130 @(cpxs_strap2 … (ⓐV1a.ⓓ{q}V0.T0)) [ /4 width=7 by cpx_zeta, lifts_bind, lifts_flat/ ] -V -V1 -T1
131 @(cpxs_strap2 … (ⓓ{q}V0.ⓐV2a.T0)) /3 width=3 by cpxs_pair_sn, cpxs_bind_dx, cpx_theta/
137 (* Basic_1: was just: pr3_iso_appls_cast *)
138 lemma cpxs_fwd_cast_vector: ∀h,G,L,Vs,W,T,X2. ⦃G,L⦄ ⊢ ⒶVs.ⓝW.T ⬈*[h] X2 →
140 | ⦃G,L⦄ ⊢ ⒶVs.T ⬈*[h] X2
141 | ⦃G,L⦄ ⊢ ⒶVs.W ⬈*[h] X2.
142 #h #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_cast/
143 #V #Vs #IHVs #W #T #X2 #H
144 elim (cpxs_inv_appl1 … H) -H *
145 [ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or3_intro0/
146 | #q #W0 #T0 #HT0 #HU elim (IHVs … HT0) -IHVs -HT0 #HT0
147 [ elim (theq_inv_applv_bind_simple … HT0) //
148 | @or3_intro1 -W (**) (* explicit constructor *)
149 @(cpxs_trans … HU) -X2
150 @(cpxs_strap1 … (ⓐV.ⓛ{q}W0.T0)) /2 width=1 by cpxs_flat_dx, cpx_beta/
151 | @or3_intro2 -T (**) (* explicit constructor *)
152 @(cpxs_trans … HU) -X2
153 @(cpxs_strap1 … (ⓐV.ⓛ{q}W0.T0)) /2 width=1 by cpxs_flat_dx, cpx_beta/
155 | #q #V0 #V1 #V2 #T0 #HV0 #HV01 #HT0 #HU
156 elim (IHVs … HT0) -IHVs -HT0 #HT0
157 [ elim (theq_inv_applv_bind_simple … HT0) //
158 | @or3_intro1 -W (**) (* explicit constructor *)
159 @(cpxs_trans … HU) -X2
160 @(cpxs_strap1 … (ⓐV0.ⓓ{q}V2.T0)) /2 width=3 by cpxs_flat, cpx_theta/
161 | @or3_intro2 -T (**) (* explicit constructor *)
162 @(cpxs_trans … HU) -X2
163 @(cpxs_strap1 … (ⓐV0.ⓓ{q}V2.T0)) /2 width=3 by cpxs_flat, cpx_theta/
168 (* Basic_1: was just: nf2_iso_appls_lref *)
169 lemma cpxs_fwd_cnx_vector: ∀h,G,L,T. 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ →
170 ∀Vs,X2. ⦃G,L⦄ ⊢ ⒶVs.T ⬈*[h] X2 → ⒶVs.T ⩳ X2.
171 #h #G #L #T #H1T #H2T #Vs elim Vs -Vs [ @(cpxs_fwd_cnx … H2T) ] (**) (* /2 width=3 by cpxs_fwd_cnx/ does not work *)
173 elim (cpxs_inv_appl1 … H) -H *
174 [ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair/
175 | #p #W0 #T0 #HT0 #HU
176 lapply (IHVs … HT0) -IHVs -HT0 #HT0
177 elim (theq_inv_applv_bind_simple … HT0) //
178 | #p #V1 #V2 #V0 #T0 #HV1 #HV12 #HT0 #HU
179 lapply (IHVs … HT0) -IHVs -HT0 #HT0
180 elim (theq_inv_applv_bind_simple … HT0) //