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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 "basic_2/grammar/tstc_vector.ma".
16 include "basic_2/relocation/lift_vector.ma".
17 include "basic_2/computation/cpxs_tstc.ma".
19 (* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
21 (* Vector form of forward lemmas involving same top term constructor ********)
23 (* Basic_1: was just: nf2_iso_appls_lref *)
24 lemma cpxs_fwd_cnx_vector: ∀h,g,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ →
25 ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.T ➡*[h, g] U → ⒶVs.T ≃ U.
26 #h #g #G #L #T #H1T #H2T #Vs elim Vs -Vs [ @(cpxs_fwd_cnx … H2T) ] (**) (* /2 width=3 by cpxs_fwd_cnx/ does not work *)
28 elim (cpxs_inv_appl1 … H) -H *
29 [ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by tstc_pair/
31 lapply (IHVs … HT0) -IHVs -HT0 #HT0
32 elim (tstc_inv_bind_appls_simple … HT0) //
33 | #a #V1 #V2 #V0 #T0 #HV1 #HV12 #HT0 #HU
34 lapply (IHVs … HT0) -IHVs -HT0 #HT0
35 elim (tstc_inv_bind_appls_simple … HT0) //
39 lemma cpxs_fwd_sort_vector: ∀h,g,G,L,k,Vs,U. ⦃G, L⦄ ⊢ ⒶVs.⋆k ➡*[h, g] U →
40 ⒶVs.⋆k ≃ U ∨ ⦃G, L⦄ ⊢ ⒶVs.⋆(next h k) ➡*[h, g] U.
41 #h #g #G #L #k #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_sort/
43 elim (cpxs_inv_appl1 … H) -H *
44 [ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by tstc_pair, or_introl/
46 elim (IHVs … HT1) -IHVs -HT1 #HT1
47 [ elim (tstc_inv_bind_appls_simple … HT1) //
48 | @or_intror (**) (* explicit constructor *)
50 @(cpxs_strap1 … (ⓐV.ⓛ{a}W1.T1)) /3 width=1 by cpxs_flat_dx, cpr_cpx, cpr_beta/
52 | #a #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
53 elim (IHVs … HT1) -IHVs -HT1 #HT1
54 [ elim (tstc_inv_bind_appls_simple … HT1) //
55 | @or_intror (**) (* explicit constructor *)
57 @(cpxs_strap1 … (ⓐV1.ⓓ{a}V3.T1)) /3 width=3 by cpxs_flat, cpr_cpx, cpr_theta/
63 (* Basic_1: was just: pr3_iso_appls_beta *)
64 lemma cpxs_fwd_beta_vector: ∀h,g,a,G,L,Vs,V,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓐV.ⓛ{a}W.T ➡*[h, g] U →
65 ⒶVs. ⓐV. ⓛ{a}W. T ≃ U ∨ ⦃G, L⦄ ⊢ ⒶVs.ⓓ{a}ⓝW.V.T ➡*[h, g] U.
66 #h #g #a #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_beta/
67 #V0 #Vs #IHVs #V #W #T #U #H
68 elim (cpxs_inv_appl1 … H) -H *
69 [ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by tstc_pair, or_introl/
71 elim (IHVs … HT1) -IHVs -HT1 #HT1
72 [ elim (tstc_inv_bind_appls_simple … HT1) //
73 | @or_intror (**) (* explicit constructor *)
75 @(cpxs_strap1 … (ⓐV0.ⓛ{b}W1.T1)) /3 width=1 by cpxs_flat_dx, cpr_cpx, cpr_beta/
77 | #b #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
78 elim (IHVs … HT1) -IHVs -HT1 #HT1
79 [ elim (tstc_inv_bind_appls_simple … HT1) //
80 | @or_intror (**) (* explicit constructor *)
82 @(cpxs_strap1 … (ⓐV1.ⓓ{b}V3.T1)) /3 width=3 by cpxs_flat, cpr_cpx, cpr_theta/
87 lemma cpxs_fwd_delta_vector: ∀h,g,I,G,L,K,V1,i. ⇩[i] L ≡ K.ⓑ{I}V1 →
88 ∀V2. ⇧[0, i + 1] V1 ≡ V2 →
89 ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.#i ➡*[h, g] U →
90 ⒶVs.#i ≃ U ∨ ⦃G, L⦄ ⊢ ⒶVs.V2 ➡*[h, g] U.
91 #h #g #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs /2 width=5 by cpxs_fwd_delta/
92 #V #Vs #IHVs #U #H -K -V1
93 elim (cpxs_inv_appl1 … H) -H *
94 [ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by tstc_pair, or_introl/
96 elim (IHVs … HT0) -IHVs -HT0 #HT0
97 [ elim (tstc_inv_bind_appls_simple … HT0) //
98 | @or_intror -i (**) (* explicit constructor *)
100 @(cpxs_strap1 … (ⓐV.ⓛ{b}W0.T0)) /3 width=1 by cpxs_flat_dx, cpr_cpx, cpr_beta/
102 | #b #V0 #V1 #V3 #T0 #HV0 #HV01 #HT0 #HU
103 elim (IHVs … HT0) -IHVs -HT0 #HT0
104 [ elim (tstc_inv_bind_appls_simple … HT0) //
105 | @or_intror -i (**) (* explicit constructor *)
106 @(cpxs_trans … HU) -U
107 @(cpxs_strap1 … (ⓐV0.ⓓ{b}V3.T0)) /3 width=3 by cpxs_flat, cpr_cpx, cpr_theta/
112 (* Basic_1: was just: pr3_iso_appls_abbr *)
113 lemma cpxs_fwd_theta_vector: ∀h,g,G,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
114 ∀a,V,T,U. ⦃G, L⦄ ⊢ ⒶV1s.ⓓ{a}V.T ➡*[h, g] U →
115 ⒶV1s. ⓓ{a}V. T ≃ U ∨ ⦃G, L⦄ ⊢ ⓓ{a}V.ⒶV2s.T ➡*[h, g] U.
116 #h #g #G #L #V1s #V2s * -V1s -V2s /3 width=1 by or_intror/
117 #V1s #V2s #V1a #V2a #HV12a #HV12s #a
118 generalize in match HV12a; -HV12a
119 generalize in match V2a; -V2a
120 generalize in match V1a; -V1a
121 elim HV12s -V1s -V2s /2 width=1 by cpxs_fwd_theta/
122 #V1s #V2s #V1b #V2b #HV12b #_ #IHV12s #V1a #V2a #HV12a #V #T #U #H
123 elim (cpxs_inv_appl1 … H) -H *
124 [ -IHV12s -HV12a -HV12b #V0 #T0 #_ #_ #H destruct /2 width=1 by tstc_pair, or_introl/
125 | #b #W0 #T0 #HT0 #HU
126 elim (IHV12s … HV12b … HT0) -IHV12s -HT0 #HT0
128 elim (tstc_inv_pair1 … HT0) #V1 #T1 #H destruct
129 | @or_intror -V1s (**) (* explicit constructor *)
130 @(cpxs_trans … HU) -U
131 elim (cpxs_inv_abbr1 … HT0) -HT0 *
132 [ -HV12a -HV12b #V1 #T1 #_ #_ #H destruct
133 | -V1b #X #HT1 #H #H0 destruct
134 elim (lift_inv_bind1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
135 @(cpxs_trans … (+ⓓV.ⓐV2a.ⓛ{b}W1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2s
136 @(cpxs_strap2 … (ⓐV1a.ⓛ{b}W0.T0))
137 /4 width=7 by cpxs_beta_dx, cpx_zeta, lift_bind, lift_flat/
140 | #b #V0a #Va #V0 #T0 #HV10a #HV0a #HT0 #HU
141 elim (IHV12s … HV12b … HT0) -HV12b -IHV12s -HT0 #HT0
142 [ -HV12a -HV10a -HV0a -HU
143 elim (tstc_inv_pair1 … HT0) #V1 #T1 #H destruct
144 | @or_intror -V1s -V1b (**) (* explicit constructor *)
145 @(cpxs_trans … HU) -U
146 elim (cpxs_inv_abbr1 … HT0) -HT0 *
147 [ #V1 #T1 #HV1 #HT1 #H destruct
148 lapply (cpxs_lift … HV10a (L.ⓓV) (Ⓣ) … HV12a … HV0a) -V1a -V0a [ /2 width=1 by ldrop_drop/ ] #HV2a
149 @(cpxs_trans … (ⓓ{a}V.ⓐV2a.T1)) /3 width=1 by cpxs_bind, cpxs_pair_sn, cpxs_flat_dx, cpxs_bind_dx/
150 | #X #HT1 #H #H0 destruct
151 elim (lift_inv_bind1 … H) -H #V1 #T1 #HW01 #HT01 #H destruct
152 lapply (cpxs_lift … HV10a (L.ⓓV0) (Ⓣ) … HV12a … HV0a) -V0a [ /2 width=1 by ldrop_drop/ ] #HV2a
153 @(cpxs_trans … (+ⓓV.ⓐV2a.ⓓ{b}V1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2s
154 @(cpxs_strap2 … (ⓐV1a.ⓓ{b}V0.T0)) [ /4 width=7 by cpx_zeta, lift_bind, lift_flat/ ] -V -V1 -T1
155 @(cpxs_strap2 … (ⓓ{b}V0.ⓐV2a.T0)) /3 width=3 by cpxs_pair_sn, cpxs_bind_dx, cpr_cpx, cpr_theta/
161 (* Basic_1: was just: pr3_iso_appls_cast *)
162 lemma cpxs_fwd_cast_vector: ∀h,g,G,L,Vs,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓝW.T ➡*[h, g] U →
164 | ⦃G, L⦄ ⊢ ⒶVs.T ➡*[h, g] U
165 | ⦃G, L⦄ ⊢ ⒶVs.W ➡*[h, g] U.
166 #h #g #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_cast/
167 #V #Vs #IHVs #W #T #U #H
168 elim (cpxs_inv_appl1 … H) -H *
169 [ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by tstc_pair, or3_intro0/
170 | #b #W0 #T0 #HT0 #HU elim (IHVs … HT0) -IHVs -HT0 #HT0
171 [ elim (tstc_inv_bind_appls_simple … HT0) //
172 | @or3_intro1 -W (**) (* explicit constructor *)
173 @(cpxs_trans … HU) -U
174 @(cpxs_strap1 … (ⓐV.ⓛ{b}W0.T0)) /2 width=1 by cpxs_flat_dx, cpx_beta/
175 | @or3_intro2 -T (**) (* explicit constructor *)
176 @(cpxs_trans … HU) -U
177 @(cpxs_strap1 … (ⓐV.ⓛ{b}W0.T0)) /2 width=1 by cpxs_flat_dx, cpx_beta/
179 | #b #V0 #V1 #V2 #T0 #HV0 #HV01 #HT0 #HU
180 elim (IHVs … HT0) -IHVs -HT0 #HT0
181 [ elim (tstc_inv_bind_appls_simple … HT0) //
182 | @or3_intro1 -W (**) (* explicit constructor *)
183 @(cpxs_trans … HU) -U
184 @(cpxs_strap1 … (ⓐV0.ⓓ{b}V2.T0)) /2 width=3 by cpxs_flat, cpx_theta/
185 | @or3_intro2 -T (**) (* explicit constructor *)
186 @(cpxs_trans … HU) -U
187 @(cpxs_strap1 … (ⓐV0.ⓓ{b}V2.T0)) /2 width=3 by cpxs_flat, cpx_theta/