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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/substitution/lift_vector.ma".
17 include "basic_2/computation/cprs_tstc.ma".
19 (* CONTEXT-SENSITIVE 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 cprs_fwd_cnf_vector: ∀L,T. 𝐒⦃T⦄ → L ⊢ 𝐍⦃T⦄ → ∀Vs,U. L ⊢ ⒶVs.T ➡* U → ⒶVs.T ≃ U.
25 #L #T #H1T #H2T #Vs elim Vs -Vs [ @(cprs_fwd_cnf … H2T) ] (**) (* /2 width=3 by cprs_fwd_cnf/ does not work *)
27 elim (cprs_inv_appl1 … H) -H *
28 [ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1/
29 | #a #V0 #W0 #T0 #HV0 #HT0 #HU
30 lapply (IHVs … HT0) -IHVs -HT0 #HT0
31 elim (tstc_inv_bind_appls_simple … HT0 ?) //
32 | #a #V1 #V2 #V0 #T0 #HV1 #HV12 #HT0 #HU
33 lapply (IHVs … HT0) -IHVs -HT0 #HT0
34 elim (tstc_inv_bind_appls_simple … HT0 ?) //
38 (* Basic_1: was: pr3_iso_appls_beta *)
39 lemma cprs_fwd_beta_vector: ∀a,L,Vs,V,W,T,U. L ⊢ ⒶVs. ⓐV. ⓛ{a}W. T ➡* U →
40 ⒶVs. ⓐV. ⓛ{a}W. T ≃ U ∨ L ⊢ ⒶVs. ⓓ{a}V. T ➡* U.
41 #a #L #Vs elim Vs -Vs /2 width=1 by cprs_fwd_beta/
42 #V0 #Vs #IHVs #V #W #T #U #H
43 elim (cprs_inv_appl1 … H) -H *
44 [ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1/
45 | #b #V1 #W1 #T1 #HV01 #HT1 #HU
46 elim (IHVs … HT1) -IHVs -HT1 #HT1
47 [ elim (tstc_inv_bind_appls_simple … HT1 ?) //
48 | @or_intror -W (**) (* explicit constructor *)
50 @(cprs_strap1 … (ⓐV1.ⓛ{b}W1.T1)) [ /2 width=1/ ] -V -V0 -Vs -T /3 width=1/
52 | #b #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 -W (**) (* explicit constructor *)
57 @(cprs_strap1 … (ⓐV1.ⓓ{b}V3.T1)) [ /2 width=1/ ] -V -V0 -Vs -T /3 width=3/
62 lemma cprs_fwd_delta_vector: ∀L,K,V1,i. ⇩[0, i] L ≡ K. ⓓV1 →
63 ∀V2. ⇧[0, i + 1] V1 ≡ V2 →
64 ∀Vs,U. L ⊢ ⒶVs.#i ➡* U →
65 ⒶVs.#i ≃ U ∨ L ⊢ ⒶVs.V2 ➡* U.
66 #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs /2 width=4 by cprs_fwd_delta/
67 #V #Vs #IHVs #U #H -K -V1
68 elim (cprs_inv_appl1 … H) -H *
69 [ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1/
70 | #b #V0 #W0 #T0 #HV0 #HT0 #HU
71 elim (IHVs … HT0) -IHVs -HT0 #HT0
72 [ elim (tstc_inv_bind_appls_simple … HT0 ?) //
73 | @or_intror -i (**) (* explicit constructor *)
75 @(cprs_strap1 … (ⓐV0.ⓛ{b}W0.T0)) [ /2 width=1/ ] -V -V2 -Vs /3 width=1/
77 | #b #V0 #V1 #V3 #T0 #HV0 #HV01 #HT0 #HU
78 elim (IHVs … HT0) -IHVs -HT0 #HT0
79 [ elim (tstc_inv_bind_appls_simple … HT0 ?) //
80 | @or_intror -i (**) (* explicit constructor *)
82 @(cprs_strap1 … (ⓐV0.ⓓ{b}V3.T0)) [ /2 width=1/ ] -V -V2 -Vs /3 width=3/
87 (* Basic_1: was: pr3_iso_appls_abbr *)
88 lemma cprs_fwd_theta_vector: ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
89 ∀a,V,T,U. L ⊢ ⒶV1s. ⓓ{a}V. T ➡* U →
90 ⒶV1s. ⓓ{a}V. T ≃ U ∨ L ⊢ ⓓ{a}V. ⒶV2s. T ➡* U.
91 #L #V1s #V2s * -V1s -V2s /3 width=1/
92 #V1s #V2s #V1a #V2a #HV12a #HV12s #a
93 generalize in match HV12a; -HV12a
94 generalize in match V2a; -V2a
95 generalize in match V1a; -V1a
96 elim HV12s -V1s -V2s /2 width=1 by cprs_fwd_theta/
97 #V1s #V2s #V1b #V2b #HV12b #_ #IHV12s #V1a #V2a #HV12a #V #T #U #H
98 elim (cprs_inv_appl1 … H) -H *
99 [ -IHV12s -HV12a -HV12b #V0 #T0 #_ #_ #H destruct /2 width=1/
100 | #b #V0a #W0 #T0 #HV10a #HT0 #HU
101 elim (IHV12s … HV12b … HT0) -IHV12s -HT0 #HT0
102 [ -HV12a -HV12b -HV10a -HU
103 elim (tstc_inv_pair1 … HT0) #V1 #T1 #H destruct
104 | @or_intror -V1s (**) (* explicit constructor *)
105 @(cprs_trans … HU) -U
106 elim (cprs_inv_abbr1 … HT0) -HT0 *
107 [ -HV12a -HV12b -HV10a #V1 #T1 #_ #_ #H destruct
108 | -V1b #X #HT1 #H #H0 destruct
109 elim (lift_inv_bind1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
110 @(cprs_trans … (+ⓓV.ⓐV2a.ⓛ{b}W1.T1)) [ /3 width=1/ ] -T -V2b -V2s
111 @(cprs_strap2 … (ⓐV1a.ⓛ{b}W0.T0)) [ /5 width=7/ ] -V -V2a -W1 -T1
112 @(cprs_strap2 … (ⓓ{b}V1a.T0)) [ /3 width=1/ ] -W0 /2 width=1/
115 | #b #V0a #Va #V0 #T0 #HV10a #HV0a #HT0 #HU
116 elim (IHV12s … HV12b … HT0) -HV12b -IHV12s -HT0 #HT0
117 [ -HV12a -HV10a -HV0a -HU
118 elim (tstc_inv_pair1 … HT0) #V1 #T1 #H destruct
119 | @or_intror -V1s -V1b (**) (* explicit constructor *)
120 @(cprs_trans … HU) -U
121 elim (cprs_inv_abbr1 … HT0) -HT0 *
122 [ #V1 #T1 #HV1 #HT1 #H destruct
123 lapply (cprs_lift (L.ⓓV) … HV12a … HV10a … HV0a) -V1a -V0a [ /2 width=1/ ] #HV2a
124 @(cprs_trans … (ⓓ{a}V.ⓐV2a.T1)) [ /3 width=1/ ] -T -V2b -V2s /3 width=1/
125 | #X #HT1 #H #H0 destruct
126 elim (lift_inv_bind1 … H) -H #V1 #T1 #HW01 #HT01 #H destruct
127 lapply (cprs_lift (L.ⓓV0) … HV12a … HV10a … HV0a) -V0a [ /2 width=1/ ] #HV2a
128 @(cprs_trans … (+ⓓV.ⓐV2a.ⓓ{b}V1.T1)) [ /3 width=1/ ] -T -V2b -V2s
129 @(cprs_strap2 … (ⓐV1a.ⓓ{b}V0.T0)) [ /5 width=7/ ] -V -V1 -T1
130 @(cprs_strap2 … (ⓓ{b}V0.ⓐV2a.T0)) [ /3 width=3/ ] -V1a /3 width=1/
136 (* Basic_1: was: pr3_iso_appls_cast *)
137 lemma cprs_fwd_tau_vector: ∀L,Vs,W,T,U. L ⊢ ⒶVs. ⓝW. T ➡* U →
138 ⒶVs. ⓝW. T ≃ U ∨ L ⊢ ⒶVs. T ➡* U.
139 #L #Vs elim Vs -Vs /2 width=1 by cprs_fwd_tau/
140 #V #Vs #IHVs #W #T #U #H
141 elim (cprs_inv_appl1 … H) -H *
142 [ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1/
143 | #b #V0 #W0 #T0 #HV0 #HT0 #HU
144 elim (IHVs … HT0) -IHVs -HT0 #HT0
145 [ elim (tstc_inv_bind_appls_simple … HT0 ?) //
146 | @or_intror -W (**) (* explicit constructor *)
147 @(cprs_trans … HU) -U
148 @(cprs_strap1 … (ⓐV0.ⓛ{b}W0.T0)) [ /2 width=1/ ] -V -Vs -T /3 width=1/
150 | #b #V0 #V1 #V2 #T0 #HV0 #HV01 #HT0 #HU
151 elim (IHVs … HT0) -IHVs -HT0 #HT0
152 [ elim (tstc_inv_bind_appls_simple … HT0 ?) //
153 | @or_intror -W (**) (* explicit constructor *)
154 @(cprs_trans … HU) -U
155 @(cprs_strap1 … (ⓐV0.ⓓ{b}V2.T0)) [ /2 width=1/ ] -V -Vs -T /3 width=3/