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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/reduction/llpr_llpr.ma".
16 include "basic_2/computation/cprs_lift.ma".
18 (* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
20 (* Main properties **********************************************************)
22 (* Basic_1: was: pr3_t *)
23 (* Basic_1: includes: pr1_t *)
24 theorem cprs_trans: ∀G,L. Transitive … (cprs G L).
25 normalize /2 width=3 by trans_TC/ qed-.
27 (* Basic_1: was: pr3_confluence *)
28 (* Basic_1: includes: pr1_confluence *)
29 theorem cprs_conf: ∀G,L. confluent2 … (cprs G L) (cprs G L).
30 normalize /3 width=3 by cpr_conf, TC_confluent2/ qed-.
32 theorem cprs_bind: ∀a,I,G,L,V1,V2,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
33 ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
34 #a #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2
35 /3 width=5 by cprs_trans, cprs_bind_dx/
38 (* Basic_1: was: pr3_flat *)
39 theorem cprs_flat: ∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
40 ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡* ⓕ{I}V2.T2.
41 #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2
42 /3 width=3 by cprs_flat_dx, cprs_strap1, cpr_pair_sn/
45 theorem cprs_beta_rc: ∀a,G,L,V1,V2,W1,W2,T1,T2.
46 ⦃G, L⦄ ⊢ V1 ➡ V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ W1 ➡* W2 →
47 ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
48 #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cprs_ind … H) -W2 /2 width=1 by cprs_beta_dx/
49 #W #W2 #_ #HW2 #IHW1 (**) (* fulla uto too slow 14s *)
50 @(cprs_trans … IHW1) -IHW1 /3 width=1 by cprs_flat_dx, cprs_bind/
53 theorem cprs_beta: ∀a,G,L,V1,V2,W1,W2,T1,T2.
54 ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
55 ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
56 #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cprs_ind … H) -V2 /2 width=1 by cprs_beta_rc/
58 @(cprs_trans … IHV1) -IHV1 /3 width=1 by cprs_flat_sn, cprs_bind/
61 theorem cprs_theta_rc: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
62 ⦃G, L⦄ ⊢ V1 ➡ V → ⇧[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
63 ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
64 #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H elim H -W2
65 /3 width=5 by cprs_trans, cprs_theta_dx, cprs_bind_dx/
68 theorem cprs_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
69 ⇧[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
70 ⦃G, L⦄ ⊢ V1 ➡* V → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
71 #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1
72 /3 width=3 by cprs_trans, cprs_theta_rc, cprs_flat_dx/
75 (* Advanced inversion lemmas ************************************************)
77 (* Basic_1: was pr3_gen_appl *)
78 lemma cprs_inv_appl1: ∀G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡* U2 →
79 ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L⦄ ⊢ T1 ➡* T2 &
81 | ∃∃a,W,T. ⦃G, L⦄ ⊢ T1 ➡* ⓛ{a}W.T &
82 ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ➡* U2
83 | ∃∃a,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ➡* V0 & ⇧[0,1] V0 ≡ V2 &
84 ⦃G, L⦄ ⊢ T1 ➡* ⓓ{a}V.T &
85 ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡* U2.
86 #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
88 [ #V0 #T0 #HV10 #HT10 #H destruct
89 elim (cpr_inv_appl1 … HU2) -HU2 *
90 [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5 by cprs_strap1, or3_intro0, ex3_2_intro/
91 | #a #V2 #W #W2 #T #T2 #HV02 #HW2 #HT2 #H1 #H2 destruct
92 lapply (cprs_strap1 … HV10 … HV02) -V0 #HV12
93 lapply (lsubr_cpr_trans … HT2 (L.ⓓⓝW.V1) ?) -HT2
94 /5 width=5 by cprs_flat_dx, cpr_cprs, cprs_bind, lsubr_abst, ex2_3_intro, or3_intro1/
95 | #a #V #V2 #W0 #W2 #T #T2 #HV0 #HV2 #HW02 #HT2 #H1 #H2 destruct
96 @or3_intro2 @(ex4_5_intro … HV2 HT10) /3 width=3 by cprs_flat_sn, cprs_strap1, cpr_cprs, cprs_bind/ (**) (* full auto is too slow 11s *)
98 | /4 width=9 by cprs_strap1, or3_intro1, ex2_3_intro/
99 | /4 width=11 by cprs_strap1, or3_intro2, ex4_5_intro/
103 (* Properties concerning sn parallel reduction on local environments ********)
105 (* Basic_1: was just: pr3_pr2_pr2_t *)
106 (* Basic_1: includes: pr3_pr0_pr2_t *)
107 lemma llpr_cpr_trans: ∀G. s_r_transitive … (cpr G) (llpr G 0).
108 #G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
110 | #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
111 elim (llpr_inv_lref_ge_dx … HL12 … HLK2) -L2
112 /5 width=7 by cprs_delta, cprs_strap2, llpr_cpr_conf/
113 | #a #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
114 elim (llpr_inv_bind_O … HL12) -HL12 /4 width=1 by cprs_bind/
115 | #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
116 elim (llpr_inv_flat … HL12) -HL12 /3 width=1 by cprs_flat/
117 | #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #HL12
118 elim (llpr_inv_bind_O … HL12) /3 width=3 by cprs_zeta/
119 | #G #L2 #V2 #T1 #T2 #HT12 #IHT12 #L1 #HL12
120 elim (llpr_inv_flat … HL12) /3 width=1 by cprs_tau/
121 | #a #G #L2 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #HL12
122 elim (llpr_inv_flat … HL12) -HL12 #HV1 #HL12
123 elim (llpr_inv_bind_O … HL12) /3 width=3 by cprs_beta/
124 | #a #G #L2 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L1 #HL12
125 elim (llpr_inv_flat … HL12) -HL12 #HV1 #HL12
126 elim (llpr_inv_bind_O … HL12) /3 width=3 by cprs_theta/
130 lemma cpr_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡ T2 →
131 ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
132 /4 width=9 by llpr_cpr_trans, cprs_bind_dx, llpr_bind_repl_O/ qed.
134 (* Advanced properties ******************************************************)
136 (* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *)
137 lemma cprs_llpr_trans: ∀G. s_rs_transitive … (cpr G) (llpr G 0).
138 /3 width=6 by llpr_cpr_trans, llpr_cpr_conf, s_r_trans_LTC1/ qed-.
140 (* Basic_1: was: pr3_strip *)
141 (* Basic_1: includes: pr1_strip *)
142 lemma cprs_strip: ∀G,L. confluent2 … (cprs G L) (cpr G L).
143 normalize /4 width=3 by cpr_conf, TC_strip1/ qed-.
145 lemma cprs_llpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 →
146 ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
147 #G #L0 #T0 #T1 #H @(cprs_ind_dx … T0 H) -T0 /2 width=3 by ex2_intro/
148 #T0 #T #HT0 #_ #IHT1 #L1 #HL01
149 elim (IHT1 … L1) /2 by llpr_cpr_conf/ -IHT1 #T2 #HT12 #HT2
150 elim (llpr_cpr_conf_dx … HT0 … HL01) -L0 #T3 #HT03 #HT3
151 elim (cprs_strip … HT2 … HT3) -T
152 /4 width=5 by cprs_strap2, cprs_strap1, ex2_intro/
155 lemma cprs_llpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
156 ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 →
157 ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
158 #G #L0 #T0 #T1 #HT01 #L1 #HL01
159 elim (cprs_llpr_conf_dx … HT01 … HL01)
160 /4 width=5 by cprs_llpr_trans, cprs_llpr_conf, ex2_intro/
163 lemma cprs_bind2_dx: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 →
164 ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
165 ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
166 /4 width=9 by cprs_llpr_trans, cprs_bind_dx, llpr_bind_repl_O/ qed.