1 (**************************************************************************)
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_2/lib/ltc.ma".
16 include "basic_2/notation/relations/predstar_6.ma".
17 include "basic_2/notation/relations/predstar_5.ma".
18 include "basic_2/rt_transition/cpm.ma".
20 (* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
22 (* Basic_2A1: uses: scpds *)
23 definition cpms (h) (G) (L): relation3 nat term term ≝
24 ltc … plus … (cpm h G L).
27 "t-bound context-sensitive parallel rt-computarion (term)"
28 'PRedStar n h G L T1 T2 = (cpms h G L n T1 T2).
31 "context-sensitive parallel r-computation (term)"
32 'PRedStar h G L T1 T2 = (cpms h G L O T1 T2).
34 (* Basic eliminators ********************************************************)
36 lemma cpms_ind_sn (h) (G) (L) (T2) (Q:relation2 …):
38 (∀n1,n2,T1,T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T → ⦃G, L⦄ ⊢ T ➡*[n2, h] T2 → Q n2 T → Q (n1+n2) T1) →
39 ∀n,T1. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → Q n T1.
40 #h #G #L #T2 #Q @ltc_ind_sn_refl //
43 lemma cpms_ind_dx (h) (G) (L) (T1) (Q:relation2 …):
45 (∀n1,n2,T,T2. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T → Q n1 T → ⦃G, L⦄ ⊢ T ➡[n2, h] T2 → Q (n1+n2) T2) →
46 ∀n,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → Q n T2.
47 #h #G #L #T1 #Q @ltc_ind_dx_refl //
50 (* Basic inversion lemmas ***************************************************)
52 lemma cpms_inv_sort1 (n) (h) (G) (L): ∀X2,s. ⦃G, L⦄ ⊢ ⋆s ➡*[n, h] X2 → X2 = ⋆(((next h)^n) s).
53 #n #h #G #L #X2 #s #H @(cpms_ind_dx … H) -X2 //
54 #n1 #n2 #X #X2 #_ #IH #HX2 destruct
55 elim (cpm_inv_sort1 … HX2) -HX2 #H #_ destruct //
58 (* Basic properties *********************************************************)
60 (* Basic_1: includes: pr1_pr0 *)
61 (* Basic_1: uses: pr3_pr2 *)
62 (* Basic_2A1: includes: cpr_cprs *)
63 lemma cpm_cpms (h) (G) (L): ∀n,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2.
64 /2 width=1 by ltc_rc/ qed.
66 lemma cpms_step_sn (h) (G) (L): ∀n1,T1,T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T →
67 ∀n2,T2. ⦃G, L⦄ ⊢ T ➡*[n2, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n1+n2, h] T2.
68 /2 width=3 by ltc_sn/ qed-.
70 lemma cpms_step_dx (h) (G) (L): ∀n1,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T →
71 ∀n2,T2. ⦃G, L⦄ ⊢ T ➡[n2, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n1+n2, h] T2.
72 /2 width=3 by ltc_dx/ qed-.
74 (* Basic_2A1: uses: cprs_bind_dx *)
75 lemma cpms_bind_dx (n) (h) (G) (L):
76 ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
77 ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡*[n, h] T2 →
78 ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n, h] ⓑ{p,I}V2.T2.
79 #n #h #G #L #V1 #V2 #HV12 #I #T1 #T2 #H #a @(cpms_ind_sn … H) -T1
80 /3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind/ qed.
82 lemma cpms_appl_dx (n) (h) (G) (L):
83 ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
84 ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 →
85 ⦃G, L⦄ ⊢ ⓐV1.T1 ➡*[n, h] ⓐV2.T2.
86 #n #h #G #L #V1 #V2 #HV12 #T1 #T2 #H @(cpms_ind_sn … H) -T1
87 /3 width=3 by cpms_step_sn, cpm_cpms, cpm_appl/
90 lemma cpms_zeta (n) (h) (G) (L):
92 ∀V,T2. ⦃G, L⦄ ⊢ T ➡*[n, h] T2 → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[n, h] T2.
93 #n #h #G #L #T1 #T #HT1 #V #T2 #H @(cpms_ind_dx … H) -T2
94 /3 width=3 by cpms_step_dx, cpm_cpms, cpm_zeta/
97 (* Basic_2A1: uses: cprs_zeta *)
98 lemma cpms_zeta_dx (n) (h) (G) (L):
100 ∀V,T1. ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[n, h] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[n, h] T2.
101 #n #h #G #L #T2 #T #HT2 #V #T1 #H @(cpms_ind_sn … H) -T1
102 /3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind, cpm_zeta/
105 (* Basic_2A1: uses: cprs_eps *)
106 lemma cpms_eps (n) (h) (G) (L):
107 ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 →
108 ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[n, h] T2.
109 #n #h #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1
110 /3 width=3 by cpms_step_sn, cpm_cpms, cpm_eps/
113 lemma cpms_ee (n) (h) (G) (L):
114 ∀U1,U2. ⦃G, L⦄ ⊢ U1 ➡*[n, h] U2 →
115 ∀T. ⦃G, L⦄ ⊢ ⓝU1.T ➡*[↑n, h] U2.
116 #n #h #G #L #U1 #U2 #H @(cpms_ind_sn … H) -U1 -n
117 [ /3 width=1 by cpm_cpms, cpm_ee/
118 | #n1 #n2 #U1 #U #HU1 #HU2 #_ #T >plus_S1
119 /3 width=3 by cpms_step_sn, cpm_ee/
123 (* Basic_2A1: uses: cprs_beta_dx *)
124 lemma cpms_beta_dx (n) (h) (G) (L):
125 ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
126 ∀W1,W2. ⦃G, L⦄ ⊢ W1 ➡[h] W2 →
127 ∀T1,T2. ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[n, h] T2 →
128 ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n, h] ⓓ{p}ⓝW2.V2.T2.
129 #n #h #G #L #V1 #V2 #HV12 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2
130 /4 width=7 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_beta/
133 (* Basic_2A1: uses: cprs_theta_dx *)
134 lemma cpms_theta_dx (n) (h) (G) (L):
135 ∀V1,V. ⦃G, L⦄ ⊢ V1 ➡[h] V →
137 ∀W1,W2. ⦃G, L⦄ ⊢ W1 ➡[h] W2 →
138 ∀T1,T2. ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[n, h] T2 →
139 ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n, h] ⓓ{p}W2.ⓐV2.T2.
140 #n #h #G #L #V1 #V #HV1 #V2 #HV2 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2
141 /4 width=9 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_theta/
144 (* Basic properties with r-transition ***************************************)
146 (* Basic_1: was: pr3_refl *)
147 lemma cprs_refl: ∀h,G,L. reflexive … (cpms h G L 0).
148 /2 width=1 by cpm_cpms/ qed.
150 (* Basic_2A1: removed theorems 5:
151 sta_cprs_scpds lstas_scpds scpds_strap1 scpds_fwd_cprs