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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/notation/relations/predstar_6.ma".
16 include "basic_2/reduction/cnx.ma".
17 include "basic_2/computation/cprs.ma".
19 (* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
21 definition cpxs: ∀h. sd h → relation4 genv lenv term term ≝
22 λh,g,G. LTC … (cpx h g G).
24 interpretation "extended context-sensitive parallel computation (term)"
25 'PRedStar h g G L T1 T2 = (cpxs h g G L T1 T2).
27 (* Basic eliminators ********************************************************)
29 lemma cpxs_ind: ∀h,g,G,L,T1. ∀R:predicate term. R T1 →
30 (∀T,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T → ⦃G, L⦄ ⊢ T ➡[h, g] T2 → R T → R T2) →
31 ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → R T2.
32 #h #g #L #G #T1 #R #HT1 #IHT1 #T2 #HT12
33 @(TC_star_ind … HT1 IHT1 … HT12) //
36 lemma cpxs_ind_dx: ∀h,g,G,L,T2. ∀R:predicate term. R T2 →
37 (∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T → ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → R T → R T1) →
38 ∀T1. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → R T1.
39 #h #g #G #L #T2 #R #HT2 #IHT2 #T1 #HT12
40 @(TC_star_ind_dx … HT2 IHT2 … HT12) //
43 (* Basic properties *********************************************************)
45 lemma cpxs_refl: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ➡*[h, g] T.
48 lemma cpx_cpxs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
51 lemma cpxs_strap1: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T →
52 ∀T2. ⦃G, L⦄ ⊢ T ➡[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
53 normalize /2 width=3/ qed.
55 lemma cpxs_strap2: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T →
56 ∀T2. ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
57 normalize /2 width=3/ qed.
59 lemma lsubr_cpxs_trans: ∀h,g,G. lsub_trans … (cpxs h g G) lsubr.
60 /3 width=5 by lsubr_cpx_trans, TC_lsub_trans/
63 lemma cprs_cpxs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
64 #h #g #G #L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=3/
67 lemma cpxs_bind_dx: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
68 ∀I,T1,T2. ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ➡*[h, g] T2 →
69 ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
70 #h #g #G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cpxs_ind_dx … HT12) -T1
71 /3 width=1/ /3 width=3/
74 lemma cpxs_flat_dx: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
75 ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
76 ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, g] ⓕ{I}V2.T2.
77 #h #g #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cpxs_ind … HT12) -T2 /3 width=1/ /3 width=5/
80 lemma cpxs_flat_sn: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 →
81 ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
82 ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, g] ⓕ{I}V2.T2.
83 #h #g #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cpxs_ind … H) -V2 /3 width=1/ /3 width=5/
86 lemma cpxs_zeta: ∀h,g,G,L,V,T1,T,T2. ⇧[0, 1] T2 ≡ T →
87 ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[h, g] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[h, g] T2.
88 #h #g #G #L #V #T1 #T #T2 #HT2 #H @(TC_ind_dx … T1 H) -T1 /3 width=3/
91 lemma cpxs_tau: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
92 ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[h, g] T2.
93 #h #g #G #L #T1 #T2 #H elim H -T2 /2 width=3/ /3 width=1/
96 lemma cpxs_ti: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
97 ∀T. ⦃G, L⦄ ⊢ ⓝV1.T ➡*[h, g] V2.
98 #h #g #G #L #V1 #V2 #H elim H -V2 /2 width=3/ /3 width=1/
101 lemma cpxs_beta_dx: ∀h,g,a,G,L,V1,V2,W1,W2,T1,T2.
102 ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 →
103 ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h, g] ⓓ{a}ⓝW2.V2.T2.
104 #h #g #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 * -T2 /3 width=1/
105 /4 width=7 by cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_beta/ (**) (* auto too slow without trace *)
108 lemma cpxs_theta_dx: ∀h,g,a,G,L,V1,V,V2,W1,W2,T1,T2.
109 ⦃G, L⦄ ⊢ V1 ➡[h, g] V → ⇧[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, g] T2 →
110 ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, g] ⓓ{a}W2.ⓐV2.T2.
111 #h #g #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2 [ /3 width=3/ ]
112 /4 width=9 by cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_theta/ (**) (* auto too slow without trace *)
115 (* Basic inversion lemmas ***************************************************)
117 lemma cpxs_inv_sort1: ∀h,g,G,L,U2,k. ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] U2 →
118 ∃∃n,l. deg h g k (n+l) & U2 = ⋆((next h)^n k).
119 #h #g #G #L #U2 #k #H @(cpxs_ind … H) -U2
120 [ elim (deg_total h g k) #l #Hkl
121 @(ex2_2_intro … 0 … Hkl) -Hkl //
122 | #U #U2 #_ #HU2 * #n #l #Hknl #H destruct
123 elim (cpx_inv_sort1 … HU2) -HU2
124 [ #H destruct /2 width=4/
125 | * #l0 #Hkl0 #H destruct -l
126 @(ex2_2_intro … (n+1) l0) /2 width=1 by deg_inv_prec/ >iter_SO //
131 lemma cpxs_inv_cast1: ∀h,g,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[h, g] U2 →
132 ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 & ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 & U2 = ⓝW2.T2
133 | ⦃G, L⦄ ⊢ T1 ➡*[h, g] U2
134 | ⦃G, L⦄ ⊢ W1 ➡*[h, g] U2.
135 #h #g #G #L #W1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5/
136 #U2 #U #_ #HU2 * /3 width=3/ *
137 #W #T #HW1 #HT1 #H destruct
138 elim (cpx_inv_cast1 … HU2) -HU2 /3 width=3/ *
139 #W2 #T2 #HW2 #HT2 #H destruct
140 lapply (cpxs_strap1 … HW1 … HW2) -W
141 lapply (cpxs_strap1 … HT1 … HT2) -T /3 width=5/
144 lemma cpxs_inv_cnx1: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T ➡*[h, g] U → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → T = U.
145 #h #g #G #L #T #U #H @(cpxs_ind_dx … H) -T //
146 #T0 #T #H1T0 #_ #IHT #H2T0
147 lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1/
150 lemma cpxs_neq_inv_step_sn: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) →
151 ∃∃T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T & T1 = T → ⊥ & ⦃G, L⦄ ⊢ T ➡*[h, g] T2.
152 #h #g #G #L #T1 #T2 #H @(cpxs_ind_dx … H) -T1
154 | #T1 #T #H1 #H2 #IH2 #H12 elim (eq_term_dec T1 T) #H destruct
155 [ -H1 -H2 /3 width=1 by/
156 | -IH2 /3 width=4 by ex3_intro/ (**) (* auto fails without clear *)