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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "Basic_2/grammar/cl_shift.ma".
16 include "Basic_2/unfold/tpss.ma".
17 include "Basic_2/reduction/tpr.ma".
19 (* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
21 (* Basic_1: includes: pr2_delta1 *)
22 definition cpr: lenv → relation term ≝
23 λL,T1,T2. ∃∃T. T1 ⇒ T & L ⊢ T [0, |L|] ≫* T2.
26 "context-sensitive parallel reduction (term)"
27 'PRed L T1 T2 = (cpr L T1 T2).
29 (* Basic properties *********************************************************)
31 (* Basic_1: was by definition: pr2_free *)
32 lemma cpr_pr: ∀T1,T2. T1 ⇒ T2 → ∀L. L ⊢ T1 ⇒ T2.
35 lemma cpr_tpss: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫* T2 → L ⊢ T1 ⇒ T2.
38 lemma cpr_refl: ∀L,T. L ⊢ T ⇒ T.
41 (* Note: new property *)
42 (* Basic_1: was only: pr2_thin_dx *)
43 lemma cpr_flat: ∀I,L,V1,V2,T1,T2.
44 L ⊢ V1 ⇒ V2 → L ⊢ T1 ⇒ T2 → L ⊢ 𝕗{I} V1. T1 ⇒ 𝕗{I} V2. T2.
45 #I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 * /3 width=5/
48 lemma cpr_cast: ∀L,V,T1,T2.
49 L ⊢ T1 ⇒ T2 → L ⊢ 𝕔{Cast} V. T1 ⇒ T2.
53 (* Basic inversion lemmas ***************************************************)
55 (* Basic_1: was: pr2_gen_csort *)
56 lemma cpr_inv_atom: ∀T1,T2. ⋆ ⊢ T1 ⇒ T2 → T1 ⇒ T2.
57 #T1 #T2 * #T #HT normalize #HT2
58 <(tpss_inv_refl_O2 … HT2) -HT2 //
61 (* Basic_1: was: pr2_gen_sort *)
62 lemma cpr_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ⇒ T2 → T2 = ⋆k.
64 >(tpr_inv_atom1 … H) -H #H
65 >(tpss_inv_sort1 … H) -H //
68 (* Basic_1: was: pr2_gen_cast *)
69 lemma cpr_inv_cast1: ∀L,V1,T1,U2. L ⊢ 𝕔{Cast} V1. T1 ⇒ U2 → (
70 ∃∃V2,T2. L ⊢ V1 ⇒ V2 & L ⊢ T1 ⇒ T2 &
73 #L #V1 #T1 #U2 * #X #H #HU2
74 elim (tpr_inv_cast1 … H) -H /3/
75 * #V #T #HV1 #HT1 #H destruct -X;
76 elim (tpss_inv_flat1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct -U2 /4 width=5/
79 (* Basic_1: removed theorems 5:
80 pr2_head_1 pr2_head_2 pr2_cflat pr2_gen_cflat clear_pr2_trans
81 Basic_1: removed local theorems 3:
82 pr2_free_free pr2_free_delta pr2_delta_delta
90 pr2/subst1 pr2_gen_cabbr