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 "basic_2/grammar/tstc.ma".
16 include "basic_2/computation/cprs_lift.ma".
17 include "basic_2/computation/cprs_lcprs.ma".
19 (* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
21 (* Forward lemmas involving same top term constructor ***********************)
23 lemma cprs_fwd_cnf: ∀L,T. L ⊢ 𝐍⦃T⦄ → ∀U. L ⊢ T ➡* U → T ≃ U.
25 >(cprs_inv_cnf1 … H HT) -L -T //
28 (* Basic_1: was: pr3_iso_beta *)
29 lemma cprs_fwd_beta: ∀L,V,W,T,U. L ⊢ ⓐV. ⓛW. T ➡* U →
30 ⓐV. ⓛW. T ≃ U ∨ L ⊢ ⓓV. T ➡* U.
32 elim (cprs_inv_appl1 … H) -H *
33 [ #V0 #T0 #_ #_ #H destruct /2 width=1/
34 | #V0 #W0 #T0 #HV0 #HT0 #HU
35 elim (cprs_inv_abst1 Abbr V … HT0) -HT0 #W1 #T1 #_ #HT1 #H destruct -W1
37 @(cprs_trans … HU) -U /2 width=1/ (**) (* explicit constructor *)
38 | #V1 #V2 #V0 #T1 #_ #_ #HT1 #_
39 elim (cprs_inv_abst1 Abbr V … HT1) -HT1 #W2 #T2 #_ #_ #H destruct
43 (* Note: probably this is an inversion lemma *)
44 lemma cprs_fwd_delta: ∀L,K,V1,i. ⇩[0, i] L ≡ K. ⓓV1 →
45 ∀V2. ⇧[0, i + 1] V1 ≡ V2 →
48 #L #K #V1 #i #HLK #V2 #HV12 #U #H
49 elim (cprs_inv_lref1 … H) -H /2 width=1/
50 * #K0 #V0 #U0 #HLK0 #HVU0 #HU0 #_
51 lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct
52 lapply (ldrop_fwd_ldrop2 … HLK) -HLK /3 width=9/
55 lemma cprs_fwd_theta: ∀L,V1,V,T,U. L ⊢ ⓐV1. ⓓV. T ➡* U →
56 ∀V2. ⇧[0, 1] V1 ≡ V2 → ⓐV1. ⓓV. T ≃ U ∨
58 #L #V1 #V #T #U #H #V2 #HV12
59 elim (cprs_inv_appl1 … H) -H *
60 [ -HV12 #V0 #T0 #_ #_ #H destruct /2 width=1/
61 | #V0 #W #T0 #HV10 #HT0 #HU
62 elim (cprs_inv_abbr1 … HT0) -HT0 *
63 [ #V3 #T3 #_ #_ #H destruct
65 elim (lift_inv_bind1 … H) -H #W2 #T2 #HW2 #HT02 #H destruct
66 @or_intror @(cprs_trans … HU) -U (**) (* explicit constructor *)
67 @(cprs_trans … (ⓓV.ⓐV2.ⓛW2.T2)) [ /3 width=1/ ] -T
68 @(cprs_strap2 … (ⓐV1.ⓛW.T0)) [ /5 width=3/ ] -V -V2 -W2 -T2
69 @(cprs_strap2 … (ⓓV1.T0)) [ /3 width=1/ ] -W /2 width=1/
71 | #V3 #V4 #V0 #T0 #HV13 #HV34 #HT0 #HU
72 @or_intror @(cprs_trans … HU) -U (**) (* explicit constructor *)
73 elim (cprs_inv_abbr1 … HT0) -HT0 *
74 [ #V5 #T5 #HV5 #HT5 #H destruct
75 lapply (cprs_lift (L.ⓓV) … HV12 … HV13 … HV34) -V1 -V3 /2 width=1/
78 elim (lift_inv_bind1 … H) -H #V5 #T5 #HV05 #HT05 #H destruct
79 lapply (cprs_lift (L.ⓓV0) … HV12 … HV13 … HV34) -V3 /2 width=1/ #HV24
80 @(cprs_trans … (ⓓV.ⓐV2.ⓓV5.T5)) [ /3 width=1/ ] -T
81 @(cprs_strap2 … (ⓐV1.ⓓV0.T0)) [ /5 width=3/ ] -V -V5 -T5
82 @(cprs_strap2 … (ⓓV0.ⓐV2.T0)) [ /3 width=3/ ] -V1 /3 width=9/
87 lemma cprs_fwd_tau: ∀L,W,T,U. L ⊢ ⓝW. T ➡* U →
88 ⓝW. T ≃ U ∨ L ⊢ T ➡* U.
90 elim (cprs_inv_cast1 … H) -H /2 width=1/ *
91 #W0 #T0 #_ #_ #H destruct /2 width=1/