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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "basic_2/reduction/lpx_ldrop.ma".
16 include "basic_2/computation/cpxs_lift.ma".
18 (* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
20 (* Main properties **********************************************************)
22 theorem cpxs_trans: ∀h,g,L. Transitive … (cpxs h g L).
23 #h #g #L #T1 #T #HT1 #T2 @trans_TC @HT1 qed-. (**) (* auto /3 width=3/ does not work because a δ-expansion gets in the way *)
25 theorem cpxs_bind: ∀h,g,a,I,L,V1,V2,T1,T2. ⦃h, L.ⓑ{I}V1⦄ ⊢ T1 ➡*[g] T2 →
26 ⦃h, L⦄ ⊢ V1 ➡*[g] V2 →
27 ⦃h, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[g] ⓑ{a,I}V2.T2.
28 #h #g #a #I #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2 /2 width=1/
30 @(cpxs_trans … IHV1) -V1 /2 width=1/
33 theorem cpxs_flat: ∀h,g,I,L,V1,V2,T1,T2. ⦃h, L⦄ ⊢ T1 ➡*[g] T2 →
34 ⦃h, L⦄ ⊢ V1 ➡*[g] V2 →
35 ⦃h, L⦄ ⊢ ⓕ{I} V1.T1 ➡*[g] ⓕ{I} V2.T2.
36 #h #g #I #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2 /2 width=1/
38 @(cpxs_trans … IHV1) -IHV1 /2 width=1/
41 theorem cpxs_beta: ∀h,g,a,L,V1,V2,W,T1,T2.
42 ⦃h, L.ⓛW⦄ ⊢ T1 ➡*[g] T2 → ⦃h, L⦄ ⊢ V1 ➡*[g] V2 →
43 ⦃h, L⦄ ⊢ ⓐV1.ⓛ{a}W.T1 ➡*[g] ⓓ{a}V2.T2.
44 #h #g #a #L #V1 #V2 #W #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2 /2 width=1/
46 @(cpxs_trans … IHV1) /2 width=1/
49 theorem cpxs_theta_rc: ∀h,g,a,L,V1,V,V2,W1,W2,T1,T2.
50 ⦃h, L⦄ ⊢ V1 ➡[g] V → ⇧[0, 1] V ≡ V2 →
51 ⦃h, L.ⓓW1⦄ ⊢ T1 ➡*[g] T2 → ⦃h, L⦄ ⊢ W1 ➡*[g] W2 →
52 ⦃h, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[g] ⓓ{a}W2.ⓐV2.T2.
53 #h #g #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H elim H -W2 /2 width=3/
55 @(cpxs_trans … IHW1) /2 width=1/
58 theorem cpxs_theta: ∀h,g,a,L,V1,V,V2,W1,W2,T1,T2.
59 ⇧[0, 1] V ≡ V2 → ⦃h, L⦄ ⊢ W1 ➡*[g] W2 →
60 ⦃h, L.ⓓW1⦄ ⊢ T1 ➡*[g] T2 → ⦃h, L⦄ ⊢ V1 ➡*[g] V →
61 ⦃h, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[g] ⓓ{a}W2.ⓐV2.T2.
62 #h #g #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1 /2 width=3/
63 #V1 #V0 #HV10 #_ #IHV0
64 @(cpxs_trans … IHV0) /2 width=1/
67 (* Properties on sn extended parallel reduction for local environments ******)
69 lemma lpx_cpx_trans: ∀h,g. s_r_trans … (cpx h g) (lpx h g).
70 #h #g #L2 #T1 #T2 #HT12 elim HT12 -L2 -T1 -T2
73 | #I #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
74 elim (lpx_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
75 elim (lpx_inv_pair2 … H) -H #K1 #V1 #HK12 #HV10 #H destruct
76 lapply (IHV02 … HK12) -K2 #HV02
77 lapply (cpxs_strap2 … HV10 … HV02) -V0 /2 width=7/
78 | #a #I #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
79 lapply (IHT12 (L1.ⓑ{I}V1) ?) -IHT12 /2 width=1/ /3 width=1/
81 | #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #HL12
82 lapply (IHT1 (L1.ⓓV2) ?) -IHT1 /2 width=1/ /2 width=3/
83 | #a #L2 #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
84 lapply (IHT12 (L1.ⓛW) ?) -IHT12 /2 width=1/ /3 width=1/
85 | #a #L2 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L1 #HL12
86 lapply (IHT12 (L1.ⓓW1) ?) -IHT12 /2 width=1/ /3 width=3/
90 lemma cpx_bind2: ∀h,g,L,V1,V2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 →
91 ∀I,T1,T2. ⦃h, L.ⓑ{I}V2⦄ ⊢ T1 ➡[g] T2 →
92 ∀a. ⦃h, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[g] ⓑ{a,I}V2.T2.
93 #h #g #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
94 lapply (lpx_cpx_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/
97 (* Advanced properties ******************************************************)
99 lemma lpx_cpxs_trans: ∀h,g. s_rs_trans … (cpx h g) (lpx h g).
100 /3 width=5 by s_r_trans_TC1, lpx_cpx_trans/ qed-.
102 lemma cpxs_bind2_dx: ∀h,g,L,V1,V2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 →
103 ∀I,T1,T2. ⦃h, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[g] T2 →
104 ∀a. ⦃h, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[g] ⓑ{a,I}V2.T2.
105 #h #g #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
106 lapply (lpx_cpxs_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/