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 *)
13 (**************************************************************************)
15 include "static_2/static/fsle_drops.ma".
16 include "static_2/static/rex_fsle.ma".
17 include "basic_2/rt_transition/rpx_length.ma".
18 include "basic_2/rt_transition/rpx_drops.ma".
19 include "basic_2/rt_transition/rpx_fqup.ma".
21 (* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
23 (* Forward lemmas with free variables inclusion for restricted closures *****)
25 (* Note: "⦃L2, T1⦄ ⊆ ⦃L2, T0⦄" does not hold *)
26 (* Note: Take L0 = K0.ⓓ(ⓝW.V), L2 = K0.ⓓW, T0 = #0, T1 = ⬆*[1]V *)
27 (* Note: This invalidates rpxs_cpx_conf: "∀h,G. s_r_confluent1 … (cpx h G) (rpxs h G)" *)
28 lemma rpx_cpx_conf_fsge (h) (G): ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
29 ∀L2. ⦃G, L0⦄ ⊢⬈[h, T0] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T0⦄.
30 #h #G0 #L0 #T0 @(fqup_wf_ind_eq (Ⓕ) … G0 L0 T0) -G0 -L0 -T0
31 #G #L #T #IH #G0 #L0 * *
32 [ #s #HG #HL #HT #X #HX #Y #HY destruct -IH
33 elim (cpx_inv_sort1 … HX) -HX #H destruct
34 lapply (rpx_fwd_length … HY) -HY #H0
35 /2 width=1 by fsle_sort_bi/
36 | * [| #i ] #HG #HL #HT #X #HX #Y #HY destruct
37 [ elim (cpx_inv_zero1 … HX) -HX
39 elim (rpx_inv_zero_length … HY) -HY *
40 [ #H1 #H2 destruct -IH //
41 | #I #K0 #K2 #V0 #V2 #HK02 #HV02 #H1 #H2 destruct
42 lapply (rpx_fwd_length … HK02) #H0
43 /4 width=4 by fsle_pair_bi, fqu_fqup, fqu_lref_O/
44 | #I #K0 #K2 #HK02 #H1 #H2 destruct -IH
45 /2 width=1 by fsle_unit_bi/
47 | * #I0 #K0 #V0 #V1 #HV01 #HV1X #H destruct
48 elim (rpx_inv_zero_pair_sn … HY) -HY #K2 #V2 #HK02 #HV02 #H destruct
49 lapply (rpx_fwd_length … HK02) #H0
50 /4 width=4 by fsle_lifts_SO_sn, fqu_fqup, fqu_lref_O/
52 | elim (cpx_inv_lref1 … HX) -HX
54 elim (rpx_inv_lref … HY) -HY *
56 | #I0 #I2 #K0 #K2 #HK02 #H1 #H2 destruct
57 lapply (rpx_fwd_length … HK02) #H0
58 /4 width=5 by fsle_lifts_SO, fqu_fqup/
60 | * #I0 #K0 #V1 #HV1 #HV1X #H0 destruct
61 elim (rpx_inv_lref_bind_sn … HY) -HY #I2 #K2 #HK02 #H destruct
62 lapply (rpx_fwd_length … HK02) #H0
63 /4 width=5 by fsle_lifts_SO, fqu_fqup/
66 | #l #HG #HL #HT #X #HX #Y #HY destruct -IH
67 >(cpx_inv_gref1 … HX) -X
68 lapply (rpx_fwd_length … HY) -HY #H0
69 /2 width=1 by fsle_gref_bi/
70 | #p #I #V0 #T0 #HG #HL #HT #X #HX #Y #HY destruct
71 elim (rpx_inv_bind … V0 ? HY) -HY #HV0 #HT0
72 elim (cpx_inv_bind1 … HX) -HX *
73 [ #V1 #T1 #HV01 #HT01 #H destruct
74 lapply (rpx_fwd_length … HV0) #H0
75 /4 width=6 by fsle_bind_eq, fsle_fwd_pair_sn/
76 | #T #H2T0 #HTX #H1 #H2 destruct
77 lapply (rpx_inv_lifts_bi … HT0 (Ⓣ) … H2T0) -HT0 [6:|*: /3 width=2 by drops_refl, drops_drop/ ] #HT
78 lapply (rpx_fwd_length … HV0) #H0
79 /5 width=6 by fsle_bind_dx_dx, fsle_lifts_dx, fqup_zeta/
81 | #I #V0 #X0 #HG #HL #HT #X #HX #Y #HY destruct
82 elim (rex_inv_flat … HY) -HY #HV0 #HX0
83 elim (cpx_inv_flat1 … HX) -HX *
84 [ #V1 #T1 #HV01 #HT01 #H destruct
85 /3 width=4 by fsle_flat/
87 /4 width=4 by fsle_flat_dx_dx/
89 /4 width=4 by fsle_flat_dx_sn/
90 | #p #V1 #W0 #W1 #T0 #T1 #HV01 #HW01 #HT01 #H1 #H2 #H3 destruct
91 elim (rpx_inv_bind … W0 ? HX0) -HX0 #HW0 #HT0
92 lapply (rpx_fwd_length … HV0) #H0
93 lapply (IH … HV01 … HV0) -HV01 -HV0 // #HV
94 lapply (IH … HW01 … HW0) -HW01 -HW0 // #HW
95 lapply (IH … HT01 … HT0) -HT01 -HT0 -IH // #HT
96 lapply (fsle_fwd_pair_sn … HT) -HT #HT
98 [ /4 width=1 by fsle_flat_sn, fsle_flat_dx_dx, fsle_flat_dx_sn, fsle_bind_dx_sn/
99 | /3 width=1 by fsle_flat_dx_dx, fsle_shift/
101 | #p #V1 #X1 #W0 #W1 #T0 #T1 #HV01 #HVX1 #HW01 #HT01 #H1 #H2 #H3 destruct
102 elim (rpx_inv_bind … W0 ? HX0) -HX0 #HW0 #HT0
103 lapply (rpx_fwd_length … HV0) #H0
104 lapply (IH … HV01 … HV0) -HV01 -HV0 // #HV
105 lapply (IH … HW01 … HW0) -HW01 -HW0 // #HW
106 lapply (IH … HT01 … HT0) -HT01 -HT0 -IH // #HT
107 lapply (fsle_fwd_pair_sn … HT) -HT #HT
109 [ /3 width=1 by fsle_flat_dx_dx, fsle_bind_dx_sn/
110 | /4 width=3 by fsle_flat_sn, fsle_flat_dx_sn, fsle_flat_dx_dx, fsle_shift, fsle_lifts_sn/
116 lemma rpx_fsge_comp (h) (G): rex_fsge_compatible (cpx h G).
117 /2 width=4 by rpx_cpx_conf_fsge/ qed-.
119 (**) (* this section concerns cpx *)
120 (* Properties with generic extension on referred entries ********************)
122 (* Basic_2A1: uses: cpx_frees_trans *)
123 lemma cpx_fsge_comp (h) (G): R_fsge_compatible (cpx h G).
124 /2 width=4 by rpx_cpx_conf_fsge/ qed-.
126 (* Note: lemma 1000 *)
127 (* Basic_2A1: uses: cpx_llpx_sn_conf *)
128 lemma cpx_rex_conf (R) (h) (G): s_r_confluent1 … (cpx h G) (rex R).
129 /3 width=3 by fsge_rex_trans, cpx_fsge_comp/ qed-.
131 (* Advanced properties ******************************************************)
133 lemma rpx_cpx_conf (h) (G): s_r_confluent1 … (cpx h G) (rpx h G).
134 /2 width=5 by cpx_rex_conf/ qed-.
136 lemma rpx_cpx_conf_fsge_dx (h) (G): ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
137 ∀L2. ⦃G, L0⦄ ⊢⬈[h, T0] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T1⦄.
138 /3 width=5 by rpx_cpx_conf, rpx_fsge_comp/ qed-.