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 "basic_2/rt_transition/lfpx_fsle.ma".
17 lemma R_fle_comp_LTC: ∀R. R_fle_compatible R → R_fle_compatible (LTC … R).
18 #R #HR #L #T1 #T2 #H elim H -T2
19 /3 width=3 by fle_trans_dx/
23 (* Note: "⦃L2, T1⦄ ⊆ ⦃L0, T1⦄" may not hold *)
24 axiom lfpx_cpx_conf_fsle4: ∀h,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
25 ∀L2. ⦃G, L0⦄ ⊢⬈[h, T0] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T1⦄.
27 #h #G0 #L0 #T0 @(fqup_wf_ind_eq (Ⓕ) … G0 L0 T0) -G0 -L0 -T0
28 #G #L #T #IH #G0 #L0 * *
29 [ #s #HG #HL #HT #X #HX #Y #HY destruct -IH
30 lapply (lfxs_fwd_length … HY) -HY #H0
31 elim (cpx_inv_sort1 … HX) -HX #H destruct
32 /3 width=1 by fsle_sort_length, and3_intro/
33 | * [| #i ] #HG #HL #HT #X #HX #Y #HY destruct
34 [ elim (cpx_inv_zero1 … HX) -HX
36 elim (lfxs_inv_zero … HY) -HY *
37 [ #H1 #H2 destruct -IH /2 width=1 by and3_intro/
38 | #I #K0 #K2 #V0 #V2 #HK02 #HV02 #H1 #H2 destruct
39 lapply (lfxs_fwd_length … HK02) #HK
41 [ <(H2R G0) in HV02; -H2R #HV02
42 elim (IH … HV02 … HK02) /2 width=2 by fqu_fqup, fqu_lref_O/ -IH -HV02 -HK02 #H1V #H2V #_
43 /4 width=1 by fsle_trans_tc, fsle_zero_bi, and3_intro/
44 | lapply (H2R … HV02 … HK02) -H2R -HV02 -HK02 -IH #HKV20
45 /3 width=1 by fsle_zero_bi, and3_intro/
47 | #f #I #K0 #K2 #Hf #HK02 #H1 #H2 destruct
49 | * #I0 #K0 #V0 #V1 #HV01 #HV1X #H destruct
50 elim (lfxs_inv_zero_pair_sn … HY) -HY #K2 #V2 #HK02 #HV02 #H destruct
52 | elim (cpx_inv_lref1 … HX) -HX
54 elim (lfxs_inv_lref … HY) -HY *
55 [ #H0 #H1 destruct /2 width=1 by and3_intro/
56 | #I0 #I2 #K0 #K2 #HK02 #H1 #H2 destruct
57 lapply (lfxs_fwd_length … HK02) #HK
58 elim (IH … HK02) [|*: /2 width=4 by fqu_fqup/ ] -IH -HK02
59 /3 width=5 by and3_intro, fsle_lifts_SO/
61 | * #I0 #K0 #V1 #HV1 #HV1X #H0 destruct
62 elim (lfxs_inv_lref_bind_sn … HY) -HY #I2 #K2 #HK02 #H destruct
63 lapply (lfxs_fwd_length … HK02) #HK
64 elim (IH … HK02) [|*: /2 width=4 by fqu_fqup/ ] -IH -HV1 -HK02
65 /3 width=5 by fsle_lifts_SO, and3_intro/
68 | #l #HG #HL #HT #X #HX #Y #HY destruct -IH
69 lapply (lfxs_fwd_length … HY) -HY #H0
70 >(cpx_inv_gref1 … HX) -X
71 /3 width=1 by fsle_gref_length, and3_intro/
72 | #p #I #V0 #T0 #HG #HL #HT #X #HX #Y #HY destruct
73 lapply (lfxs_fwd_length … HY) #H0
74 elim (lfxs_inv_bind … V0 ? HY) -HY // #HV0 #HT0
75 elim (cpx_inv_bind1 … HX) -HX *
76 [ #V1 #T1 #HV01 #HT01 #H destruct
77 elim (IH … HV01 … HV0) -HV01 -HV0 // #H1V #H2V #H3V
78 elim (IH … HT01 … HT0) -HT01 -HT0 -IH // #H1T #H2T #H3T
79 /4 width=3 by fsle_bind_eq, fsle_fwd_pair_sn, and3_intro/
80 | #T #HT #HXT #H1 #H2 destruct
81 elim (IH G0 … V0… V0 … HV0) -HV0 // #H1V #H2V #H3V
82 elim (IH … HT … HT0) -HT -HT0 -IH // #H1T #H2T #H3T
83 /3 width=5 by fsle_bind, fsle_inv_lifts_sn, and3_intro/
85 | #I #V0 #X0 #HG #HL #HT #X #HX #Y #HY destruct
86 elim (lfxs_inv_flat … HY) -HY #HV0 #HX0
87 elim (cpx_inv_flat1 … HX) -HX *
88 [ #V1 #T1 #HV01 #HT01 #H destruct
89 elim (IH … HV01 … HV0) -HV01 -HV0 // #H1V #H2V #H3V
90 elim (IH … HT01 … HX0) -HT01 -HX0 -IH // #H1T #H2T #H3T
91 /3 width=3 by fsle_flat, and3_intro/
93 elim (IH G0 … V0… V0 … HV0) -HV0 // #H1V #H2V #H3V
94 elim (IH … HX … HX0) -HX -HX0 -IH // #H1T #H2T #H3T
95 /4 width=3 by fsle_flat_sn, fsle_flat_dx_dx, fsle_flat_dx_sn, and3_intro/
97 elim (IH … HX … HV0) -HX -HV0 // #H1V #H2V #H3V
98 elim (IH G0 … X0… X0 … HX0) -HX0 -IH // #H1T #H2T #H3T
99 /4 width=3 by fsle_flat_sn, fsle_flat_dx_dx, fsle_flat_dx_sn, and3_intro/
100 | #p #V1 #W0 #W1 #T0 #T1 #HV01 #HW01 #HT01 #H1 #H2 #H3 destruct
101 elim (lfxs_inv_bind … W0 ? HX0) -HX0 // #HW0 #HT0
102 elim (IH … HV01 … HV0) -HV01 -HV0 // #H1V #H2V #H3V
103 elim (IH … HW01 … HW0) -HW01 -HW0 // #H1W #H2W #H3W
104 elim (IH … HT01 … HT0) -HT01 -HT0 -IH // #H1T #H2T #H3T
105 lapply (fsle_fwd_pair_sn … H2T) -H2T #H2T
106 lapply (fsle_fwd_pair_sn … H3T) -H3T #H3T
107 @and3_intro [ /3 width=5 by fsle_flat, fsle_bind/ ] (**) (* full auto too slow *)
108 @fsle_bind_sn_ge /4 width=1 by fsle_shift, fsle_flat_sn, fsle_flat_dx_dx, fsle_flat_dx_sn, fsle_bind_dx_sn/
109 | #p #V1 #X1 #W0 #W1 #T0 #T1 #HV01 #HVX1 #HW01 #HT01 #H1 #H2 #H3 destruct
110 elim (lfxs_inv_bind … W0 ? HX0) -HX0 // #HW0 #HT0
111 elim (IH … HV01 … HV0) -HV01 -HV0 // #H1V #H2V #H3V
112 elim (IH … HW01 … HW0) -HW01 -HW0 // #H1W #H2W #H3W
113 elim (IH … HT01 … HT0) -HT01 -HT0 -IH // #H1T #H2T #H3T
114 lapply (fsle_fwd_pair_sn … H2T) -H2T #H2T
115 lapply (fsle_fwd_pair_sn … H3T) -H3T #H3T
116 @and3_intro[ /3 width=5 by fsle_flat, fsle_bind/ ] (**) (* full auto too slow *)
118 [1,3: /3 width=1 by fsle_flat_dx_dx, fsle_bind_dx_sn/
119 |2,4: /4 width=3 by fsle_flat_sn, fsle_flat_dx_sn, fsle_flat_dx_dx, fsle_shift, fsle_lifts_sn/