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/syntax/lveq_length.ma".
16 include "basic_2/relocation/lexs_length.ma".
17 include "basic_2/relocation/drops_lexs.ma".
18 include "basic_2/static/frees_drops.ma".
19 include "basic_2/static/lsubf_frees.ma".
20 include "basic_2/static/lfxs.ma".
21 include "basic_2/rt_transition/cpx_drops.ma".
22 include "basic_2/rt_transition/cpx_ext.ma".
24 (* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
26 (* Properties with context-sensitive free variables *************************)
28 (* Basic_2A1: uses: lpx_cpx_frees_trans *)
29 lemma cpx_frees_conf_lexs: ∀h,G,L1,T1,f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 →
30 ∀L2. L1 ⪤*[cpx_ext h G, cfull, f1] L2 →
31 ∀T2. ⦃G, L1⦄ ⊢ T1 ⬈[h] T2 →
32 ∃∃f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ f1.
33 #h #G #L1 #T1 @(fqup_wf_ind_eq (Ⓣ) … G L1 T1) -G -L1 -T1
34 #G0 #L0 #U0 #IH #G #L1 * *
35 [ -IH #s #HG #HL #HU #g1 #H1 #L2 #_ #U2 #H0 destruct
36 lapply (frees_inv_sort … H1) -H1 #Hg1
37 elim (cpx_inv_sort1 … H0) -H0 #H destruct
38 /3 width=3 by frees_sort, sle_refl, ex2_intro/
39 | #i #HG #HL #HU #g1 #H1 #L2 #H2 #U2 #H0 destruct
40 elim (frees_inv_lref_drops … H1) -H1 *
41 [ -IH #f1 #HL1 #Hf1 #H destruct
42 elim (cpx_inv_lref1_drops … H0) -H0
44 /4 width=9 by frees_atom_drops, drops_atom2_lexs_conf, sle_refl, ex2_intro/
45 | * -H2 -Hf1 #I #K1 #V1 #V2 #HLK1
46 lapply (drops_TF … HLK1) -HLK1 #HLK1
47 lapply (drops_mono … HLK1 … HL1) -L1 #H destruct
49 | #f1 #I #K1 #V1 #Hf1 #HLK1 #H destruct
50 elim (cpx_inv_lref1_drops … H0) -H0
52 elim (lexs_drops_conf_next … H2 … HLK1) -H2 [ |*: // ] #Z #K2 #HLK2 #HK12 #H
53 elim (ext2_inv_pair_sn … H) -H #V2 #HV12 #H destruct
54 elim (IH … Hf1 … HK12 … HV12) /2 width=2 by fqup_lref/ -L1 -K1 -V1 #f2 #Hf2 #Hf21
55 /4 width=7 by frees_lref_pushs, frees_pair_drops, drops_refl, sle_pushs, sle_next, ex2_intro/
56 | * #Z #Y #X #V2 #H #HV12 #HVU2
57 lapply (drops_mono … H … HLK1) -H #H destruct
58 elim (lexs_drops_conf_next … H2 … HLK1) -H2 [ |*: // ] #I2 #K2 #HLK2 #HK12 #H
59 elim (ext2_inv_pair_sn … H) -H #V0 #_ #H destruct
60 lapply (drops_isuni_fwd_drop2 … HLK2) // -V0 #HLK2
61 elim (IH … Hf1 … HK12 … HV12) /2 width=2 by fqup_lref/ -I -L1 -K1 -V1 #f2 #Hf2 #Hf21
62 lapply (frees_lifts … Hf2 … HLK2 … HVU2 ??) /4 width=7 by sle_weak, ex2_intro, sle_pushs/
64 | #f1 #I #K1 #HLK1 #Hf1 #H destruct
65 elim (cpx_inv_lref1_drops … H0) -H0
67 elim (lexs_drops_conf_next … H2 … HLK1) -H2 -HLK1 [ |*: // ] #Z #K2 #HLK2 #_ #H
68 lapply (ext2_inv_unit_sn … H) -H #H destruct
69 /3 width=3 by frees_unit_drops, sle_refl, ex2_intro/
70 | * -H2 -Hf1 #Z #Y1 #X1 #X2 #HLY1
71 lapply (drops_mono … HLK1 … HLY1) -L1 #H destruct
74 | -IH #l #HG #HL #HU #g1 #H1 #L2 #_ #U2 #H0 destruct
75 lapply (frees_inv_gref … H1) -H1 #Hg1
76 lapply (cpx_inv_gref1 … H0) -H0 #H destruct
77 /3 width=3 by frees_gref, sle_refl, ex2_intro/
78 | #p #I #V1 #T1 #HG #HL #HU #g1 #H1 #L2 #H2 #U2 #H0 destruct
79 elim (frees_inv_bind … H1) -H1 #gV1 #gT1 #HgV1 #HgT1 #Hg1
80 elim (cpx_inv_bind1 … H0) -H0 *
81 [ #V2 #T2 #HV12 #HT12 #H destruct
82 lapply (sle_lexs_trans … H2 gV1 ?) /2 width=2 by sor_inv_sle_sn/ #HL12V
83 lapply (sle_lexs_trans … H2 (⫱gT1) ?) /2 width=2 by sor_inv_sle_dx/ -H2 #HL12T
84 lapply (lexs_inv_tl … (BPair I V1) (BPair I V2) … HL12T ??) /2 width=1 by ext2_pair/ -HL12T #HL12T
85 elim (IH … HgV1 … HL12V … HV12) // -HgV1 -HL12V -HV12 #gV2 #HgV2 #HgV21
86 elim (IH … HgT1 … HL12T … HT12) // -IH -HgT1 -HL12T -HT12 #gT2 #HgT2 #HgT21
87 elim (sor_isfin_ex gV2 (⫱gT2)) /3 width=3 by frees_fwd_isfin, isfin_tl/
88 /4 width=10 by frees_bind, monotonic_sle_sor, sle_tl, ex2_intro/
89 | #T2 #HT12 #HUT2 #H0 #H1 destruct -HgV1
90 lapply (sle_lexs_trans … H2 (⫱gT1) ?) /2 width=2 by sor_inv_sle_dx/ -H2 #HL12T
91 lapply (lexs_inv_tl … (BPair Abbr V1) (BPair Abbr V1) … HL12T ??) /2 width=1 by ext2_pair/ -HL12T #HL12T
92 elim (IH … HgT1 … HL12T … HT12) // -L1 -T1 #gT2 #HgT2 #HgT21
93 lapply (frees_inv_lifts_SO (Ⓣ) … HgT2 … L2 … HUT2) [ /3 width=1 by drops_refl, drops_drop/ ] -V1 -T2
94 /5 width=6 by sor_inv_sle_dx, sle_trans, sle_tl, ex2_intro/
96 | #I #V1 #T0 #HG #HL #HU #g1 #H1 #L2 #H2 #U2 #H0 destruct
97 elim (frees_inv_flat … H1) -H1 #gV1 #gT0 #HgV1 #HgT0 #Hg1
98 elim (cpx_inv_flat1 … H0) -H0 *
99 [ #V2 #T2 #HV12 #HT12 #H destruct
100 lapply (sle_lexs_trans … H2 gV1 ?) /2 width=2 by sor_inv_sle_sn/ #HL12V
101 lapply (sle_lexs_trans … H2 gT0 ?) /2 width=2 by sor_inv_sle_dx/ -H2 #HL12T
102 elim (IH … HgV1 … HL12V … HV12) // -HgV1 -HL12V -HV12 #gV2 #HgV2 #HgV21
103 elim (IH … HgT0 … HL12T … HT12) // -IH -HgT0 -HL12T -HT12 #gT2 #HgT2 #HgT21
104 elim (sor_isfin_ex gV2 gT2) /2 width=3 by frees_fwd_isfin/
105 /3 width=10 by frees_flat, monotonic_sle_sor, ex2_intro/
106 | #HU2 #H destruct -HgV1
107 lapply (sle_lexs_trans … H2 gT0 ?) /2 width=2 by sor_inv_sle_dx/ -H2 #HL12T
108 elim (IH … HgT0 … HL12T … HU2) // -L1 -T0 -V1
109 /4 width=6 by sor_inv_sle_dx, sle_trans, ex2_intro/
110 | #HU2 #H destruct -HgT0
111 lapply (sle_lexs_trans … H2 gV1 ?) /2 width=2 by sor_inv_sle_sn/ -H2 #HL12V
112 elim (IH … HgV1 … HL12V … HU2) // -L1 -T0 -V1
113 /4 width=6 by sor_inv_sle_sn, sle_trans, ex2_intro/
114 | #p #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #H0 #H1 #H destruct
115 elim (frees_inv_bind … HgT0) -HgT0 #gW1 #gT1 #HgW1 #HgT1 #HgT0
116 lapply (sle_lexs_trans … H2 gV1 ?) /2 width=2 by sor_inv_sle_sn/ #HL12V
117 lapply (sle_lexs_trans … H2 gT0 ?) /2 width=2 by sor_inv_sle_dx/ -H2 #H2
118 lapply (sle_lexs_trans … H2 gW1 ?) /2 width=2 by sor_inv_sle_sn/ #HL12W
119 lapply (sle_lexs_trans … H2 (⫱gT1) ?) /2 width=2 by sor_inv_sle_dx/ -H2 #HL12T
120 lapply (lexs_inv_tl … (BPair Abst W1) (BPair Abst W2) … HL12T ??) /2 width=1 by ext2_pair, I/ -HL12T #HL12T
121 elim (sor_isfin_ex gV1 gW1) /2 width=3 by frees_fwd_isfin/ #g0 #Hg0 #_
122 lapply (sor_assoc_sn … Hg1 … HgT0 … Hg0) -Hg1 -HgT0 #Hg1
123 lapply (sor_comm … Hg0) -Hg0 #Hg0
124 elim (IH … HgV1 … HL12V … HV12) // -HgV1 -HL12V -HV12 #gV2 #HgV2 #HgV21
125 elim (IH … HgW1 … HL12W … HW12) // -HgW1 -HL12W -HW12 #gW2 #HgW2 #HgW21
126 elim (IH … HgT1 … HL12T … HT12) // -IH -HgT1 -HL12T -HT12 #gT2 #HgT2 #HgT21
127 elim (sor_isfin_ex (⫱gT2) gV2) /3 width=3 by frees_fwd_isfin, isfin_tl/ #gVT2 #HgVT2 #_
128 elim (lsubf_beta_tl_dx … HgV2 … HgVT2 … W2) [ |*: /1 width=3 by lsubf_refl/ ] #gT0 #HL2 #HgT02
129 lapply (lsubf_frees_trans … HgT2 … HL2) -HgT2 -HL2 #HgT0
130 lapply (sor_comm … HgVT2) -HgVT2 #HgVT2 (**) (* this should be removed *)
131 elim (sor_isfin_ex gW2 gV2) /2 width=3 by frees_fwd_isfin/ #gV0 #HgV0 #H
132 elim (sor_isfin_ex gV0 (⫱gT0)) /3 width=3 by frees_fwd_isfin, isfin_tl/ -H #g2 #Hg2 #_
133 @(ex2_intro … g2) /3 width=5 by frees_flat, frees_bind/ -h -p -G -L1 -L2 -V1 -V2 -W1 -W2 -T1 -T2
134 @(sor_inv_sle … Hg2) -Hg2
135 [ /3 width=11 by sor_inv_sle_sn_trans, monotonic_sle_sor/
136 | @(sle_trans … HgT02) -HgT02
137 /3 width=8 by monotonic_sle_sor, sor_inv_sle_dx_trans, sle_tl/
138 ] (**) (* full auto too slow *)
139 | #p #V2 #V #W1 #W2 #T1 #T2 #HV12 #HV2 #HW12 #HT12 #H0 #H1 #H destruct
140 elim (frees_inv_bind … HgT0) -HgT0 #gW1 #gT1 #HgW1 #HgT1 #HgT0
141 lapply (sle_lexs_trans … H2 gV1 ?) /2 width=2 by sor_inv_sle_sn/ #HL12V
142 lapply (sle_lexs_trans … H2 gT0 ?) /2 width=2 by sor_inv_sle_dx/ -H2 #H2
143 lapply (sle_lexs_trans … H2 gW1 ?) /2 width=2 by sor_inv_sle_sn/ #HL12W
144 lapply (sle_lexs_trans … H2 (⫱gT1) ?) /2 width=2 by sor_inv_sle_dx/ -H2 #HL12T
145 lapply (lexs_inv_tl … (BPair Abbr W1) (BPair Abbr W2) … HL12T ??) /2 width=1 by ext2_pair, I/ -HL12T #HL12T
146 elim (sor_isfin_ex gV1 gW1) /2 width=3 by frees_fwd_isfin/ #g0 #Hg0 #_
147 lapply (sor_assoc_sn … Hg1 … HgT0 … Hg0) -Hg1 -HgT0 #Hg1
148 elim (IH … HgV1 … HL12V … HV12) // -HgV1 -HL12V -HV12 #gV2 #HgV2 #HgV21
149 elim (IH … HgW1 … HL12W … HW12) // -HgW1 -HL12W -HW12 #gW2 #HgW2 #HgW21
150 elim (IH … HgT1 … HL12T … HT12) // -IH -HgT1 -HL12T -HT12 #gT2 #HgT2 #HgT21
151 elim (sor_isfin_ex (↑gV2) gT2) /3 width=3 by frees_fwd_isfin, isfin_push/ #gV0 #HgV0 #H
152 elim (sor_isfin_ex gW2 (⫱gV0)) /3 width=3 by frees_fwd_isfin, isfin_tl/ -H #g2 #Hg2 #_
153 elim (sor_isfin_ex gW2 gV2) /2 width=3 by frees_fwd_isfin/ #g #Hg #_
154 lapply (sor_assoc_sn … Hg2 … (⫱gT2) … Hg) /2 width=1 by sor_tl/ #Hg2
155 lapply (frees_lifts (Ⓣ) … HgV2 … (L2.ⓓW2) … HV2 ??)
156 [4: lapply (sor_comm … Hg) |*: /3 width=3 by drops_refl, drops_drop/ ] -V2 (**) (* full auto too slow *)
157 /4 width=10 by frees_flat, frees_bind, monotonic_sle_sor, sle_tl, ex2_intro/
162 (* Basic_2A1: uses: cpx_frees_trans *)
163 lemma cpx_fle_comp: ∀h,G. R_fle_compatible (cpx h G).
164 #h #G #L #T1 #T2 #HT12
165 elim (frees_total L T1) #f1 #Hf1
166 elim (cpx_frees_conf_lexs … Hf1 L … HT12) -HT12
167 /3 width=8 by lexs_refl, ext2_refl, ex4_4_intro/
170 (* Basic_2A1: uses: lpx_frees_trans *)
171 lemma lfpx_fle_comp: ∀h,G. lfxs_fle_compatible (cpx h G).
172 #h #G #L1 #L2 #T * #f1 #Hf1 #HL12
173 elim (cpx_frees_conf_lexs h … Hf1 … HL12 T) // #f2 #Hf2
174 lapply (lexs_fwd_length … HL12)
175 /3 width=8 by lveq_length_eq, ex4_4_intro/ (**) (* full auto fails *)
178 (* Properties with generic extension on referred entries ********************)
180 (* Note: lemma 1000 *)
181 (* Basic_2A1: uses: cpx_llpx_sn_conf *)
182 lemma cpx_lfxs_conf: ∀R,h,G. s_r_confluent1 … (cpx h G) (lfxs R).
183 #R #h #G #L1 #T1 #T2 #H #L2 * #f1 #Hf1 elim (cpx_frees_conf_lexs … Hf1 L1 … H) -T1
184 /3 width=5 by lexs_refl, ext2_refl, sle_lexs_trans, ex2_intro/