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 "ground_2/xoa/ex_4_1_props.ma".
16 include "basic_2/dynamic/cnv_cpm_tdeq.ma".
18 (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
20 definition IH_cnv_cpm_tdeq_conf_lpr (a) (h): relation3 genv lenv term ≝
21 λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![a,h] →
22 ∀n1,T1. ⦃G,L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 →
23 ∀n2,T2. ⦃G,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
24 ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ➡[h] L2 →
25 ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡[n2-n1,h] T & T1 ≛ T & ⦃G,L2⦄ ⊢ T2 ➡[n1-n2,h] T & T2 ≛ T.
27 (* Diamond propery with restricted rt-transition for terms ******************)
29 fact cnv_cpm_tdeq_conf_lpr_atom_atom_aux (h) (G0) (L1) (L2) (I):
30 ∃∃T. ⦃G0,L1⦄ ⊢ ⓪{I} ➡[h] T & ⓪{I} ≛ T & ⦃G0,L2⦄ ⊢ ⓪{I} ➡[h] T & ⓪{I} ≛ T.
32 /2 width=5 by ex4_intro/
35 fact cnv_cpm_tdeq_conf_lpr_atom_ess_aux (h) (G0) (L1) (L2) (s):
36 ∃∃T. ⦃G0,L1⦄ ⊢ ⋆s ➡[1,h] T & ⋆s ≛ T & ⦃G0,L2⦄ ⊢ ⋆(⫯[h]s) ➡[h] T & ⋆(⫯[h]s) ≛ T.
38 /3 width=5 by tdeq_sort, ex4_intro/
41 fact cnv_cpm_tdeq_conf_lpr_bind_bind_aux (a) (h) (p) (I) (G0) (L0) (V0) (T0):
42 (∀G,L,T. ⦃G0,L0,ⓑ{p,I}V0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cnv_cpm_tdeq_conf_lpr a h G L T) →
43 ⦃G0,L0⦄ ⊢ ⓑ{p,I}V0.T0 ![a,h] →
44 ∀n1,T1. ⦃G0,L0.ⓑ{I}V0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 →
45 ∀n2,T2. ⦃G0,L0.ⓑ{I}V0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
46 ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
47 ∃∃T. ⦃G0,L1⦄ ⊢ ⓑ{p,I}V0.T1 ➡[n2-n1,h] T & ⓑ{p,I}V0.T1 ≛ T & ⦃G0,L2⦄ ⊢ ⓑ{p,I}V0.T2 ➡[n1-n2,h] T & ⓑ{p,I}V0.T2 ≛ T.
48 #a #h #p #I #G0 #L0 #V0 #T0 #IH #H0
49 #n1 #T1 #H1T01 #H2T01 #n2 #T2 #H1T02 #H2T02
51 elim (cnv_inv_bind … H0) -H0 #_ #HT0
52 elim (IH … H1T01 H2T01 … H1T02 H2T02 (L1.ⓑ{I}V0) … (L2.ⓑ{I}V0)) [|*: /2 width=1 by lpr_bind_refl_dx/ ]
53 #T #H1T1 #H2T1 #H1T2 #H2T2 -L0 -T0
54 /3 width=7 by cpm_bind, tdeq_pair, ex4_intro/
57 fact cnv_cpm_tdeq_conf_lpr_appl_appl_aux (a) (h) (G0) (L0) (V0) (T0):
58 (∀G,L,T. ⦃G0,L0,ⓐV0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cnv_cpm_tdeq_conf_lpr a h G L T) →
59 ⦃G0,L0⦄ ⊢ ⓐV0.T0 ![a,h] →
60 ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 →
61 ∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
62 ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
63 ∃∃T. ⦃G0,L1⦄ ⊢ ⓐV0.T1 ➡[n2-n1,h] T & ⓐV0.T1 ≛ T & ⦃G0,L2⦄ ⊢ ⓐV0.T2 ➡[n1-n2,h] T & ⓐV0.T2 ≛ T.
64 #a #h #G0 #L0 #V0 #T0 #IH #H0
65 #n1 #T1 #H1T01 #H2T01 #n2 #T2 #H1T02 #H2T02
67 elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #_ #HT0 #_ #_ -n0 -p0 -X01 -X02
68 elim (IH … H1T01 H2T01 … H1T02 H2T02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
69 #T #H1T1 #H2T1 #H1T2 #H2T2 -L0 -T0
70 /3 width=7 by cpm_appl, tdeq_pair, ex4_intro/
73 fact cnv_cpm_tdeq_conf_lpr_cast_cast_aux (a) (h) (G0) (L0) (V0) (T0):
74 (∀G,L,T. ⦃G0,L0,ⓝV0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cnv_cpm_tdeq_conf_lpr a h G L T) →
75 ⦃G0,L0⦄ ⊢ ⓝV0.T0 ![a,h] →
76 ∀n1,V1. ⦃G0,L0⦄ ⊢ V0 ➡[n1,h] V1 → V0 ≛ V1 →
77 ∀n2,V2. ⦃G0,L0⦄ ⊢ V0 ➡[n2,h] V2 → V0 ≛ V2 →
78 ∀T1. ⦃G0,L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 →
79 ∀T2. ⦃G0,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
80 ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
81 ∃∃T. ⦃G0,L1⦄ ⊢ ⓝV1.T1 ➡[n2-n1,h] T & ⓝV1.T1 ≛ T & ⦃G0,L2⦄ ⊢ ⓝV2.T2 ➡[n1-n2,h] T & ⓝV2.T2 ≛ T.
82 #a #h #G0 #L0 #V0 #T0 #IH #H0
83 #n1 #V1 #H1V01 #H2V01 #n2 #V2 #H1V02 #H2V02 #T1 #H1T01 #H2T01 #T2 #H1T02 #H2T02
85 elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #_ #_ -X0
86 elim (IH … H1V01 H2V01 … H1V02 H2V02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
87 elim (IH … H1T01 H2T01 … H1T02 H2T02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ]
88 #T #H1T1 #H2T1 #H1T2 #H2T2 #V #H1V1 #H2V1 #H1V2 #H2V2 -L0 -V0 -T0
89 /3 width=7 by cpm_cast, tdeq_pair, ex4_intro/
92 fact cnv_cpm_tdeq_conf_lpr_aux (a) (h) (G0) (L0) (T0):
93 (∀G,L,T. ⦃G0,L0,T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cnv_cpm_tdeq_conf_lpr a h G L T) →
94 ∀G,L,T. G0 = G → L0 = L → T0 = T → IH_cnv_cpm_tdeq_conf_lpr a h G L T.
95 #a #h #G0 #L0 #T0 #IH1 #G #L * [| * [| * ]]
96 [ #I #HG0 #HL0 #HT0 #HT #n1 #X1 #H1X1 #H2X1 #n2 #X2 #H1X2 #H2X2 #L1 #HL1 #L2 #HL2 destruct
97 elim (cpm_tdeq_inv_atom_sn … H1X1 H2X1) -H1X1 -H2X1 *
98 elim (cpm_tdeq_inv_atom_sn … H1X2 H2X2) -H1X2 -H2X2 *
99 [ #H21 #H22 #H11 #H12 destruct -a -L
101 /2 width=1 by cnv_cpm_tdeq_conf_lpr_atom_atom_aux/
102 | #s2 #H21 #H22 #H23 #H11 #H12 destruct -a -L
103 <minus_O_n <minus_n_O
104 /2 width=1 by cnv_cpm_tdeq_conf_lpr_atom_ess_aux/
105 | #H21 #H22 #s1 #H11 #H12 #H13 destruct -a -L
106 <minus_O_n <minus_n_O
107 @ex4_commute /2 width=1 by cnv_cpm_tdeq_conf_lpr_atom_ess_aux/
108 | #s2 #H21 #H22 #H23 #s1 #H11 #H12 #H13 destruct -a -L
110 /2 width=1 by cnv_cpm_tdeq_conf_lpr_atom_atom_aux/
112 | #p #I #V #T #HG0 #HL0 #HT0 #HT #n1 #X1 #H1X1 #H2X1 #n2 #X2 #H1X2 #H2X2 #L1 #HL1 #L2 #HL2 destruct
113 elim (cpm_tdeq_inv_bind_sn … HT … H1X1 H2X1) -H1X1 -H2X1
114 elim (cpm_tdeq_inv_bind_sn … HT … H1X2 H2X2) -H1X2 -H2X2
115 #T2 #_ #_ #H1T2 #H2T2 #H21 #T1 #_ #_ #H1T1 #H2T1 #H11 destruct
116 @(cnv_cpm_tdeq_conf_lpr_bind_bind_aux … IH1) -IH1 /1 width=1 by/
117 | #V #T #HG0 #HL0 #HT0 #HT #n1 #X1 #H1X1 #H2X1 #n2 #X2 #H1X2 #H2X2 #L1 #HL1 #L2 #HL2 destruct
118 elim (cpm_tdeq_inv_appl_sn … HT … H1X1 H2X1) -H1X1 -H2X1
119 elim (cpm_tdeq_inv_appl_sn … HT … H1X2 H2X2) -H1X2 -H2X2
120 #m2 #q2 #W2 #U2 #T2 #_ #_ #_ #_ #_ #H1T2 #H2T2 #H21 #m1 #q1 #W1 #U1 #T1 #_ #_ #_ #_ #_ #H1T1 #H2T1 #H11 destruct -m1 -m2 -q1 -q2 -W1 -W2 -U1 -U2
121 @(cnv_cpm_tdeq_conf_lpr_appl_appl_aux … IH1) -IH1 /1 width=1 by/
122 | #V #T #HG0 #HL0 #HT0 #HT #n1 #X1 #H1X1 #H2X1 #n2 #X2 #H1X2 #H2X2 #L1 #HL1 #L2 #HL2 destruct
123 elim (cpm_tdeq_inv_cast_sn … HT … H1X1 H2X1) -H1X1 -H2X1
124 elim (cpm_tdeq_inv_cast_sn … HT … H1X2 H2X2) -H1X2 -H2X2
125 #U2 #V2 #T2 #_ #_ #_ #H1V2 #H2V2 #_ #H1T2 #H2T2 #H21 #U1 #V1 #T1 #_ #_ #_ #H1V1 #H2V1 #_ #H1T1 #H2T1 #H11 destruct -U1 -U2
126 @(cnv_cpm_tdeq_conf_lpr_cast_cast_aux … IH1) -IH1 /1 width=1 by/
130 lemma cnv_cpm_tdeq_conf_lpr (a) (h) (G0) (L0) (T0):
131 IH_cnv_cpm_tdeq_conf_lpr a h G0 L0 T0.
133 @(fqup_wf_ind (Ⓣ) … G0 L0 T0) -G0 -L0 -T0 #G0 #L0 #T0 #IH
134 /3 width=17 by cnv_cpm_tdeq_conf_lpr_aux/