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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 *)
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15 include "basic_2/dynamic/cnv_cpm_tdeq_conf.ma".
16 include "basic_2/dynamic/cnv_cpms_tdeq.ma".
18 include "basic_2/dynamic/cnv_cpm_trans.ma".
19 include "basic_2/dynamic/cnv_cpm_conf.ma".
22 (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
24 (* Sub confluence propery with t-bound rt-computation for terms *************)
26 fact cnv_cpms_tdeq_strip_lpr_aux (a) (h) (o) (G0) (L0) (T0):
27 (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
28 (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
29 ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ⦃G0,L0⦄ ⊢ T0 ![a,h] → T0 ≛[h,o] T1 →
30 ∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛[h,o] T2 →
31 ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
32 ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡[n2-n1,h] T & T1 ≛[h,o] T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-n2,h] T & T2 ≛[h,o] T.
33 #a #h #o #G #L0 #T0 #IH2 #IH1 #n1 #T1 #H1T01 #H0T0 #H2T01
34 @(cpms_tdeq_ind_sn … H1T01 H0T0 H2T01 IH1 IH2) -n1 -T0
35 [ #H0T1 #n2 #T2 #H1T12 #H2T12 #L1 #HL01 #L2 #HL02
37 elim (cnv_cpm_tdeq_conf_lpr … H0T1 0 T1 … H1T12 H2T12 … HL01 … HL02) // -L0 -H2T12
38 <minus_O_n <minus_n_O #T #H1T1 #H2T1 #H1T2 #H2T2
39 /3 width=5 by cpm_cpms, ex4_intro/
40 | #m1 #m2 #T0 #T3 #H1T03 #H0T0 #H2T03 #_ #_ #_ #IH
41 #n2 #T2 #H1T02 #H2T02 #L1 #HL01 #L2 #HL02
42 elim (cnv_cpm_tdeq_conf_lpr … H0T0 … H1T03 H2T03 … H1T02 H2T02 … L0 … HL02) -T0 //
43 #T0 #H1T30 #H2T30 #H1T20 #H2T20
44 elim (IH … H1T30 H2T30 … HL01 … HL02) -L0 -T3
45 #T3 #H1T13 #H2T13 #H1T03 #H2T03
47 /3 width=7 by cpms_step_sn, tdeq_trans, ex4_intro/
51 fact cnv_cpms_tdeq_conf_lpr_aux (a) (h) (o) (G0) (L0) (T0):
52 (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
53 (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
54 ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ⦃G0,L0⦄ ⊢ T0 ![a,h] → T0 ≛[h,o] T1 →
55 ∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡*[n2,h] T2 → T0 ≛[h,o] T2 →
56 ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
57 ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & T1 ≛[h,o] T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-n2,h] T & T2 ≛[h,o] T.
58 #a #h #o #G #L0 #T0 #IH2 #IH1 #n1 #T1 #H1T01 #H0T0 #H2T01
59 generalize in match IH1; generalize in match IH2;
60 @(cpms_tdeq_ind_sn … H1T01 H0T0 H2T01 IH1 IH2) -n1 -T0
61 [ #H0T1 #IH2 #IH1 #n2 #T2 #H1T12 #H2T12 #L1 #HL01 #L2 #HL02
63 elim (cnv_cpms_tdeq_strip_lpr_aux … IH2 IH1 … H1T12 H0T1 H2T12 0 T1 … HL02 … HL01) // -L0 -H2T12
64 <minus_O_n <minus_n_O #T #H1T2 #H2T2 #H1T1 #H2T1
65 /3 width=5 by cpm_cpms, ex4_intro/
66 | #m1 #m2 #T0 #T3 #H1T03 #H0T0 #H2T03 #_ #_ #_ #IH #IH2 #IH1
67 #n2 #T2 #H1T02 #H2T02 #L1 #HL01 #L2 #HL02
68 elim (cnv_cpms_tdeq_strip_lpr_aux … IH2 IH1 … H1T02 H0T0 H2T02 … H1T03 H2T03 … HL02 L0) -H0T0 -H2T03 //
69 #T4 #H1T24 #H2T24 #H1T34 #H2T34
70 elim (IH … H1T34 H2T34 … HL01 … HL02) [|*: /4 width=5 by cpm_fpbq, fpbq_fpbg_trans/ ] -L0 -T0 -T3 (**)
71 #T3 #H1T13 #H2T13 #H1T43 #H2T43
73 /3 width=7 by cpms_step_sn, tdeq_trans, ex4_intro/
78 fact cnv_cpms_conf_lpr_refl_refl_aux (h) (G0) (L1) (L2) (T0:term):
79 ∃∃T. ⦃G0,L1⦄ ⊢ T0 ➡*[h] T & ⦃G0,L2⦄ ⊢ T0 ➡*[h] T.
80 /2 width=3 by ex2_intro/ qed-.
82 fact cnv_cpms_conf_lpr_refl_step_aux (a) (h) (o) (G0) (L0) (T0) (m21) (m22):
83 (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
84 (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
86 ∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛[h,o] X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
87 ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
88 ∃∃T. ⦃G0,L1⦄ ⊢ T0 ➡*[m21+m22,h] T& ⦃G0,L2⦄ ⊢ T2 ➡*[h] T.
89 #a #h #o #G0 #L0 #T0 #m21 #m22 #IH2 #IH1 #H0
90 #X2 #HX02 #HnX02 #T2 #HXT2
92 lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … HX02 … L0 ?) // #HX2
93 elim (cnv_cpm_conf_lpr_aux … IH2 IH1 … HX02 … 0 T0 … L0 … HL01) //
94 <minus_n_O <minus_O_n #Y1 #HXY1 #HTY1
95 elim (cnv_cpms_strip_lpr_sub … IH1 … HXT2 0 X2 … HL02 L0) [|*: /4 width=2 by fpb_fpbg, cpm_fpb/ ]
96 <minus_n_O <minus_O_n #Y2 #HTY2 #HXY2 -HXT2
97 elim (IH1 … HXY1 … HXY2 … HL01 … HL02) [|*: /4 width=2 by fpb_fpbg, cpm_fpb/ ]
98 -a -o -L0 -X2 <minus_n_O <minus_O_n #Y #HY1 #HY2
99 lapply (cpms_trans … HTY1 … HY1) -Y1 #HT0Y
100 lapply (cpms_trans … HTY2 … HY2) -Y2 #HT2Y
101 /2 width=3 by ex2_intro/
104 fact cnv_cpms_conf_lpr_step_step_aux (a) (h) (o) (G0) (L0) (T0) (m11) (m12) (m21) (m22):
105 (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
106 (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
107 ⦃G0,L0⦄ ⊢ T0 ![a,h] →
108 ∀X1. ⦃G0,L0⦄ ⊢ T0 ➡[m11,h] X1 → (T0 ≛[h,o] X1 → ⊥) → ∀T1. ⦃G0,L0⦄ ⊢ X1 ➡*[m12,h] T1 →
109 ∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛[h,o] X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
110 ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
111 ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[m21+m22-(m11+m12),h] T& ⦃G0,L2⦄ ⊢ T2 ➡*[m11+m12-(m21+m22),h] T.
112 #a #h #o #G0 #L0 #T0 #m11 #m12 #m21 #m22 #IH2 #IH1 #H0
113 #X1 #HX01 #HnX01 #T1 #HXT1 #X2 #HX02 #HnX02 #T2 #HXT2
115 lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … HX01 … L0 ?) // #HX1
116 lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … HX02 … L0 ?) // #HX2
117 elim (cnv_cpm_conf_lpr_aux … IH2 IH1 … HX01 … HX02 … L0 … L0) // #Z0 #HXZ10 #HXZ20
118 cut (⦃G0,L0,T0⦄ >[h,o] ⦃G0,L0,X1⦄) [ /4 width=5 by cpms_fwd_fpbs, cpm_fpb, ex2_3_intro/ ] #H1fpbg (**) (* cut *)
119 lapply (fpbg_fpbs_trans ??? G0 ? L0 ? Z0 ? … H1fpbg) [ /2 width=2 by cpms_fwd_fpbs/ ] #H2fpbg
120 lapply (cnv_cpms_trans_lpr_sub … IH2 … HXZ10 … L0 ?) // #HZ0
121 elim (IH1 … HXT1 … HXZ10 … L1 … L0) [|*: /4 width=2 by fpb_fpbg, cpm_fpb/ ] -HXT1 -HXZ10 #Z1 #HTZ1 #HZ01
122 elim (IH1 … HXT2 … HXZ20 … L2 … L0) [|*: /4 width=2 by fpb_fpbg, cpm_fpb/ ] -HXT2 -HXZ20 #Z2 #HTZ2 #HZ02
123 elim (IH1 … HZ01 … HZ02 L1 … L2) // -L0 -T0 -X1 -X2 -Z0 #Z #HZ01 #HZ02
124 lapply (cpms_trans … HTZ1 … HZ01) -Z1 <arith_l4 #HT1Z
125 lapply (cpms_trans … HTZ2 … HZ02) -Z2 <arith_l4 #HT2Z
126 /2 width=3 by ex2_intro/
129 fact cnv_cpms_conf_lpr_aux (a) (h) (o):
130 ∀G0,L0,T0. 𝐏[h,o]⦃T0⦄ →
131 (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
132 (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
133 ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_cnv_cpms_conf_lpr a h G1 L1 T1.
134 #a #h #o #G #L #T #H0 #IH2 #IH1 #G0 #L0 #T0 #HG #HL #HT
135 #HT0 #n1 #T1 #HT01 #n2 #T2 #HT02 #L1 #HL01 #L2 #HL02 destruct
136 elim (cpms_fwd_tdpos_sn … HT0 H0 … HT01) *
137 elim (cpms_fwd_tdpos_sn … HT0 H0 … HT02) *
139 [ #H21 #H22 #H11 #H12 destruct -a -o -L0
141 /2 width=1 by cnv_cpms_conf_lpr_refl_refl_aux/
142 | #m21 #m22 #X2 #HX02 #HnX02 #HXT2 #H2 #H11 #H12 destruct
143 <minus_n_O <minus_O_n
144 @(cnv_cpms_conf_lpr_refl_step_aux … IH2 IH1) -IH2 -IH1 /2 width=4 by/
145 | #H21 #H22 #m11 #m12 #X1 #HX01 #HnX01 #HXT1 #H1 destruct
146 <minus_n_O <minus_O_n
147 @ex2_commute @(cnv_cpms_conf_lpr_refl_step_aux … IH2 IH1) -IH2 -IH1 /2 width=4 by/
148 | #m21 #m22 #X2 #HX02 #HnX02 #HXT2 #H2 #m11 #m12 #X1 #HX01 #HnX01 #HXT1 #H1 destruct
149 @(cnv_cpms_conf_lpr_step_step_aux … IH2 IH1) -IH2 -IH1 /2 width=4 by/