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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 *)
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15 include "basic_2/dynamic/cnv_cpm_conf.ma".
16 include "basic_2/dynamic/cnv_cpms_tdeq_conf.ma".
18 (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
20 (* Sub confluence propery with t-bound rt-computation for terms *************)
22 fact cnv_cpms_conf_lpr_tdeq_tdeq_aux (a) (h) (o) (G0) (L0) (T0):
23 (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
24 (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
26 ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → T0 ≛[h,o] T1 →
27 ∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡*[n2,h] T2 → T0 ≛[h,o] T2 →
28 ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
29 ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
30 #a #h #o #G #L0 #T0 #IH2 #IH1 #HT0
31 #n1 #T1 #H1T01 #H2T01 #n2 #T2 #H1T02 #H2T02
33 elim (cnv_cpms_tdeq_conf_lpr_aux … IH2 IH1 … H1T01 … H1T02 … HL01 … HL02) -IH2 -IH1 -H1T01 -H1T02 -HL01 -HL02
34 /2 width=3 by ex2_intro/
37 fact cnv_cpms_conf_lpr_refl_tdneq_sub (a) (h) (o) (G0) (L0) (T0) (m21) (m22):
38 (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
39 (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
41 ∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛[h,o] X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
42 ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
43 ∃∃T. ⦃G0,L1⦄ ⊢ T0 ➡*[m21+m22,h] T& ⦃G0,L2⦄ ⊢ T2 ➡*[h] T.
44 #a #h #o #G0 #L0 #T0 #m21 #m22 #IH2 #IH1 #H0
45 #X2 #HX02 #HnX02 #T2 #HXT2
47 lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … HX02 … L0 ?) // #HX2
48 elim (cnv_cpm_conf_lpr_aux … IH2 IH1 … HX02 … 0 T0 … L0 … HL01) //
49 <minus_n_O <minus_O_n #Y1 #HXY1 #HTY1
50 elim (cnv_cpms_strip_lpr_sub … IH1 … HXT2 0 X2 … HL02 L0) [|*: /4 width=2 by fpb_fpbg, cpm_fpb/ ]
51 <minus_n_O <minus_O_n #Y2 #HTY2 #HXY2 -HXT2
52 elim (IH1 … HXY1 … HXY2 … HL01 … HL02) [|*: /4 width=2 by fpb_fpbg, cpm_fpb/ ]
53 -a -o -L0 -X2 <minus_n_O <minus_O_n #Y #HY1 #HY2
54 lapply (cpms_trans … HTY1 … HY1) -Y1 #HT0Y
55 lapply (cpms_trans … HTY2 … HY2) -Y2 #HT2Y
56 /2 width=3 by ex2_intro/
59 fact cnv_cpms_conf_lpr_step_tdneq_sub (a) (h) (o) (G0) (L0) (T0) (m11) (m12) (m21) (m22):
60 (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
61 (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
63 ∀X1. ⦃G0,L0⦄ ⊢ T0 ➡[m11,h] X1 → T0 ≛[h,o] X1 → ∀T1. ⦃G0,L0⦄ ⊢ X1 ➡*[m12,h] T1 → X1 ≛[h,o] T1 →
64 ∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛[h,o] X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
65 ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
66 ((∀G,L,T. ⦃G0,L0,X1⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
67 (∀G,L,T. ⦃G0,L0,X1⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
69 ∀X2. ⦃G0,L0⦄ ⊢ X1 ➡[m21,h] X2 → (X1 ≛[h,o] X2 → ⊥) →
70 ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
71 ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
72 ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[m21+m22-m12,h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[m12-(m21+m22),h]T
74 ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[m21+m22-(m11+m12),h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[m11+m12-(m21+m22),h] T.
75 #a #h #o #G0 #L0 #T0 #m11 #m12 #m21 #m22 #IH2 #IH1 #HT0
76 #X1 #H1X01 #H2X01 #T1 #H1XT1 #H2XT1 #X2 #H1X02 #H2X02 #T2 #HXT2
77 #L1 #HL01 #L2 #HL02 #IH
78 lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … H1X01 … L0 ?) // #HX1
79 lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … H1X02 … L0 ?) // #HX2
80 elim (cnv_cpm_conf_lpr_aux … IH2 IH1 … H1X01 … H1X02 … L0 … L0) // #Z0 #HXZ10 #HXZ20
81 cut (⦃G0,L0,T0⦄ >[h,o] ⦃G0,L0,X2⦄) [ /4 width=5 by cpms_fwd_fpbs, cpm_fpb, ex2_3_intro/ ] #H1fpbg (**) (* cut *)
82 lapply (fpbg_fpbs_trans ??? G0 ? L0 ? Z0 ? … H1fpbg) [ /2 width=2 by cpms_fwd_fpbs/ ] #H2fpbg
83 lapply (cnv_cpms_trans_lpr_sub … IH2 … HXZ20 … L0 ?) // #HZ0
84 elim (IH1 … HXT2 … HXZ20 … L2 … L0) [|*: /4 width=2 by fpb_fpbg, cpm_fpb/ ] -HXT2 -HXZ20 #Z2 #HTZ2 #HZ02
85 elim (tdeq_dec h o X1 Z0) #H2XZ
87 elim (cnv_cpms_conf_lpr_tdeq_tdeq_aux … HX1 … H1XT1 H2XT1 … HXZ10 H2XZ … L1 … L0) [2,3: // |4,5: /4 width=5 by cpm_fpbq, fpbq_fpbg_trans/ ]
89 elim (cpms_tdneq_fwd_step_sn_aux … HXZ10 HX1 H2XZ) [|*: /4 width=5 by cpm_fpbq, fpbq_fpbg_trans/ ]
90 -HXZ10 -H2XZ #n1 #n2 #X0 #H1X10 #H2X10 #HXZ0 #Hn
91 elim (IH … H1X10 H2X10 … HXZ0 … L1 … L0) [2,3: // |4,5: /4 width=5 by cpm_fpbq, fpbq_fpbg_trans/ ]
95 elim (IH1 … HZ01 … HZ02 L1 … L2) // -L0 -T0 -X1 -X2 -Z0 #Z #HZ01 #HZ02
96 lapply (cpms_trans … HTZ1 … HZ01) -Z1 <arith_l4 #HT1Z
97 lapply (cpms_trans … HTZ2 … HZ02) -Z2 <arith_l4 #HT2Z
98 /2 width=3 by ex2_intro/
101 fact cnv_cpms_conf_lpr_tdeq_tdneq_aux (a) (h) (o) (G0) (L0) (T0) (n1) (m21) (m22):
102 (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
103 (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
104 ⦃G0,L0⦄ ⊢ T0 ![a,h] →
105 ∀T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → T0 ≛[h,o] T1 →
106 ∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛[h,o] X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
107 ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
108 ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[m21+m22-n1,h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-(m21+m22),h] T.
109 #a #h #o #G0 #L0 #T0 #n1 #m21 #m22 #IH2 #IH1 #HT0
111 generalize in match m22; generalize in match m21; -m21 -m22
112 generalize in match IH1; generalize in match IH2;
113 @(cpms_tdeq_ind_sn … H1T01 HT0 H2T01 IH1 IH2) -n1 -T0
114 [ #HT1 #IH2 #IH1 #m21 #m22
115 #X2 #HX02 #HnX02 #T2 #HXT2 #L1 #HL01 #L2 #HL02
116 <minus_O_n <minus_n_O
117 @(cnv_cpms_conf_lpr_refl_tdneq_sub … IH2 IH1) -IH2 -IH1 /2 width=4 by/
118 | #m11 #m12 #T0 #X1 #H1X01 #HT0 #H2X01 #H1XT1 #_ #H2XT1 #IH #IH2 #IH1 #m21 #m22
119 #X2 #HX02 #HnX02 #T2 #HXT2 #L1 #HL01 #L2 #HL02
120 @(cnv_cpms_conf_lpr_step_tdneq_sub … IH2 IH1 … IH) -IH2 -IH1 -IH /2 width=4 by/
124 fact cnv_cpms_conf_lpr_tdneq_tdneq_aux (a) (h) (o) (G0) (L0) (T0) (m11) (m12) (m21) (m22):
125 (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
126 (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
127 ⦃G0,L0⦄ ⊢ T0 ![a,h] →
128 ∀X1. ⦃G0,L0⦄ ⊢ T0 ➡[m11,h] X1 → (T0 ≛[h,o] X1 → ⊥) → ∀T1. ⦃G0,L0⦄ ⊢ X1 ➡*[m12,h] T1 →
129 ∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛[h,o] X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
130 ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
131 ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[m21+m22-(m11+m12),h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[m11+m12-(m21+m22),h] T.
132 #a #h #o #G0 #L0 #T0 #m11 #m12 #m21 #m22 #IH2 #IH1 #H0
133 #X1 #HX01 #HnX01 #T1 #HXT1 #X2 #HX02 #HnX02 #T2 #HXT2
135 lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … HX01 … L0 ?) // #HX1
136 lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … HX02 … L0 ?) // #HX2
137 elim (cnv_cpm_conf_lpr_aux … IH2 IH1 … HX01 … HX02 … L0 … L0) // #Z0 #HXZ10 #HXZ20
138 cut (⦃G0,L0,T0⦄ >[h,o] ⦃G0,L0,X1⦄) [ /4 width=5 by cpms_fwd_fpbs, cpm_fpb, ex2_3_intro/ ] #H1fpbg (**) (* cut *)
139 lapply (fpbg_fpbs_trans ??? G0 ? L0 ? Z0 ? … H1fpbg) [ /2 width=2 by cpms_fwd_fpbs/ ] #H2fpbg
140 lapply (cnv_cpms_trans_lpr_sub … IH2 … HXZ10 … L0 ?) // #HZ0
141 elim (IH1 … HXT1 … HXZ10 … L1 … L0) [|*: /4 width=2 by fpb_fpbg, cpm_fpb/ ] -HXT1 -HXZ10 #Z1 #HTZ1 #HZ01
142 elim (IH1 … HXT2 … HXZ20 … L2 … L0) [|*: /4 width=2 by fpb_fpbg, cpm_fpb/ ] -HXT2 -HXZ20 #Z2 #HTZ2 #HZ02
143 elim (IH1 … HZ01 … HZ02 L1 … L2) // -L0 -T0 -X1 -X2 -Z0 #Z #HZ01 #HZ02
144 lapply (cpms_trans … HTZ1 … HZ01) -Z1 <arith_l4 #HT1Z
145 lapply (cpms_trans … HTZ2 … HZ02) -Z2 <arith_l4 #HT2Z
146 /2 width=3 by ex2_intro/
149 fact cnv_cpms_conf_lpr_aux (a) (h) (o) (G0) (L0) (T0):
150 (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
151 (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
152 ∀G,L,T. G0 = G → L0 = L → T0 = T → IH_cnv_cpms_conf_lpr a h G L T.
153 #a #h #o #G #L #T #IH2 #IH1 #G0 #L0 #T0 #HG #HL #HT
154 #HT0 #n1 #T1 #HT01 #n2 #T2 #HT02 #L1 #HL01 #L2 #HL02 destruct
155 elim (tdeq_dec h o T0 T1) #H2T01
156 elim (tdeq_dec h o T0 T2) #H2T02
157 [ @(cnv_cpms_conf_lpr_tdeq_tdeq_aux … IH2 IH1) -IH2 -IH1 /2 width=1 by/
158 | elim (cpms_tdneq_fwd_step_sn_aux … HT02 HT0 H2T02 IH1 IH2) -HT02 -H2T02
159 #m21 #m22 #X2 #HX02 #HnX02 #HXT2 #H2 destruct
160 @(cnv_cpms_conf_lpr_tdeq_tdneq_aux … IH2 IH1) -IH2 -IH1 /2 width=4 by/
161 | elim (cpms_tdneq_fwd_step_sn_aux … HT01 HT0 H2T01 IH1 IH2) -HT01 -H2T01
162 #m11 #m12 #X1 #HX01 #HnX01 #HXT1 #H1 destruct
163 @ex2_commute @(cnv_cpms_conf_lpr_tdeq_tdneq_aux … IH2 IH1) -IH2 -IH1 /2 width=4 by/
164 | elim (cpms_tdneq_fwd_step_sn_aux … HT01 HT0 H2T01 IH1 IH2) -HT01 -H2T01
165 elim (cpms_tdneq_fwd_step_sn_aux … HT02 HT0 H2T02 IH1 IH2) -HT02 -H2T02
166 #m21 #m22 #X2 #HX02 #HnX02 #HXT2 #H2 #m11 #m12 #X1 #HX01 #HnX01 #HXT1 #H1 destruct
167 @(cnv_cpms_conf_lpr_tdneq_tdneq_aux … IH2 IH1) -IH2 -IH1 /2 width=4 by/