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15 include "basic_2/dynamic/cnv_cpm_teqx_conf.ma".
16 include "basic_2/dynamic/cnv_cpms_teqx.ma".
18 (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
20 (* Sub confluence propery with restricted rt-transition for terms ***********)
22 fact cnv_cpms_teqx_strip_lpr_aux (h) (a) (G0) (L0) (T0):
23 (∀G,L,T. ❪G0,L0,T0❫ > ❪G,L,T❫ → IH_cnv_cpm_trans_lpr h a G L T) →
24 (∀G,L,T. ❪G0,L0,T0❫ > ❪G,L,T❫ → IH_cnv_cpms_conf_lpr h a G L T) →
25 ∀n1,T1. ❪G0,L0❫ ⊢ T0 ➡*[h,n1] T1 → ❪G0,L0❫ ⊢ T0 ![h,a] → T0 ≛ T1 →
26 ∀n2,T2. ❪G0,L0❫ ⊢ T0 ➡[h,n2] T2 → T0 ≛ T2 →
27 ∀L1. ❪G0,L0❫ ⊢ ➡[h,0] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h,0] L2 →
28 ∃∃T. ❪G0,L1❫ ⊢ T1 ➡[h,n2-n1] T & T1 ≛ T & ❪G0,L2❫ ⊢ T2 ➡*[h,n1-n2] T & T2 ≛ T.
29 #h #a #G #L0 #T0 #IH2 #IH1 #n1 #T1 #H1T01 #H0T0 #H2T01
30 @(cpms_teqx_ind_sn … H1T01 H0T0 H2T01 IH1 IH2) -n1 -T0
31 [ #H0T1 #n2 #T2 #H1T12 #H2T12 #L1 #HL01 #L2 #HL02
33 elim (cnv_cpm_teqx_conf_lpr … H0T1 0 T1 … H1T12 H2T12 … HL01 … HL02) // -L0 -H2T12
34 <minus_O_n <minus_n_O #T #H1T1 #H2T1 #H1T2 #H2T2
35 /3 width=5 by cpm_cpms, ex4_intro/
36 | #m1 #m2 #T0 #T3 #H1T03 #H0T0 #H2T03 #_ #_ #_ #IH
37 #n2 #T2 #H1T02 #H2T02 #L1 #HL01 #L2 #HL02
38 elim (cnv_cpm_teqx_conf_lpr … H0T0 … H1T03 H2T03 … H1T02 H2T02 … L0 … HL02) -T0 //
39 #T0 #H1T30 #H2T30 #H1T20 #H2T20
40 elim (IH … H1T30 H2T30 … HL01 … HL02) -L0 -T3
41 #T3 #H1T13 #H2T13 #H1T03 #H2T03
43 /3 width=7 by cpms_step_sn, teqx_trans, ex4_intro/
47 fact cnv_cpms_teqx_conf_lpr_aux (h) (a) (G0) (L0) (T0):
48 (∀G,L,T. ❪G0,L0,T0❫ > ❪G,L,T❫ → IH_cnv_cpm_trans_lpr h a G L T) →
49 (∀G,L,T. ❪G0,L0,T0❫ > ❪G,L,T❫ → IH_cnv_cpms_conf_lpr h a G L T) →
50 ∀n1,T1. ❪G0,L0❫ ⊢ T0 ➡*[h,n1] T1 → ❪G0,L0❫ ⊢ T0 ![h,a] → T0 ≛ T1 →
51 ∀n2,T2. ❪G0,L0❫ ⊢ T0 ➡*[h,n2] T2 → T0 ≛ T2 →
52 ∀L1. ❪G0,L0❫ ⊢ ➡[h,0] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h,0] L2 →
53 ∃∃T. ❪G0,L1❫ ⊢ T1 ➡*[h,n2-n1] T & T1 ≛ T & ❪G0,L2❫ ⊢ T2 ➡*[h,n1-n2] T & T2 ≛ T.
54 #h #a #G #L0 #T0 #IH2 #IH1 #n1 #T1 #H1T01 #H0T0 #H2T01
55 generalize in match IH1; generalize in match IH2;
56 @(cpms_teqx_ind_sn … H1T01 H0T0 H2T01 IH1 IH2) -n1 -T0
57 [ #H0T1 #IH2 #IH1 #n2 #T2 #H1T12 #H2T12 #L1 #HL01 #L2 #HL02
59 elim (cnv_cpms_teqx_strip_lpr_aux … IH2 IH1 … H1T12 H0T1 H2T12 0 T1 … HL02 … HL01) // -L0 -H2T12
60 <minus_O_n <minus_n_O #T #H1T2 #H2T2 #H1T1 #H2T1
61 /3 width=5 by cpm_cpms, ex4_intro/
62 | #m1 #m2 #T0 #T3 #H1T03 #H0T0 #H2T03 #_ #_ #_ #IH #IH2 #IH1
63 #n2 #T2 #H1T02 #H2T02 #L1 #HL01 #L2 #HL02
64 elim (cnv_cpms_teqx_strip_lpr_aux … IH2 IH1 … H1T02 H0T0 H2T02 … H1T03 H2T03 … HL02 L0) -H0T0 -H2T03 //
65 #T4 #H1T24 #H2T24 #H1T34 #H2T34
66 elim (IH … H1T34 H2T34 … HL01 … HL02) [|*: /4 width=5 by cpm_fpbq, fpbq_fpbg_trans/ ] -L0 -T0 -T3 (**)
67 #T3 #H1T13 #H2T13 #H1T43 #H2T43
69 /3 width=7 by cpms_step_sn, teqx_trans, ex4_intro/