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4 (* ||A|| A project by Andrea Asperti *)
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15 include "basic_2/dynamic/cnv_cpm_tdeq_trans.ma".
17 (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
19 (* Properties with restricted rt-computation for terms **********************)
21 fact cpms_tdneq_fwd_step_sn_aux (a) (h) (n) (o) (G) (L) (T1):
22 ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G, L⦄ ⊢ T1 ![a,h] → (T1 ≛[h,o] T2 → ⊥) →
23 (∀G0,L0,T0. ⦃G,L,T1⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
24 (∀G0,L0,T0. ⦃G,L,T1⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) →
25 ∃∃n1,n2,T0. ⦃G, L⦄ ⊢ T1 ➡[n1,h] T0 & T1 ≛[h,o] T0 → ⊥ & ⦃G, L⦄ ⊢ T0 ➡*[n2,h] T2 & n1+n2 = n.
26 #a #h #n #o #G #L #T1 #T2 #H
27 @(cpms_ind_sn … H) -n -T1
28 [ #_ #H2T2 elim H2T2 -H2T2 //
29 | #n1 #n2 #T1 #T #H1T1 #H1T2 #IH #H0T1 #H2T12 #IH2 #IH1
30 elim (tdeq_dec h o T1 T) #H2T1
31 [ elim (tdeq_dec h o T T2) #H2T2
32 [ -IH -IH2 -IH1 -H0T1 /4 width=7 by tdeq_trans, ex4_3_intro/
33 | lapply (cnv_cpm_trans_lpr_aux … IH2 IH1 … H1T1 L ?) [6:|*: // ] -H1T2 -H2T12 #H0T
34 elim (IH H0T H2T2) [|*: /4 width=5 by cpm_fpbq, fpbq_fpbg_trans/ ] -IH -IH2 -H0T -H2T2 (**)
35 #m1 #m2 #T0 #H1T0 #H2T0 #H1T02 #H destruct
36 elim (cnv_cpm_tdeq_cpm_trans_aux … IH1 … H0T1 … H1T1 H2T1 … H1T0) -IH1 -H0T1 -H1T1 -H1T0
37 #T3 #H1T13 #H1T30 #H2T30
38 @(ex4_3_intro … H1T13) /4 width=3 by cpms_step_sn, tdeq_canc_sn/ (**) (* explicit constructor *)
40 | -IH -IH2 -IH1 -H2T12 /3 width=7 by tdeq_trans, ex4_3_intro/
45 fact cpms_tdeq_ind_sn (a) (h) (o) (G) (L) (T2) (Q:relation2 …):
46 (⦃G, L⦄ ⊢ T2 ![a,h] → Q 0 T2) →
47 (∀n1,n2,T1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → ⦃G, L⦄ ⊢ T1 ![a,h] → T1 ≛[h,o] T → ⦃G, L⦄ ⊢ T ➡*[n2,h] T2 → ⦃G, L⦄ ⊢ T ![a,h] → T ≛[h,o] T2 → Q n2 T → Q (n1+n2) T1) →
48 ∀n,T1. ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G, L⦄ ⊢ T1 ![a,h] → T1 ≛[h,o] T2 →
49 (∀G0,L0,T0. ⦃G,L,T1⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
50 (∀G0,L0,T0. ⦃G,L,T1⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) →
52 #a #h #o #G #L #T2 #Q #IB1 #IB2 #n #T1 #H
53 @(cpms_ind_sn … H) -n -T1
54 [ -IB2 #H0T2 #_ #_ #_ /2 width=1 by/
55 | #n1 #n2 #T1 #T #H1T1 #H1T2 #IH #H0T1 #H2T12 #IH2 #IH1 -IB1
56 elim (tdeq_dec h o T1 T) #H2T1
57 [ lapply (cnv_cpm_trans_lpr_aux … IH2 IH1 … H1T1 L ?) [6:|*: // ] #H0T
58 lapply (tdeq_canc_sn … H2T1 … H2T12) -H2T12 #H2T2
59 /6 width=7 by cpm_fpbq, fpbq_fpbg_trans/ (**)
61 elim (cnv_fpbg_refl_false … o … H0T1) -a -Q
62 /3 width=8 by cpm_tdneq_cpm_cpms_tdeq_sym_fwd_fpbg/