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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "ground_2/xoa/ex_3_2.ma".
16 include "ground_2/steps/rtc_max.ma".
17 include "ground_2/steps/rtc_ist.ma".
19 (* RT-TRANSITION COUNTER ****************************************************)
21 (* Properties with test for t-transition counter ****************************)
23 lemma ist_max: ∀n1,n2,c1,c2. 𝐓❪n1,c1❫ → 𝐓❪n2,c2❫ → 𝐓❪n1∨n2,c1∨c2❫.
24 #n1 #n2 #c1 #c2 #H1 #H2 destruct //
27 lemma ist_max_O1: ∀n,c1,c2. 𝐓❪0,c1❫ → 𝐓❪n,c2❫ → 𝐓❪n,c1∨c2❫.
28 /2 width=1 by ist_max/ qed.
30 lemma ist_max_O2: ∀n,c1,c2. 𝐓❪n,c1❫ → 𝐓❪0,c2❫ → 𝐓❪n,c1∨c2❫.
31 #n #c1 #c2 #H1 #H2 >(max_O2 n) /2 width=1 by ist_max/
34 lemma ist_max_idem1: ∀n,c1,c2. 𝐓❪n,c1❫ → 𝐓❪n,c2❫ → 𝐓❪n,c1∨c2❫.
35 #n #c1 #c2 #H1 #H2 >(idempotent_max n) /2 width=1 by ist_max/
38 (* Inversion properties with test for t-transition counter ******************)
41 ∀n,c1,c2. 𝐓❪n,c1 ∨ c2❫ →
42 ∃∃n1,n2. 𝐓❪n1,c1❫ & 𝐓❪n2,c2❫ & (n1 ∨ n2) = n.
44 elim (max_inv_dx … H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #H1 #H2 #H3 #H4 #H5 #H6 destruct
45 elim (max_inv_O3 … H1) -H1 #H11 #H12 destruct
46 elim (max_inv_O3 … H2) -H2 #H21 #H22 destruct
47 elim (max_inv_O3 … H3) -H3 #H31 #H32 destruct
48 /2 width=5 by ex3_2_intro/
51 lemma ist_O_inv_max: ∀c1,c2. 𝐓❪0,c1 ∨ c2❫ → ∧∧ 𝐓❪0,c1❫ & 𝐓❪0,c2❫.
53 elim (ist_inv_max … H) -H #n1 #n2 #Hn1 #Hn2 #H
54 elim (max_inv_O3 … H) -H #H1 #H2 destruct
58 lemma ist_inv_max_O_dx: ∀n,c1,c2. 𝐓❪n,c1 ∨ c2❫ → 𝐓❪0,c2❫ → 𝐓❪n,c1❫.
60 elim (ist_inv_max … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct //