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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/substitution/tps_tps.ma".
16 include "Basic-2/unfold/tpss_lift.ma".
18 (* PARTIAL UNFOLD ON TERMS **************************************************)
20 (* Advanced properties ******************************************************)
22 lemma tpss_tps: ∀L,T1,T2,d. L ⊢ T1 [d, 1] ≫* T2 → L ⊢ T1 [d, 1] ≫ T2.
23 #L #T1 #T2 #d #H @(tpss_ind … H) -H T2 //
25 lapply (tps_trans_ge … IHT1 … HT2 ?) //
28 lemma tpss_strip_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 [d1, e1] ≫* T1 →
29 ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫ T2 →
30 ∃∃T. L ⊢ T1 [d2, e2] ≫ T & L ⊢ T2 [d1, e1] ≫* T.
33 lemma tpss_strip_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 [d1, e1] ≫* T1 →
34 ∀L2,T2,d2,e2. L2 ⊢ T0 [d2, e2] ≫ T2 →
35 (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
36 ∃∃T. L2 ⊢ T1 [d2, e2] ≫ T & L1 ⊢ T2 [d1, e1] ≫* T.
39 lemma tpss_strap1_down: ∀L,T1,T0,d1,e1. L ⊢ T1 [d1, e1] ≫* T0 →
40 ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫ T2 → d2 + e2 ≤ d1 →
41 ∃∃T. L ⊢ T1 [d2, e2] ≫ T & L ⊢ T [d1, e1] ≫* T2.
44 lemma tpss_strap2_down: ∀L,T1,T0,d1,e1. L ⊢ T1 [d1, e1] ≫ T0 →
45 ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫* T2 → d2 + e2 ≤ d1 →
46 ∃∃T. L ⊢ T1 [d2, e2] ≫* T & L ⊢ T [d1, e1] ≫ T2.
49 lemma tpss_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫* T2 →
50 ∀i. d ≤ i → i ≤ d + e →
51 ∃∃T. L ⊢ T1 [d, i - d] ≫* T & L ⊢ T [i, d + e - i] ≫* T2.
52 #L #T1 #T2 #d #e #H #i #Hdi #Hide @(tpss_ind … H) -H T2
54 | #T #T2 #_ #HT12 * #T3 #HT13 #HT3
55 elim (tps_split_up … HT12 … Hdi Hide) -HT12 Hide #T0 #HT0 #HT02
56 elim (tpss_strap1_down … HT3 … HT0 ?) -T [2: <plus_minus_m_m_comm // ]
57 /3 width=7 by ex2_1_intro, step/ (**) (* just /3 width=7/ is too slow *)
61 lemma tpss_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫* U2 →
62 ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
63 d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
64 ∃∃T2. K ⊢ T1 [d, dt + et - (d + e)] ≫* T2 & ↑[d, e] T2 ≡ U2.
65 #L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
66 elim (tpss_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
67 lapply (tpss_weak … HU1 d e ? ?) -HU1 // <plus_minus_m_m_comm // -Hddt Hdtde #HU1
68 lapply (tpss_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct -U1;
69 elim (tpss_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 HLK HTU1 // <minus_plus_m_m /2/
72 (* Main properties **********************************************************)
74 theorem tpss_conf_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 [d1, e1] ≫* T1 →
75 ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫* T2 →
76 ∃∃T. L ⊢ T1 [d2, e2] ≫* T & L ⊢ T2 [d1, e1] ≫* T.
79 theorem tpss_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 [d1, e1] ≫* T1 →
80 ∀L2,T2,d2,e2. L2 ⊢ T0 [d2, e2] ≫* T2 →
81 (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
82 ∃∃T. L2 ⊢ T1 [d2, e2] ≫* T & L1 ⊢ T2 [d1, e1] ≫* T.
85 theorem tpss_trans_eq: ∀L,T1,T,T2,d,e.
86 L ⊢ T1 [d, e] ≫* T → L ⊢ T [d, e] ≫* T2 →
90 theorem tpss_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 [d1, e1] ≫* T0 →
91 ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫* T2 → d2 + e2 ≤ d1 →
92 ∃∃T. L ⊢ T1 [d2, e2] ≫* T & L ⊢ T [d1, e1] ≫* T2.