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15 include "basic_2/unfold/tpss_tpss.ma".
16 include "basic_2/unfold/delift.ma".
18 (* INVERSE BASIC TERM RELOCATION *******************************************)
20 (* Properties on partial unfold on terms ************************************)
22 lemma delift_tpss_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
23 ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
24 ∀K. ⇩[dd, ee] L ≡ K → d + e ≤ dd →
25 ∃∃T2. K ⊢ T1 ▶* [d, e] T2 & L ⊢ ▼*[dd, ee] U2 ≡ T2.
26 #L #U1 #U2 #d #e #HU12 #T1 #dd #ee * #X1 #HUX1 #HTX1 #K #HLK #H1
27 elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
28 elim (tpss_inv_lift1_le … HXU1 … HLK … HTX1 ?) -X1 -HLK // -H1 /3 width=5/
31 lemma delift_tps_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
32 ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
33 ∀K. ⇩[dd, ee] L ≡ K → d + e ≤ dd →
34 ∃∃T2. K ⊢ T1 ▶* [d, e] T2 & L ⊢ ▼*[dd, ee] U2 ≡ T2.
37 lemma delift_tpss_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
38 ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
40 d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
41 ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
42 L ⊢ ▼*[dd, ee] U2 ≡ T2.
43 #L #U1 #U2 #d #e #HU12 #T1 #dd #ee * #X1 #HUX1 #HTX1 #K #HLK #H1 #H2 #H3
44 elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
45 elim (tpss_inv_lift1_le_up … HXU1 … HLK … HTX1 ? ? ?) -X1 -HLK // -H1 -H2 -H3 /3 width=5/
48 lemma delift_tps_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
49 ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
51 d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
52 ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
53 L ⊢ ▼*[dd, ee] U2 ≡ T2.
56 lemma delift_tpss_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
57 ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
58 ∀K. ⇩[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
59 ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
60 L ⊢ ▼*[dd, ee] U2 ≡ T2.
61 #L #U1 #U2 #d #e #HU12 #T1 #dd #ee * #X1 #HUX1 #HTX1 #K #HLK #H1 #H2
62 elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
63 elim (tpss_inv_lift1_be … HXU1 … HLK … HTX1 ? ?) -X1 -HLK // -H1 -H2 /3 width=5/
66 lemma delift_tps_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
67 ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
68 ∀K. ⇩[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
69 ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
70 L ⊢ ▼*[dd, ee] U2 ≡ T2.
73 lemma delift_tpss_conf_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
74 ∀T. L ⊢ ▼*[d, e] U1 ≡ T → L ⊢ ▼*[d, e] U2 ≡ T.
75 #L #U1 #U2 #d #e #HU12 #T * #X1 #HUX1 #HTX1
76 elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
77 lapply (tpss_inv_lift1_eq … HXU1 … HTX1) -HXU1 #H destruct /2 width=3/
80 lemma delift_tps_conf_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
81 ∀T. L ⊢ ▼*[d, e] U1 ≡ T → L ⊢ ▼*[d, e] U2 ≡ T.
84 lemma tpss_delift_trans_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
85 ∀T. L ⊢ ▼*[d, e] U2 ≡ T → L ⊢ ▼*[d, e] U1 ≡ T.
86 #L #U1 #U2 #d #e #HU12 #T * #X1 #HUX1 #HTX1
87 lapply (tpss_trans_eq … HU12 … HUX1) -U2 /2 width=3/
90 lemma tps_delift_trans_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
91 ∀T. L ⊢ ▼*[d, e] U2 ≡ T → L ⊢ ▼*[d, e] U1 ≡ T.