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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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11 (* v GNU General Public License Version 2 *)
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15 include "lambda-delta/substitution/tps_lift.ma".
16 include "lambda-delta/reduction/tpr.ma".
18 (* Relocation properties ****************************************************)
20 lemma tpr_lift: ∀T1,T2. T1 ⇒ T2 →
21 ∀d,e,U1. ↑[d, e] T1 ≡ U1 → ∀U2. ↑[d, e] T2 ≡ U2 → U1 ⇒ U2.
22 #T1 #T2 #H elim H -H T1 T2
23 [ #k #d #e #U1 #HU1 #U2 #HU2
24 lapply (lift_mono … HU1 … HU2) -HU1 #H destruct -U1;
25 lapply (lift_inv_sort1 … HU2) -HU2 #H destruct -U2 //
26 | #i #d #e #U1 #HU1 #U2 #HU2
27 lapply (lift_mono … HU1 … HU2) -HU1 #H destruct -U1;
28 lapply (lift_inv_lref1 … HU2) * * #Hid #H destruct -U2 //
29 | #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X1 #HX1 #X2 #HX2
30 elim (lift_inv_bind1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct -X1;
31 elim (lift_inv_bind1 … HX2) -HX2 #W2 #U2 #HVW2 #HTU2 #HX2 destruct -X2 /3/
32 | #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X1 #HX1 #X2 #HX2
33 elim (lift_inv_flat1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct -X1;
34 elim (lift_inv_flat1 … HX2) -HX2 #W2 #U2 #HVW2 #HTU2 #HX2 destruct -X2 /3/
35 | #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X1 #HX1 #X2 #HX2
36 elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct -X1;
37 elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct -X;
38 elim (lift_inv_bind1 … HX2) -HX2 #V3 #T3 #HV23 #HT23 #HX2 destruct -X2 /3/
39 | #V1 #V2 #T1 #T2 #T0 #HV12 #HT12 #HT2 #IHV12 #IHT12 #d #e #X1 #HX1 #X2 #HX2
40 elim (lift_inv_bind1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct -X1;
41 elim (lift_inv_bind1 … HX2) -HX2 #W2 #U0 #HVW2 #HTU0 #HX2 destruct -X2;
42 elim (lift_total T2 (d + 1) e) #U2 #HTU2
44 [4: @(tps_lift_le … HT2 … HTU2 HTU0 ?) /2/ |1: skip |2: /2/ |3: /2/ ] (**) (*/3. is too slow *)
45 | #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #d #e #X1 #HX1 #X2 #HX2
46 elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct -X1;
47 elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct -X;
48 elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct -X2;
49 elim (lift_inv_flat1 … HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct -X;
50 elim (lift_trans_ge … HV2 … HV3 ?) -HV2 HV3 V // /3/
51 | #V #T #T1 #T2 #HT1 #_ #IHT12 #d #e #X #HX #T0 #HT20
52 elim (lift_inv_bind1 … HX) -HX #V3 #T3 #_ #HT3 #HX destruct -X;
53 elim (lift_trans_ge … HT1 … HT3 ?) -HT1 HT3 T // /3 width=6/
54 | #V #T1 #T2 #_ #IHT12 #d #e #X #HX #T #HT2
55 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #_ #HT0 #HX destruct -X /3/
59 lemma tpr_inv_lift: ∀T1,T2. T1 ⇒ T2 →
60 ∀d,e,U1. ↑[d, e] U1 ≡ T1 →
61 ∃∃U2. ↑[d, e] U2 ≡ T2 & U1 ⇒ U2.
62 #T1 #T2 #H elim H -H T1 T2
64 lapply (lift_inv_sort2 … HU1) -HU1 #H destruct -U1 /2/
66 lapply (lift_inv_lref2 … HU1) -HU1 * * #Hid #H destruct -U1 /3/
67 | #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X #HX
68 elim (lift_inv_bind2 … HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct -X;
69 elim (IHV12 … HV01) -IHV12 HV01;
70 elim (IHT12 … HT01) -IHT12 HT01 /3 width=5/
71 | #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X #HX
72 elim (lift_inv_flat2 … HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct -X;
73 elim (IHV12 … HV01) -IHV12 HV01;
74 elim (IHT12 … HT01) -IHT12 HT01 /3 width=5/
75 | #V1 #V2 #W1 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X #HX
76 elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct -X;
77 elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct -Y;
78 elim (IHV12 … HV01) -IHV12 HV01;
79 elim (IHT12 … HT01) -IHT12 HT01 /3 width=5/
80 | #V1 #V2 #T1 #T2 #T0 #_ #_ #HT20 #IHV12 #IHT12 #d #e #X #HX
81 elim (lift_inv_bind2 … HX) -HX #W1 #U1 #HWV1 #HUT1 #HX destruct -X;
82 elim (IHV12 … HWV1) -IHV12 HWV1 #W2 #HWV2 #HW12
83 elim (IHT12 … HUT1) -IHT12 HUT1 #U2 #HUT2 #HU12
84 elim (tps_inv_lift1_le … HT20 … HUT2 ?) -HT20 HUT2 // [3: /2 width=5/ |2: skip ] #U0 #HU20 #HUT0
85 @ex2_1_intro [2: /2/ |1: skip |3: /2/ ] (**) (* /3 width=5/ is slow *)
86 | #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #d #e #X #HX
87 elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct -X;
88 elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct -Y;
89 elim (IHV12 … HV01) -IHV12 HV01 #V3 #HV32 #HV03
90 elim (IHW12 … HW01) -IHW12 HW01 #W3 #HW32 #HW03
91 elim (IHT12 … HT01) -IHT12 HT01 #T3 #HT32 #HT03
92 elim (lift_trans_le … HV32 … HV2 ?) -HV32 HV2 V2 // #V2 #HV32 #HV2
93 @ex2_1_intro [2: /3/ |1: skip |3: /2/ ] (**) (* /4 width=5/ is slow *)
94 | #V #T #T1 #T2 #HT1 #_ #IHT12 #d #e #X #HX
95 elim (lift_inv_bind2 … HX) -HX #V0 #T0 #_ #HT0 #H destruct -X;
96 elim (lift_div_le … HT1 … HT0 ?) -HT1 HT0 T // #T #HT0 #HT1
97 elim (IHT12 … HT1) -IHT12 HT1 /3 width=5/
98 | #V #T1 #T2 #_ #IHT12 #d #e #X #HX
99 elim (lift_inv_flat2 … HX) -HX #V0 #T0 #_ #HT01 #H destruct -X;
100 elim (IHT12 … HT01) -IHT12 HT01 /3/