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14
15 include "basic_2/substitution/ldrop_ldrop.ma".
16 include "basic_2/static/ssta.ma".
17
18 (* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
19
20 (* Properties on relocation *************************************************)
21
22 (* Basic_1: was just: sty0_lift *)
23 lemma ssta_lift: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
24                  ∀L2,d,e. ⇩[d, e] L2 ≡ L1 → ∀T2. ⇧[d, e] T1 ≡ T2 →
25                  ∀U2. ⇧[d, e] U1 ≡ U2 → ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄.
26 #h #g #L1 #T1 #U1 #l #H elim H -L1 -T1 -U1 -l
27 [ #L1 #k #l #Hkl #L2 #d #e #HL21 #X1 #H1 #X2 #H2
28   >(lift_inv_sort1 … H1) -X1
29   >(lift_inv_sort1 … H2) -X2 /2 width=1/
30 | #L1 #K1 #V1 #W1 #W #i #l #HLK1 #_ #HW1 #IHVW1 #L2 #d #e #HL21 #X #H #U2 #HWU2
31   elim (lift_inv_lref1 … H) * #Hid #H destruct
32   [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // #W2 #HW12 #HWU2
33     elim (ldrop_trans_le … HL21 … HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H
34     elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
35     /3 width=8/
36   | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W // /2 width=1/ #HW1U2
37     lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/
38   ]
39 | #L1 #K1 #W1 #V1 #W #i #l #HLK1 #_ #HW1 #IHWV1 #L2 #d #e #HL21 #X #H #U2 #HWU2
40   elim (lift_inv_lref1 … H) * #Hid #H destruct
41   [ elim (lift_trans_ge … HW1 … HWU2 ?) -W // <minus_plus #W #HW1 #HWU2
42     elim (ldrop_trans_le … HL21 … HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H
43     elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K2 #W2 #HK21 #HW12 #H destruct
44     lapply (lift_mono … HW1 … HW12) -HW1 #H destruct
45     elim (lift_total V1 (d-i-1) e) /3 width=8/
46   | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W // /2 width=1/ #HW1U2
47     lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/
48   ]
49 | #a #I #L1 #V1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL21 #X1 #H1 #X2 #H2
50   elim (lift_inv_bind1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
51   elim (lift_inv_bind1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
52   lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=5/
53 | #L1 #V1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL21 #X1 #H1 #X2 #H2
54   elim (lift_inv_flat1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
55   elim (lift_inv_flat1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
56   lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=5/
57 | #L1 #W1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL21 #X #H #U2 #HU12
58   elim (lift_inv_flat1 … H) -H #W2 #T2 #HW12 #HT12 #H destruct /3 width=5/
59 ]
60 qed.
61
62 (* Note: apparently this was missing in basic_1 *)
63 lemma ssta_inv_lift1: ∀h,g,L2,T2,U2,l. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ →
64                       ∀L1,d,e. ⇩[d, e] L2 ≡ L1 → ∀T1. ⇧[d, e] T1 ≡ T2 →
65                       ∃∃U1. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ & ⇧[d, e] U1 ≡ U2.
66 #h #g #L2 #T2 #U2 #l #H elim H -L2 -T2 -U2 -l
67 [ #L2 #k #l #Hkl #L1 #d #e #_ #X #H
68   >(lift_inv_sort2 … H) -X /3 width=3/
69 | #L2 #K2 #V2 #W2 #W #i #l #HLK2 #HVW2 #HW2 #IHVW2 #L1 #d #e #HL21 #X #H
70   elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HVW2 | -IHVW2 ]
71   [ elim (ldrop_conf_lt … HL21 … HLK2 ?) -L2 // #K1 #V1 #HLK1 #HK21 #HV12
72     elim (IHVW2 … HK21 … HV12) -K2 -V2 #W1 #HVW1 #HW12
73     elim (lift_trans_le … HW12 … HW2 ?) -W2 // >minus_plus <plus_minus_m_m // -Hid /3 width=6/
74   | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
75     elim (le_inv_plus_l … Hid) -Hid #Hdie #ei
76     elim (lift_split … HW2 d (i-e+1) ? ? ?) -HW2 // [3: /2 width=1/ ]
77     [ #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /2 width=1/ /3 width=6/
78     | <le_plus_minus_comm //
79     ]
80   ]
81 | #L2 #K2 #W2 #V2 #W #i #l #HLK2 #HWV2 #HW2 #IHWV2 #L1 #d #e #HL21 #X #H
82   elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HWV2 | -IHWV2 ]
83   [ elim (ldrop_conf_lt … HL21 … HLK2 ?) -L2 // #K1 #W1 #HLK1 #HK21 #HW12
84     elim (IHWV2 … HK21 … HW12) -K2 #V1 #HWV1 #_
85     elim (lift_trans_le … HW12 … HW2 ?) -W2 // >minus_plus <plus_minus_m_m // -Hid /3 width=6/
86   | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
87     elim (le_inv_plus_l … Hid) -Hid #Hdie #ei
88     elim (lift_split … HW2 d (i-e+1) ? ? ?) -HW2 // [3: /2 width=1/ ]
89     [ #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /2 width=1/ /3 width=6/
90     | <le_plus_minus_comm //
91     ]
92   ]
93 | #a #I #L2 #V2 #T2 #U2 #l #_ #IHTU2 #L1 #d #e #HL21 #X #H
94   elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
95   elim (IHTU2 (L1.ⓑ{I}V1) … HT12) -IHTU2 -HT12 /2 width=1/ -HL21 /3 width=5/
96 | #L2 #V2 #T2 #U2 #l #_ #IHTU2 #L1 #d #e #HL21 #X #H
97   elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
98   elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=5/
99 | #L2 #W2 #T2 #U2 #l #_ #IHTU2 #L1 #d #e #HL21 #X #H
100   elim (lift_inv_flat2 … H) -H #W1 #T1 #HW12 #HT12 #H destruct
101   elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=3/
102 ]
103 qed.
104
105 (* Advanced forvard lemmas **************************************************)
106
107 (* Basic_1: was just: sty0_correct *)
108 lemma ssta_fwd_correct: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ →
109                         ∃T0. ⦃h, L⦄ ⊢ U •[g] ⦃l-1, T0⦄.
110 #h #g #L #T #U #l #H elim H -L -T -U -l
111 [ /4 width=2/
112 | #L #K #V #W #W0 #i #l #HLK #_ #HW0 * #V0 #HWV0
113   lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
114   elim (lift_total V0 0 (i+1)) /3 width=10/
115 | #L #K #W #V #V0 #i #l #HLK #HWV #HWV0 #_
116   lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
117   elim (lift_total V 0 (i+1)) /3 width=10/
118 | #a #I #L #V #T #U #l #_ * /3 width=2/
119 | #L #V #T #U #l #_ * #T0 #HUT0 /3 width=2/
120 | #L #W #T #U #l #_ * /2 width=2/
121 ]
122 qed-.