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14
15 include "basic_2/equivalence/lsubss.ma".
16
17 (* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED STATIC TYPE ASSIGNMENT *******)
18
19 (* Properties concerning basic local environment slicing ********************)
20
21 (* Note: the constant 0 cannot be generalized *)
22 lemma lsubss_ldrop_O1_conf: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 →
23                             ∀K1,e. ⇩[0, e] L1 ≡ K1 →
24                             ∃∃K2. h ⊢ K1 •⊑[g] K2 & ⇩[0, e] L2 ≡ K2.
25 #h #g #L1 #L2 #H elim H -L1 -L2
26 [ /2 width=3/
27 | #I #L1 #L2 #V #_ #IHL12 #K1 #e #H
28   elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
29   [ destruct
30     elim (IHL12 L1 0) -IHL12 // #X #HL12 #H
31     <(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
32   | elim (IHL12 … HLK1) -L1 /3 width=3/
33   ]
34 | #L1 #L2 #V1 #V2 #W1 #W2 #l #HW12 #HVW1 #HWV2 #_ #IHL12 #K1 #e #H
35   elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
36   [ destruct
37     elim (IHL12 L1 0) -IHL12 // #X #HL12 #H
38     <(ldrop_inv_O2 … H) in HL12; -H /3 width=6/
39   | elim (IHL12 … HLK1) -L1 /3 width=3/
40   ]
41 ]
42 qed-.
43
44 (* Note: the constant 0 cannot be generalized *)
45 lemma lsubss_ldrop_O1_trans: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 →
46                              ∀K2,e. ⇩[0, e] L2 ≡ K2 →
47                              ∃∃K1. h ⊢ K1 •⊑[g] K2 & ⇩[0, e] L1 ≡ K1.
48 #h #g #L1 #L2 #H elim H -L1 -L2
49 [ /2 width=3/
50 | #I #L1 #L2 #V #_ #IHL12 #K2 #e #H
51   elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
52   [ destruct
53     elim (IHL12 L2 0) -IHL12 // #X #HL12 #H
54     <(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
55   | elim (IHL12 … HLK2) -L2 /3 width=3/
56   ]
57 | #L1 #L2 #V1 #V2 #W1 #W2 #l #HW12 #HVW1 #HWV2 #_ #IHL12 #K2 #e #H
58   elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
59   [ destruct
60     elim (IHL12 L2 0) -IHL12 // #X #HL12 #H
61     <(ldrop_inv_O2 … H) in HL12; -H /3 width=6/
62   | elim (IHL12 … HLK2) -L2 /3 width=3/
63   ]
64 ]
65 qed-.