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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/dynamic/nta.ma".
17 (* LOCAL ENVIRONMENT REFINEMENT FOR NATIVE TYPE ASSIGNMENT ******************)
19 (* Note: may not be transitive *)
20 inductive lsubn (h:sh): relation lenv ≝
21 | lsubn_atom: lsubn h (⋆) (⋆)
22 | lsubn_pair: ∀I,L1,L2,W. lsubn h L1 L2 → lsubn h (L1. ⓑ{I} W) (L2. ⓑ{I} W)
23 | lsubn_abbr: ∀L1,L2,V,W. ⦃h, L1⦄ ⊢ V : W → ⦃h, L2⦄ ⊢ V : W →
24 lsubn h L1 L2 → lsubn h (L1. ⓓV) (L2. ⓛW)
28 "local environment refinement (native type assigment)"
29 'CrSubEqN h L1 L2 = (lsubn h L1 L2).
31 (* Basic inversion lemmas ***************************************************)
33 fact lsubn_inv_atom1_aux: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → L1 = ⋆ → L2 = ⋆.
36 | #I #L1 #L2 #V #_ #H destruct
37 | #L1 #L2 #V #W #_ #_ #_ #H destruct
41 lemma lsubn_inv_atom1: ∀h,L2. h ⊢ ⋆ :⊑ L2 → L2 = ⋆.
44 fact lsubn_inv_pair1_aux: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V →
45 (∃∃K2. h ⊢ K1 :⊑ K2 & L2 = K2. ⓑ{I} V) ∨
46 ∃∃K2,W. ⦃h, K1⦄ ⊢ V : W & ⦃h, K2⦄ ⊢ V : W &
47 h ⊢ K1 :⊑ K2 & L2 = K2. ⓛW & I = Abbr.
49 [ #I #K1 #V #H destruct
50 | #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
51 | #L1 #L2 #V #W #H1VW #H2VW #HL12 #I #K1 #V1 #H destruct /3 width=7/
55 lemma lsubn_inv_pair1: ∀h,I,K1,L2,V. h ⊢ K1. ⓑ{I} V :⊑ L2 →
56 (∃∃K2. h ⊢ K1 :⊑ K2 & L2 = K2. ⓑ{I} V) ∨
57 ∃∃K2,W. ⦃h, K1⦄ ⊢ V : W & ⦃h, K2⦄ ⊢ V : W &
58 h ⊢ K1 :⊑ K2 & L2 = K2. ⓛW & I = Abbr.
61 fact lsubn_inv_atom2_aux: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → L2 = ⋆ → L1 = ⋆.
64 | #I #L1 #L2 #V #_ #H destruct
65 | #L1 #L2 #V #W #_ #_ #_ #H destruct
69 lemma lsubc_inv_atom2: ∀h,L1. h ⊢ L1 :⊑ ⋆ → L1 = ⋆.
72 fact lsubn_inv_pair2_aux: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W →
73 (∃∃K1. h ⊢ K1 :⊑ K2 & L1 = K1. ⓑ{I} W) ∨
74 ∃∃K1,V. ⦃h, K1⦄ ⊢ V : W & ⦃h, K2⦄ ⊢ V : W &
75 h ⊢ K1 :⊑ K2 & L1 = K1. ⓓV & I = Abst.
77 [ #I #K2 #W #H destruct
78 | #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
79 | #L1 #L2 #V #W #H1VW #H2VW #HL12 #I #K2 #W2 #H destruct /3 width=7/
83 (* Basic_1: was: csubt_gen_bind *)
84 lemma lsubn_inv_pair2: ∀h,I,L1,K2,W. h ⊢ L1 :⊑ K2. ⓑ{I} W →
85 (∃∃K1. h ⊢ K1 :⊑ K2 & L1 = K1. ⓑ{I} W) ∨
86 ∃∃K1,V. ⦃h, K1⦄ ⊢ V : W & ⦃h, K2⦄ ⊢ V : W &
87 h ⊢ K1 :⊑ K2 & L1 = K1. ⓓV & I = Abst.
90 (* Basic_forward lemmas *****************************************************)
92 lemma lsubn_fwd_lsubs1: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → L1 ≼[0, |L1|] L2.
93 #h #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
96 lemma lsubn_fwd_lsubs2: ∀h,L1,L2. h ⊢ L1 :⊑ L2 → L1 ≼[0, |L2|] L2.
97 #h #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
100 (* Basic properties *********************************************************)
102 (* Basic_1: was: csubt_refl *)
103 lemma lsubn_refl: ∀h,L. h ⊢ L :⊑ L.
104 #h #L elim L -L // /2 width=1/
107 (* Basic_1: removed theorems 6:
108 csubt_gen_flat csubt_drop_flat csubt_clear_conf
109 csubt_getl_abbr csubt_getl_abst csubt_ty3_ld