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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "basic_2/syntax/bind.ma".
17 (* EXTENSION TO BINDERS OF A RELATION FOR TERMS *****************************)
19 inductive ext2 (R:relation term): relation bind ≝
20 | ext2_unit: ∀I. ext2 R (BUnit I) (BUnit I)
21 | ext2_pair: ∀I,V1,V2. R V1 V2 → ext2 R (BPair I V1) (BPair I V2)
24 (* Basic_inversion lemmas **************************************************)
26 fact ext2_inv_unit_sn_aux: ∀R,Z1,Z2. ext2 R Z1 Z2 →
27 ∀I. Z1 = BUnit I → Z2 = BUnit I.
28 #R #Z1 #Z2 * -Z1 -Z2 #I [2: #V1 #V2 #_ ]
32 lemma ext2_inv_unit_sn: ∀R,I,Z2. ext2 R (BUnit I) Z2 → Z2 = BUnit I.
33 /2 width=4 by ext2_inv_unit_sn_aux/ qed-.
35 fact ext2_inv_pair_sn_aux: ∀R,Z1,Z2. ext2 R Z1 Z2 →
36 ∀I,V1. Z1 = BPair I V1 →
37 ∃∃V2. R V1 V2 & Z2 = BPair I V2.
38 #R #Z1 #Z2 * -Z1 -Z2 #I [2: #V1 #V2 #HV12 ]
39 #J #W1 #H destruct /2 width=3 by ex2_intro/
42 lemma ext2_inv_pair_sn: ∀R,Z2,I,V1. ext2 R (BPair I V1) Z2 →
43 ∃∃V2. R V1 V2 & Z2 = BPair I V2.
44 /2 width=3 by ext2_inv_pair_sn_aux/ qed-.
46 fact ext2_inv_unit_dx_aux: ∀R,Z1,Z2. ext2 R Z1 Z2 →
47 ∀I. Z2 = BUnit I → Z1 = BUnit I.
48 #R #Z1 #Z2 * -Z1 -Z2 #I [2: #V1 #V2 #_ ]
52 lemma ext2_inv_unit_dx: ∀R,I,Z1. ext2 R Z1 (BUnit I) → Z1 = BUnit I.
53 /2 width=4 by ext2_inv_unit_dx_aux/ qed-.
55 fact ext2_inv_pair_dx_aux: ∀R,Z1,Z2. ext2 R Z1 Z2 →
56 ∀I,V2. Z2 = BPair I V2 →
57 ∃∃V1. R V1 V2 & Z1 = BPair I V1.
58 #R #Z1 #Z2 * -Z1 -Z2 #I [2: #V1 #V2 #HV12 ]
59 #J #W2 #H destruct /2 width=3 by ex2_intro/
62 lemma ext2_inv_pair_dx: ∀R,Z1,I,V2. ext2 R Z1 (BPair I V2) →
63 ∃∃V1. R V1 V2 & Z1 = BPair I V1.
64 /2 width=3 by ext2_inv_pair_dx_aux/ qed-.
66 (* Basic properties ********************************************************)
68 lemma ext2_refl: ∀R. reflexive … R → reflexive … (ext2 R).
69 #R #HR * /2 width=1 by ext2_pair/
72 lemma ext2_sym: ∀R. symmetric … R → symmetric … (ext2 R).
73 #R #HR #T1 #T2 * /3 width=1 by ext2_unit, ext2_pair/