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4 (* ||A|| A project by Andrea Asperti *)
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15 include "static_2/syntax/tdeq.ma".
17 (* SORT-IRRELEVANT EQUIVALENCE ON TERMS *************************************)
19 (* Main properties **********************************************************)
21 theorem tdeq_trans: Transitive … tdeq.
22 #T1 #T #H elim H -T1 -T
24 elim (tdeq_inv_sort1 … H) -s /2 width=1 by tdeq_sort/
25 | #i1 #i #H <(tdeq_inv_lref1 … H) -H //
26 | #l1 #l #H <(tdeq_inv_gref1 … H) -H //
27 | #I #V1 #V #T1 #T #_ #_ #IHV #IHT #X #H
28 elim (tdeq_inv_pair1 … H) -H /3 width=1 by tdeq_pair/
32 theorem tdeq_canc_sn: left_cancellable … tdeq.
33 /3 width=3 by tdeq_trans, tdeq_sym/ qed-.
35 theorem tdeq_canc_dx: right_cancellable … tdeq.
36 /3 width=3 by tdeq_trans, tdeq_sym/ qed-.
38 theorem tdeq_repl: ∀T1,T2. T1 ≛ T2 →
39 ∀U1. T1 ≛ U1 → ∀U2. T2 ≛ U2 → U1 ≛ U2.
40 /3 width=3 by tdeq_canc_sn, tdeq_trans/ qed-.
42 (* Negated main properies ***************************************************)
44 theorem tdeq_tdneq_trans: ∀T1,T. T1 ≛ T → ∀T2. (T ≛ T2 → ⊥) → T1 ≛ T2 → ⊥.
45 /3 width=3 by tdeq_canc_sn/ qed-.
47 theorem tdneq_tdeq_canc_dx: ∀T1,T. (T1 ≛ T → ⊥) → ∀T2. T2 ≛ T → T1 ≛ T2 → ⊥.
48 /3 width=3 by tdeq_trans/ qed-.