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14
15 include "ground/xoa/ex_1_4.ma".
16 include "static_2/notation/relations/simple_1.ma".
17 include "static_2/syntax/term.ma".
18
19 (* SIMPLE (NEUTRAL) TERMS ***************************************************)
20
21 inductive simple: predicate term ≝
22    | simple_atom: ∀I. simple (⓪[I])
23    | simple_flat: ∀I,V,T. simple (ⓕ[I]V.T)
24 .
25
26 interpretation "simple (term)" 'Simple T = (simple T).
27
28 (* Basic inversion lemmas ***************************************************)
29
30 fact simple_inv_bind_aux: ∀T. 𝐒❨T❩ → ∀p,J,W,U. T = ⓑ[p,J]W.U → ⊥.
31 #T * -T
32 [ #I #p #J #W #U #H destruct
33 | #I #V #T #a #J #W #U #H destruct
34 ]
35 qed-.
36
37 lemma simple_inv_bind: ∀p,I,V,T. 𝐒❨ⓑ[p,I] V. T❩ → ⊥.
38 /2 width=7 by simple_inv_bind_aux/ qed-.
39
40 lemma simple_inv_pair: ∀I,V,T. 𝐒❨②[I]V.T❩ → ∃J. I = Flat2 J.
41 * /2 width=2 by ex_intro/
42 #p #I #V #T #H elim (simple_inv_bind … H)
43 qed-.
44
45 (* Basic properties *********************************************************)
46
47 lemma simple_dec_ex (X): ∨∨ 𝐒❨X❩ | ∃∃p,I,T,U. X = ⓑ[p,I]T.U.
48 * [ /2 width=1 by simple_atom, or_introl/ ]
49 * [| /2 width=1 by simple_flat, or_introl/ ]
50 /3 width=5 by ex1_4_intro, or_intror/
51 qed-.