matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_isuni.ma

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  14
  15 include "ground_2/notation/relations/isuniform_1.ma".
  16 include "ground_2/relocation/rtmap_isfin.ma".
  17
  18 (* RELOCATION MAP ***********************************************************)
  19
  20 inductive isuni: predicate rtmap ≝
  21 | isuni_isid: ∀f. 𝐈❪f❫ → isuni f
  22 | isuni_next: ∀f. isuni f → ∀g. ↑f = g → isuni g
  23 .
  24
  25 interpretation "test for uniformity (rtmap)"
  26    'IsUniform f = (isuni f).
  27
  28 (* Basic inversion lemmas ***************************************************)
  29
  30 lemma isuni_inv_push: ∀g. 𝐔❪g❫ → ∀f. ⫯f = g → 𝐈❪f❫.
  31 #g * -g /2 width=3 by isid_inv_push/
  32 #f #_ #g #H #x #Hx destruct elim (discr_push_next … Hx)
  33 qed-.
  34
  35 lemma isuni_inv_next: ∀g. 𝐔❪g❫ → ∀f. ↑f = g → 𝐔❪f❫.
  36 #g * -g #f #Hf
  37 [ #x #Hx elim (isid_inv_next … Hf … Hx)
  38 | #g #H #x #Hx destruct /2 width=1 by injective_push/
  39 ]
  40 qed-.
  41
  42 lemma isuni_split: ∀g. 𝐔❪g❫ → (∃∃f. 𝐈❪f❫ & ⫯f = g) ∨ (∃∃f.𝐔❪f❫ & ↑f = g).
  43 #g #H elim (pn_split g) * #f #Hf
  44 /4 width=3 by isuni_inv_next, isuni_inv_push, or_introl, or_intror, ex2_intro/
  45 qed-.
  46
  47 (* basic forward lemmas *****************************************************)
  48
  49 lemma isuni_fwd_push: ∀g. 𝐔❪g❫ → ∀f. ⫯f = g → 𝐔❪f❫.
  50 /3 width=3 by isuni_inv_push, isuni_isid/ qed-.
  51
  52 (* Forward lemmas with test for finite colength *****************************)
  53
  54 lemma isuni_fwd_isfin: ∀f. 𝐔❪f❫ → 𝐅❪f❫.
  55 #f #H elim H -f /3 width=1 by isfin_next, isfin_isid/
  56 qed-.