matita/matita/contribs/lambdadelta/ground/etc/relocation/rtmap_le.etc

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  14
  15 include "ground_2/relocation/rtmap_tl.ma".
  16
  17 (* RELOCATION MAP ***********************************************************)
  18
  19 inductive le (f1): predicate rtmap ≝
  20 | le_eq: ∀f2. f1 ≗ f2 → le f1 f2
  21 | le_tl: ∀f2,g2. le f1 f2 → ↓g2 = f2 → le f1 g2
  22 .
  23
  24 interpretation "less or equal to (rtmap)" 'leq x y = (le x y).
  25
  26 (* Basic properties *********************************************************)
  27
  28 lemma le_refl: reflexive … le.
  29 /2 width=1 by eq_refl, le_eq/ qed.
  30
  31 lemma le_eq_repl_back_dx: ∀f1. eq_repl_back (λf2. f1 ≤ f2).
  32 #f #f1 #Hf1 elim Hf1 -f1
  33 /4 width=3 by le_tl, le_eq, tl_eq_repl, eq_trans/
  34 qed-.
  35
  36 lemma le_eq_repl_fwd_dx: ∀f1. eq_repl_fwd (λf2. f1 ≤ f2).
  37 #f1 @eq_repl_sym /2 width=3 by le_eq_repl_back_dx/
  38 qed-.
  39
  40 lemma le_eq_repl_back_sn: ∀f2. eq_repl_back (λf1. f1 ≤ f2).
  41 #f #f1 #Hf1 elim Hf1 -f
  42 /4 width=3 by le_tl, le_eq, tl_eq_repl, eq_canc_sn/
  43 qed-.
  44
  45 lemma le_eq_repl_fwd_sn: ∀f2. eq_repl_fwd (λf1. f1 ≤ f2).
  46 #f2 @eq_repl_sym /2 width=3 by le_eq_repl_back_sn/
  47 qed-.
  48
  49 lemma le_tl_comp: ∀f1,f2. f1 ≤ f2 → ∀g1,g2. ↓f1 = g1 → ↓f2 = g2 → g1 ≤ g2.
  50 #f1 #f2 #H elim H -f2
  51 /3 width=3 by le_tl, le_eq, tl_eq_repl/
  52 qed.
  53
  54 (* Main properties **********************************************************)
  55
  56 theorem le_trans: Transitive … le.
  57 #f1 #f #H elim H -f
  58 /4 width=5 by le_tl_comp, le_eq_repl_fwd_sn, le_tl/
  59 qed-.