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

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  14
  15 include "ground/arith/nat_lt_plus.ma".
  16 include "ground/relocation/rtmap_basic_at.ma".
  17 include "ground/relocation/pstream_after.ma".
  18
  19 (* RELOCATION P-STREAM ******************************************************)
  20
  21 (* Properties on basic relocation (specific) **********************************)
  22
  23 lemma apply_basic_lt: ∀m,n,i. ninj i ≤ m → 𝐛❨m,n❩@❨i❩ = i.
  24 /3 width=1 by at_inv_total, gr_pat_basic_lt/ qed-.
  25
  26 lemma apply_basic_ge: ∀m,n,i. m < ninj i → 𝐛❨m,n❩@❨i❩ = i+n.
  27 /3 width=1 by at_inv_total, gr_pat_basic_ge/ qed-.
  28
  29 (* Main properties on basic relocation (specific) *****************************)
  30
  31 theorem basic_swap: ∀d1,d2. d2 ≤ d1 →
  32                     ∀h1,h2. 𝐛❨d2,h2❩∘𝐛❨d1,h1❩ ≡ 𝐛❨d1+h2,h1❩∘𝐛❨d2,h2❩.
  33 #d1 #d2 #Hd21 #h1 #h2
  34 @nstream_inv_eq
  35 @nstream_eq_inv_ext #i
  36 <compose_apply <compose_apply
  37 elim (nat_split_lt_ge d2 i) #Hd2
  38 [ elim (nat_split_lt_ge d1 i) -Hd21 #Hd1
  39   [ >(apply_basic_ge … Hd1) >(apply_basic_ge … Hd2) >apply_basic_ge
  40     [ >apply_basic_ge // >nrplus_inj_sn /2 width=1 by nlt_plus_bi_sn/
  41     | >nrplus_inj_sn /2 width=2 by nlt_plus_dx_dx/
  42     ]
  43   | >(apply_basic_lt … Hd1) >(apply_basic_ge … Hd2)
  44     >apply_basic_lt // >nrplus_inj_sn /2 width=1 by nle_plus_bi_dx/
  45   ]
  46 | lapply (nle_trans … Hd2 … Hd21) -Hd21 #Hd1
  47   >(apply_basic_lt … Hd1) >(apply_basic_lt … Hd2)
  48   >apply_basic_lt /2 width=1 by nle_plus_dx_dx/
  49 ]
  50 qed-.