matita/matita/contribs/lambdadelta/ground/arith/arith_2a.ma

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  14
  15 include "ground/notation/functions/two_0.ma".
  16 include "ground/arith/nat_le_minus_plus.ma".
  17 include "ground/arith/nat_lt.ma".
  18
  19 (* ARITHMETICAL PROPERTIES FOR λδ-2A ****************************************)
  20
  21 interpretation
  22   "zero (non-negative integers)"
  23   'Two = (ninj (psucc punit)).
  24
  25 (* Equalities ***************************************************************)
  26
  27 lemma plus_n_2: ∀n. (n + 𝟐) = n + 𝟏 + 𝟏.
  28 // qed.
  29
  30 lemma arith_b1: ∀a,b,c1. c1 ≤ b → a - c1 - (b - c1) = a - b.
  31 #a #b #c1 #H >nminus_comm <nminus_assoc_comm_23 //
  32 qed-.
  33
  34 lemma arith_b2: ∀a,b,c1,c2. c1 + c2 ≤ b → a - c1 - c2 - (b - c1 - c2) = a - b.
  35 #a #b #c1 #c2 #H
  36 >(nminus_plus_assoc ? c1 c2) >(nminus_plus_assoc ? c1 c2)
  37 /2 width=1 by arith_b1/
  38 qed-.
  39
  40 lemma arith_c1x: ∀x,a,b,c1. x + c1 + a - (b + c1) = x + a - b.
  41 #x #a #b #c1
  42 <nplus_plus_comm_23 //
  43 qed.
  44
  45 lemma arith_h1: ∀a1,a2,b,c1. c1 ≤ a1 → c1 ≤ b →
  46                 a1 - c1 + a2 - (b - c1) = a1 + a2 - b.
  47 #a1 #a2 #b #c1 #H1 #H2
  48 >nminus_plus_comm_23
  49 /2 width=1 by arith_b1/
  50 qed-.
  51
  52 lemma arith_i: ∀x,y,z. y < x → x+z-y-(𝟏) = x-y-(𝟏)+z.
  53 /2 width=1 by nminus_plus_comm_23/ qed-.
  54
  55 (* Constructions ************************************************************)
  56
  57 fact le_repl_sn_conf_aux: ∀x,y,z:nat. x ≤ z → x = y → y ≤ z.
  58 // qed-.
  59
  60 fact le_repl_sn_trans_aux: ∀x,y,z:nat. x ≤ z → y = x → y ≤ z.
  61 // qed-.
  62
  63 lemma monotonic_le_minus_l2: ∀x1,x2,y,z. x1 ≤ x2 → x1 - y - z ≤ x2 - y - z.
  64 /3 width=1 by nle_minus_bi_dx/ qed.
  65
  66 lemma arith_j: ∀x,y,z. x-y-(𝟏) ≤ x-(y-z)-𝟏.
  67 /3 width=1 by nle_minus_bi_dx, nle_minus_bi_sn/ qed.
  68
  69 lemma arith_k_sn: ∀z,x,y,n. z < x → x+n ≤ y → x-z-(𝟏)+n ≤ y-z-𝟏.
  70 #z #x #y #n #Hzx #Hxny
  71 >nminus_plus_comm_23 [2: /2 width=1 by nle_minus_bi_sn/ ]
  72 >nminus_plus_comm_23 [2: /2 width=1 by nlt_des_le/ ]
  73 /2 width=1 by monotonic_le_minus_l2/
  74 qed.
  75
  76 lemma arith_k_dx: ∀z,x,y,n. z < x → y ≤ x+n → y-z-(𝟏) ≤ x-z-(𝟏)+n.
  77 #z #x #y #n #Hzx #Hyxn
  78 >nminus_plus_comm_23 [2: /2 width=1 by nle_minus_bi_sn/ ]
  79 >nminus_plus_comm_23 [2: /2 width=1 by nlt_des_le/ ]
  80 /2 width=1 by monotonic_le_minus_l2/
  81 qed.
  82
  83 (* Inversions ***************************************************************)
  84
  85 lemma lt_plus_SO_to_le: ∀x,y. x < y + (𝟏) → x ≤ y.
  86 /2 width=1 by nle_inv_succ_bi/ qed-.
  87
  88 (* Iterators ****************************************************************)
  89
  90 lemma iter_SO: ∀B:Type[0]. ∀f:B→B. ∀b,l. (f^(l+𝟏)) b = f ((f^l) b).
  91 #B #f #b #l
  92 <niter_succ //
  93 qed.