source: src/utilities/binary/division.ma @ 2649

include "utilities/binary/Z.ma".
include "utilities/binary/positive.ma".


(* XXX: divides goes to arithmetics *)
inductive dividesP (n,m:Pos) : Prop ≝
| witness : ∀p:Pos.m = times n p → dividesP n m.
interpretation "positive divides" 'divides n m = (dividesP n m).
interpretation "positive not divides" 'ndivides n m = (Not (dividesP n m)).

definition dividesZ : Z → Z → Prop ≝
λx,y. match x with
[ OZ ⇒ False
| pos n ⇒ match y with [ OZ ⇒ True | pos m ⇒ dividesP n m | neg m ⇒ dividesP n m ]
| neg n ⇒ match y with [ OZ ⇒ True | pos m ⇒ dividesP n m | neg m ⇒ dividesP n m ]
].

interpretation "integer divides" 'divides n m = (dividesZ n m).
interpretation "integer not divides" 'ndivides n m = (Not (dividesZ n m)).




(* division *)

inductive natp : Type[0] ≝
| pzero : natp
| ppos  : Pos → natp.

definition natp0 ≝
λn. match n with [ pzero ⇒ pzero | ppos m ⇒ ppos (p0 m) ].

definition natp1 ≝
λn. match n with [ pzero ⇒ ppos one | ppos m ⇒ ppos (p1 m) ].

let rec divide (m,n:Pos) on m ≝ 
match m with 
[ one ⇒
  match n with
  [ one ⇒ 〈ppos one,pzero〉
  | _ ⇒ 〈pzero,ppos one〉
  ]
| p0 m' ⇒
  let 〈q,r〉 ≝ divide m' n in
    match r with
    [ pzero ⇒ 〈natp0 q,pzero〉
    | ppos r' ⇒
      match partial_minus (p0 r') n with
      [ MinusNeg ⇒ 〈natp0 q, ppos (p0 r')〉
      | MinusZero ⇒ 〈natp1 q, pzero〉
      | MinusPos r'' ⇒ 〈natp1 q, ppos r''〉
      ]
    ]
| p1 m' ⇒
  let 〈q,r〉 ≝ divide m' n in
    match r with
    [ pzero ⇒ match n with [ one ⇒ 〈natp1 q,pzero〉 | _ ⇒ 〈natp0 q,ppos one〉 ]
    | ppos r' ⇒
      match partial_minus (p1 r') n with
      [ MinusNeg ⇒ 〈natp0 q, ppos (p1 r')〉
      | MinusZero ⇒ 〈natp1 q, pzero〉
      | MinusPos r'' ⇒ 〈natp1 q, ppos r''〉
      ]
    ]
].

definition div ≝ λm,n. fst ?? (divide m n).
definition mod ≝ λm,n. snd ?? (divide m n).

lemma pred_minus: ∀n,m:Pos. pred n - m = pred (n-m).
@pos_elim /3/;
qed.

lemma minus_plus_distrib: ∀n,m,p:Pos. m-(n+p) = m-n-p.
@pos_elim
[ //
| #n #IH #m #p >(succ_plus_one …) >(IH m one) /2/;
] qed.

theorem plus_minus_r:
∀m,n,p:Pos. m < n → p+(n-m) = (p+n)-m.
#m #n #p #le >(commutative_plus_pos …)
>(commutative_plus_pos p ?) @plus_minus //; qed.

lemma plus_minus_le: ∀m,n,p:Pos. m≤n → m+p-n≤p.
#m #n #p elim m;/2/; qed.
(*
lemma le_to_minus: ∀m,n. m≤n → m-n = 0.
#m #n elim n;
[ <(minus_n_O …) /2/;
| #n' #IH #le inversion le; /2/; #n'' #A #B #C destruct;
    >(eq_minus_S_pred …) >(IH A) /2/
] qed.
*)
lemma minus_times_distrib_l: ∀n,m,p:Pos. n < m → p*m-p*n = p*(m-n).
#n #m #p (*elim (decidable_lt n m);*)#lt
(*[*) @(pos_elim … p) //;#p' #IH
    <(times_succn_m …) <(times_succn_m …) <(times_succn_m …)
    >(minus_plus_distrib …)
    <(plus_minus … lt) <IH
    >(plus_minus_r …) /2/;
qed.
(*|
lapply (not_lt_to_le … lt); #le
@(pos_elim … p) //; #p'
 cut (m-n = one); [ /3/ ]
  #mn >mn >(times_n_one …) >(times_n_one …)
  #H <H in ⊢ (???%)
  @sym_eq @le_n_one_to_eq <H
  >(minus_plus_distrib …) @monotonic_le_minus_l
  /2/;
] qed.

lemma S_pred_m_n: ∀m,n. m > n → S (pred (m-n)) = m-n.
#m #n #H lapply (refl ? (m-n));elim (m-n) in ⊢ (???% → %);//;
#H' lapply (minus_to_plus … H'); /2/;
<(plus_n_O …) #H'' >H'' in H #H
@False_ind @(absurd ? H ( not_le_Sn_n n))
qed.
*)

let rec natp_plus (n,m:natp) ≝
match n with
[ pzero ⇒ m
| ppos n' ⇒ match m with [ pzero ⇒ n | ppos m' ⇒ ppos (n'+m') ]
].

let rec natp_times (n,m:natp) ≝
match n with
[ pzero ⇒ pzero
| ppos n' ⇒ match m with [ pzero ⇒ pzero | ppos m' ⇒ ppos (n'*m') ]
].

inductive natp_lt : natp → Pos → Prop ≝
| plt_zero : ∀n. natp_lt pzero n
| plt_pos : ∀n,m. n < m → natp_lt (ppos n) m.

lemma lt_p0: ∀n:Pos. one < p0 n.
#n normalize; /2/; qed.

lemma lt_p1: ∀n:Pos. one < p1 n.
#n' normalize; >(?:p1 n' = succ (p0 n')) //; qed.

lemma divide_by_one: ∀m. divide m one = 〈ppos m,pzero〉.
#m elim m;
[ //;
| 2,3: #m' #IH normalize; >IH //;
] qed.

lemma pos_nonzero2: ∀n. ∀P:Pos→Type[0]. ∀Q:Type[0]. match succ n with [ one ⇒ Q | p0 p ⇒ P (p0 p) | p1 p ⇒ P (p1 p) ] = P (succ n).
#n #P #Q @succ_elim /2/; qed.

lemma partial_minus_to_Prop: ∀n,m.
  match partial_minus n m with
  [ MinusNeg ⇒ n < m
  | MinusZero ⇒ n = m
  | MinusPos r ⇒ n = m+r
  ].
#n #m lapply (pos_compare_to_Prop n m); lapply (minus_to_plus_pos n m);
normalize; cases (partial_minus n m); /2/; qed.

lemma double_lt1: ∀n,m:Pos. n<m → p1 n < p0 m.
#n #m #lt elim lt; /2/;
qed.

lemma double_lt2: ∀n,m:Pos. n<m → p1 n < p1 m.
#n #m #lt @(transitive_lt ? (p0 m) ?) /2/;
qed.

lemma double_lt3: ∀n,m:Pos. n<succ m → p0 n < p1 m.
#n #m #lt inversion lt;
[ #H >(succ_injective … H) //;
| #p #H1 #H2 #H3 >(succ_injective … H3)
    @(transitive_lt ? (p0 p) ?) /2/;
] 
qed.

lemma double_lt4: ∀n,m:Pos. n<m → p0 n < p0 m.
#n #m #lt elim lt; /2/;
qed.



lemma plt_lt: ∀n,m:Pos. natp_lt (ppos n) m → n<m.
#n #m #lt inversion lt;
[ #p #H destruct;
| #n' #m' #lt #e1 #e2 destruct @lt
] qed.

lemma lt_foo: ∀a,b:Pos. b+a < p0 b → a<b.
#a #b /2/; qed.

lemma lt_foo2: ∀a,b:Pos. b+a < p1 b → a<succ b.
#a #b >(?:p1 b = succ (p0 b)) /2/; qed.

lemma p0_plus: ∀n,m:Pos. p0 (n+m) = p0 n + p0 m.
/2/; qed.

theorem divide_ok : ∀m,n,dv,md.
 divide m n = 〈dv,md〉 →
 ppos m = natp_plus (natp_times dv (ppos n)) md ∧ natp_lt md n.
#m #n @(pos_cases … n)
[ >(divide_by_one m) #dv #md #H destruct normalize /2/
| #n' elim m;
  [ #dv #md normalize; >(pos_nonzero …) #H destruct /3/
  | #m' #IH #dv #md whd in ⊢ (??%? → ?(???%)?);
      lapply (refl ? (divide m' (succ n')));
      elim (divide m' (succ n')) in ⊢ (???% → % → ?);
      #dv' #md' #DIVr elim (IH … DIVr);
      whd in ⊢ (? → ? → ??%? → ?);
      cases md';
      [ cases dv'; normalize;
        [ #H destruct
        | #dv'' #Hr1 #Hr2 >(pos_nonzero …) #H destruct % /2/
        ]
      | cases dv'; [ 2: #dv'' ] @succ_elim
          normalize; #n #md'' #Hr1 #Hr2
          lapply (plt_lt … Hr2); #Hr2'
          lapply (partial_minus_to_Prop md'' n);
          cases (partial_minus md'' n); [ 3,6,9,12: #r'' ] normalize
          #lt #e destruct % [ normalize /3 by le_to_le_to_eq, le_n, eq_f/ | *: normalize /2 by plt_zero, plt_pos/ ] @plt_pos
          [ 1,3,5,7: @double_lt1 /2/;
          | 2,4: @double_lt3 /2/;
          | *: @double_lt2 /2/;
          ]
      ]
  | #m' #IH #dv #md whd in ⊢ (??%? → ?);
      lapply (refl ? (divide m' (succ n')));
      elim (divide m' (succ n')) in ⊢ (???% → % → ?);
      #dv' #md' #DIVr elim (IH … DIVr);
      whd in ⊢ (? → ? → ??%? → ?); cases md';
      [ cases dv'; normalize;
        [ #H destruct;
        | #dv'' #Hr1 #Hr2 #H destruct; /3/;
        ]
      | (*cases dv'; [ 2: #dv'' ] @succ_elim
          normalize; #n #md'' #Hr1 #Hr2 *) #md'' #Hr1 #Hr2
          lapply (plt_lt … Hr2); #Hr2'
          whd in ⊢ (??%? → ?);
          lapply (partial_minus_to_Prop (p0 md'') (succ n'));
          cases (partial_minus (p0 md'') (succ n')); [ 3(*,6,9,12*): #r'' ]
          cases dv' in Hr1 ⊢ %; [ 2,4,6: #dv'' ] normalize
          #Hr1 destruct [ 1,2,3: >(p0_plus ? md'') ]
          #lt #e [ 1,3,4,6: >lt ]
          destruct
          normalize in ⊢ (?(???%)?); % /2/; @plt_pos
          [ cut (succ n' + r'' < p0 (succ n')); /2/;
          | cut (succ n' + r'' < p0 (succ n')); /2/; ]]]]
qed.

lemma mod0_divides: ∀m,n,dv:Pos.
  divide n m = 〈ppos dv,pzero〉 → m∣n.
#m #n #dv #DIVIDE %{dv} lapply (divide_ok … DIVIDE) *; (* XXX: need ; or it eats normalize *)
normalize #H destruct //
qed.

lemma divides_mod0: ∀dv,m,n:Pos.
  n = dv*m → divide n m = 〈ppos dv,pzero〉.
#dv #m #n #DIV lapply (refl ? (divide n m));
elim (divide n m) in ⊢ (???% → ?); #dv' #md' #DIVIDE
lapply (divide_ok … DIVIDE); *;
cases dv' in DIVIDE ⊢ %; [ 2: #dv'' ]
cases md'; [ 2,4: #md'' ] #DIVIDE  normalize;
>DIV in ⊢ (% → ?); #H #lt destruct;
[ lapply (plus_to_minus_pos … e0);
    >(commutative_times …) >(commutative_times dv'' …)
    cut (dv'' < dv); [ cut (dv''*m < dv*m); /2/; ] #dvlt
    >(minus_times_distrib_l …) //;

 (*cut (0 < dv-dv'); [ lapply (not_le_to_lt … nle); /2/ ]
    #Hdv *) #H' cut (md'' ≥ m); /2/; lapply (plt_lt … lt); #A #B @False_ind
    @(absurd ? B (lt_to_not_le … A))

| @False_ind @(absurd ? (plt_lt … lt) ?) /2/;

| >DIVIDE cut (dv = dv''); /2/;
]
qed.

lemma dec_divides: ∀m,n:Pos. (m∣n) + ¬(m∣n).
#m #n lapply (refl ? (divide n m)); elim (divide n m) in ⊢ (???% → %);
#dv #md cases md; cases dv;
[ #DIVIDES lapply (divide_ok … DIVIDES); *; normalize; #H destruct
| #dv' #H %1 @mod0_divides /2/;
| #md' #DIVIDES %2 @nmk *; #dv'
    >(commutative_times …) #H lapply (divides_mod0 … H);
    >DIVIDES #H'
    destruct;
| #md' #dv' #DIVIDES %2 @nmk *; #dv'
    >(commutative_times …) #H lapply (divides_mod0 … H);
    >DIVIDES #H'
    destruct;
] qed.

theorem dec_dividesZ: ∀p,q:Z. (p∣q) + ¬(p∣q).
#p #q cases p;
[ cases q; normalize; [ %2 /2/; | *: #m %2 /2/; ]
| *: #n cases q; normalize; /2/;
] qed.

definition natp_to_Z ≝
λn. match n with [ pzero ⇒ OZ | ppos p ⇒ pos p ].

definition natp_to_negZ ≝
λn. match n with [ pzero ⇒ OZ | ppos p ⇒ neg p ].

(* TODO: check these definitions are right.  They are supposed to be the same
   as Zdiv/Zmod in Coq. *)
definition divZ ≝ λx,y.
  match x with
  [ OZ ⇒ OZ
  | pos n ⇒
    match y with
    [ OZ ⇒ OZ
    | pos m ⇒ natp_to_Z (fst ?? (divide n m))
    | neg m ⇒ let 〈q,r〉 ≝ divide n m in
                match r with [ pzero ⇒ natp_to_negZ q | _ ⇒ Zpred (natp_to_negZ q) ]
    ]
  | neg n ⇒
    match y with
    [ OZ ⇒ OZ
    | pos m ⇒ let 〈q,r〉 ≝ divide n m in
                match r with [ pzero ⇒ natp_to_negZ q | _ ⇒ Zpred (natp_to_negZ q) ]
    | neg m ⇒ natp_to_Z (fst ?? (divide n m))
    ]
  ].

definition modZ ≝ λx,y.
  match x with
  [ OZ ⇒ OZ
  | pos n ⇒
    match y with
    [ OZ ⇒ OZ
    | pos m ⇒ natp_to_Z (snd ?? (divide n m))
    | neg m ⇒ let 〈q,r〉 ≝ divide n m in
                match r with [ pzero ⇒ OZ | _ ⇒ y+(natp_to_Z r) ]
    ]
  | neg n ⇒
    match y with
    [ OZ ⇒ OZ
    | pos m ⇒ let 〈q,r〉 ≝ divide n m in
                match r with [ pzero ⇒ OZ | _ ⇒ y-(natp_to_Z r) ]
    | neg m ⇒ natp_to_Z (snd ?? (divide n m))
    ]
  ].

interpretation "natural division" 'divide m n = (div m n).
interpretation "natural modulus" 'module m n = (mod m n).
interpretation "integer division" 'divide m n = (divZ m n).
interpretation "integer modulus" 'module m n = (modZ m n).







theorem modZ_lt_mod: ∀x,y:Z. y>0 → 0 ≤ x \mod y ∧ x \mod y < y.
#x #y cases y;
[ #H @(False_ind … H)
| #m #_ cases x;
  [ % //;
  | #n whd in ⊢ (?(??%)(?%?)); lapply (refl ? (divide n m));
      cases (divide n m) in ⊢ (???% → %); #dv #md #H
      elim (divide_ok … H); #e #l elim l; /2/;
  | #n whd in ⊢ (?(??%)(?%?)); lapply (refl ? (divide n m));
      cases (divide n m) in ⊢ (???% → %); #dv #md #H
      elim (divide_ok … H); #e #l elim l;
      [ /2/;
      | #md' #m' #l' %
        [ whd in ⊢ (??%); >(pos_compare_n_m_m_n …)
            >(pos_compare_lt … l') //;
        | @Zminus_Zlt //;
        ]
      ]
  ]
| #m #H @(False_ind … H)
] qed.