--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / Matita is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/library_autobatch/nat/div_and_mod".
+
+include "datatypes/constructors.ma".
+include "auto/nat/minus.ma".
+
+let rec mod_aux p m n: nat \def
+match (leb m n) with
+[ true \Rightarrow m
+| false \Rightarrow
+ match p with
+ [O \Rightarrow m
+ |(S q) \Rightarrow mod_aux q (m-(S n)) n]].
+
+definition mod : nat \to nat \to nat \def
+\lambda n,m.
+match m with
+[O \Rightarrow m
+| (S p) \Rightarrow mod_aux n n p].
+
+interpretation "natural remainder" 'module x y =
+ (cic:/matita/library_autobatch/nat/div_and_mod/mod.con x y).
+
+let rec div_aux p m n : nat \def
+match (leb m n) with
+[ true \Rightarrow O
+| false \Rightarrow
+ match p with
+ [O \Rightarrow O
+ |(S q) \Rightarrow S (div_aux q (m-(S n)) n)]].
+
+definition div : nat \to nat \to nat \def
+\lambda n,m.
+match m with
+[O \Rightarrow S n
+| (S p) \Rightarrow div_aux n n p].
+
+interpretation "natural divide" 'divide x y =
+ (cic:/matita/library_autobatch/nat/div_and_mod/div.con x y).
+
+theorem le_mod_aux_m_m:
+\forall p,n,m. n \leq p \to (mod_aux p n m) \leq m.
+intro.
+elim p
+[ apply (le_n_O_elim n H (\lambda n.(mod_aux O n m) \leq m)).
+ autobatch
+ (*simplify.
+ apply le_O_n*)
+| simplify.
+ apply (leb_elim n1 m);simplify;intro
+ [ assumption
+ | apply H.
+ cut (n1 \leq (S n) \to n1-(S m) \leq n)
+ [ autobatch
+ (*apply Hcut.
+ assumption*)
+ | elim n1;simplify;autobatch
+ (*[ apply le_O_n.
+ | apply (trans_le ? n2 n)
+ [ apply le_minus_m
+ | apply le_S_S_to_le.
+ assumption
+ ]
+ ]*)
+ ]
+ ]
+]
+qed.
+
+theorem lt_mod_m_m: \forall n,m. O < m \to (n \mod m) < m.
+intros 2.
+elim m
+[ apply False_ind.
+ apply (not_le_Sn_O O H)
+| simplify.
+ autobatch
+ (*unfold lt.
+ apply le_S_S.
+ apply le_mod_aux_m_m.
+ apply le_n*)
+]
+qed.
+
+theorem div_aux_mod_aux: \forall p,n,m:nat.
+(n=(div_aux p n m)*(S m) + (mod_aux p n m)).
+intro.
+elim p;simplify
+[ elim (leb n m);autobatch
+ (*simplify;apply refl_eq.*)
+| apply (leb_elim n1 m);simplify;intro
+ [ apply refl_eq
+ | rewrite > assoc_plus.
+ elim (H (n1-(S m)) m).
+ change with (n1=(S m)+(n1-(S m))).
+ rewrite < sym_plus.
+ autobatch
+ (*apply plus_minus_m_m.
+ change with (m < n1).
+ apply not_le_to_lt.
+ exact H1*)
+ ]
+]
+qed.
+
+theorem div_mod: \forall n,m:nat. O < m \to n=(n / m)*m+(n \mod m).
+intros 2.
+elim m
+[ elim (not_le_Sn_O O H)
+| simplify.
+ apply div_aux_mod_aux
+]
+qed.
+
+inductive div_mod_spec (n,m,q,r:nat) : Prop \def
+div_mod_spec_intro: r < m \to n=q*m+r \to (div_mod_spec n m q r).
+
+(*
+definition div_mod_spec : nat \to nat \to nat \to nat \to Prop \def
+\lambda n,m,q,r:nat.r < m \land n=q*m+r).
+*)
+
+theorem div_mod_spec_to_not_eq_O: \forall n,m,q,r.(div_mod_spec n m q r) \to m \neq O.
+intros 4.
+unfold Not.
+intros.
+elim H.
+absurd (le (S r) O);autobatch.
+(*[ rewrite < H1.
+ assumption
+| exact (not_le_Sn_O r).
+]*)
+qed.
+
+theorem div_mod_spec_div_mod:
+\forall n,m. O < m \to (div_mod_spec n m (n / m) (n \mod m)).
+intros.
+autobatch.
+(*apply div_mod_spec_intro
+[ apply lt_mod_m_m.
+ assumption
+| apply div_mod.
+ assumption
+]*)
+qed.
+
+theorem div_mod_spec_to_eq :\forall a,b,q,r,q1,r1.
+(div_mod_spec a b q r) \to (div_mod_spec a b q1 r1) \to
+(eq nat q q1).
+intros.
+elim H.
+elim H1.
+apply (nat_compare_elim q q1)
+[ intro.
+ apply False_ind.
+ cut (eq nat ((q1-q)*b+r1) r)
+ [ cut (b \leq (q1-q)*b+r1)
+ [ cut (b \leq r)
+ [ apply (lt_to_not_le r b H2 Hcut2)
+ | elim Hcut.
+ assumption
+ ]
+ | apply (trans_le ? ((q1-q)*b));autobatch
+ (*[ apply le_times_n.
+ apply le_SO_minus.
+ exact H6
+ | rewrite < sym_plus.
+ apply le_plus_n
+ ]*)
+ ]
+ | rewrite < sym_times.
+ rewrite > distr_times_minus.
+ rewrite > plus_minus;autobatch
+ (*[ rewrite > sym_times.
+ rewrite < H5.
+ rewrite < sym_times.
+ apply plus_to_minus.
+ apply H3
+ | apply le_times_r.
+ apply lt_to_le.
+ apply H6
+ ]*)
+ ]
+| (* eq case *)
+ autobatch
+ (*intros.
+ assumption*)
+| (* the following case is symmetric *)
+ intro.
+ apply False_ind.
+ cut (eq nat ((q-q1)*b+r) r1)
+ [ cut (b \leq (q-q1)*b+r)
+ [ cut (b \leq r1)
+ [ apply (lt_to_not_le r1 b H4 Hcut2)
+ | elim Hcut.
+ assumption
+ ]
+ | apply (trans_le ? ((q-q1)*b));autobatch
+ (*[ apply le_times_n.
+ apply le_SO_minus.
+ exact H6
+ | rewrite < sym_plus.
+ apply le_plus_n
+ ]*)
+ ]
+ | rewrite < sym_times.
+ rewrite > distr_times_minus.
+ rewrite > plus_minus;autobatch
+ (*[ rewrite > sym_times.
+ rewrite < H3.
+ rewrite < sym_times.
+ apply plus_to_minus.
+ apply H5
+ | apply le_times_r.
+ apply lt_to_le.
+ apply H6
+ ]*)
+ ]
+]
+qed.
+
+theorem div_mod_spec_to_eq2 :\forall a,b,q,r,q1,r1.
+(div_mod_spec a b q r) \to (div_mod_spec a b q1 r1) \to
+(eq nat r r1).
+intros.
+elim H.
+elim H1.
+apply (inj_plus_r (q*b)).
+rewrite < H3.
+rewrite > (div_mod_spec_to_eq a b q r q1 r1 H H1).
+assumption.
+qed.
+
+theorem div_mod_spec_times : \forall n,m:nat.div_mod_spec ((S n)*m) (S n) m O.
+intros.
+autobatch.
+(*constructor 1
+[ unfold lt.
+ apply le_S_S.
+ apply le_O_n
+| rewrite < plus_n_O.
+ rewrite < sym_times.
+ reflexivity
+]*)
+qed.
+
+
+(*il corpo del seguente teorema non e' stato strutturato *)
+(* some properties of div and mod *)
+theorem div_times: \forall n,m:nat. ((S n)*m) / (S n) = m.
+intros.
+apply (div_mod_spec_to_eq ((S n)*m) (S n) ? ? ? O);
+[2: apply div_mod_spec_div_mod.autobatch.
+| skip
+| autobatch
+]
+(*unfold lt.apply le_S_S.apply le_O_n.
+apply div_mod_spec_times.*)
+qed.
+
+theorem div_n_n: \forall n:nat. O < n \to n / n = S O.
+intros.
+apply (div_mod_spec_to_eq n n (n / n) (n \mod n) (S O) O);autobatch.
+(*[ apply div_mod_spec_div_mod.
+ assumption
+| constructor 1
+ [ assumption
+ | rewrite < plus_n_O.
+ simplify.
+ rewrite < plus_n_O.
+ reflexivity
+ ]
+] *)
+qed.
+
+theorem eq_div_O: \forall n,m. n < m \to n / m = O.
+intros.
+apply (div_mod_spec_to_eq n m (n/m) (n \mod m) O n);autobatch.
+(*[ apply div_mod_spec_div_mod.
+ apply (le_to_lt_to_lt O n m)
+ [ apply le_O_n
+ | assumption
+ ]
+| constructor 1
+ [ assumption
+ | reflexivity
+ ]
+]*)
+qed.
+
+theorem mod_n_n: \forall n:nat. O < n \to n \mod n = O.
+intros.
+apply (div_mod_spec_to_eq2 n n (n / n) (n \mod n) (S O) O);autobatch.
+(*[ apply div_mod_spec_div_mod.
+ assumption
+| constructor 1
+ [ assumption.
+ | rewrite < plus_n_O.
+ simplify.
+ rewrite < plus_n_O.
+ reflexivity
+ ]
+]*)
+qed.
+
+theorem mod_S: \forall n,m:nat. O < m \to S (n \mod m) < m \to
+((S n) \mod m) = S (n \mod m).
+intros.
+apply (div_mod_spec_to_eq2 (S n) m ((S n) / m) ((S n) \mod m) (n / m) (S (n \mod m)))
+[ autobatch
+ (*apply div_mod_spec_div_mod.
+ assumption*)
+| constructor 1
+ [ assumption
+ | rewrite < plus_n_Sm.
+ autobatch
+ (*apply eq_f.
+ apply div_mod.
+ assumption*)
+ ]
+]
+qed.
+
+theorem mod_O_n: \forall n:nat.O \mod n = O.
+intro.
+elim n;autobatch.
+ (*simplify;reflexivity*)
+
+qed.
+
+theorem lt_to_eq_mod:\forall n,m:nat. n < m \to n \mod m = n.
+intros.
+apply (div_mod_spec_to_eq2 n m (n/m) (n \mod m) O n);autobatch.
+(*[ apply div_mod_spec_div_mod.
+ apply (le_to_lt_to_lt O n m)
+ [ apply le_O_n
+ | assumption
+ ]
+| constructor 1.
+ [ assumption
+ | reflexivity
+ ]
+]*)
+qed.
+
+(* injectivity *)
+theorem injective_times_r: \forall n:nat.injective nat nat (\lambda m:nat.(S n)*m).
+change with (\forall n,p,q:nat.(S n)*p = (S n)*q \to p=q).
+intros.
+rewrite < (div_times n).
+autobatch.
+(*rewrite < (div_times n q).
+apply eq_f2
+[ assumption
+| reflexivity
+]*)
+qed.
+
+variant inj_times_r : \forall n,p,q:nat.(S n)*p = (S n)*q \to p=q \def
+injective_times_r.
+
+theorem lt_O_to_injective_times_r: \forall n:nat. O < n \to injective nat nat (\lambda m:nat.n*m).
+simplify.
+intros 4.
+apply (lt_O_n_elim n H).
+intros.
+autobatch.
+(*apply (inj_times_r m).
+assumption.*)
+qed.
+
+variant inj_times_r1:\forall n. O < n \to \forall p,q:nat.n*p = n*q \to p=q
+\def lt_O_to_injective_times_r.
+
+theorem injective_times_l: \forall n:nat.injective nat nat (\lambda m:nat.m*(S n)).
+simplify.
+intros.
+autobatch.
+(*apply (inj_times_r n x y).
+rewrite < sym_times.
+rewrite < (sym_times y).
+assumption.*)
+qed.
+
+variant inj_times_l : \forall n,p,q:nat. p*(S n) = q*(S n) \to p=q \def
+injective_times_l.
+
+theorem lt_O_to_injective_times_l: \forall n:nat. O < n \to injective nat nat (\lambda m:nat.m*n).
+simplify.
+intros 4.
+apply (lt_O_n_elim n H).
+intros.
+autobatch.
+(*apply (inj_times_l m).
+assumption.*)
+qed.
+
+variant inj_times_l1:\forall n. O < n \to \forall p,q:nat.p*n = q*n \to p=q
+\def lt_O_to_injective_times_l.
+
+(* n_divides computes the pair (div,mod) *)
+
+(* p is just an upper bound, acc is an accumulator *)
+let rec n_divides_aux p n m acc \def
+ match n \mod m with
+ [ O \Rightarrow
+ match p with
+ [ O \Rightarrow pair nat nat acc n
+ | (S p) \Rightarrow n_divides_aux p (n / m) m (S acc)]
+ | (S a) \Rightarrow pair nat nat acc n].
+
+(* n_divides n m = <q,r> if m divides n q times, with remainder r *)
+definition n_divides \def \lambda n,m:nat.n_divides_aux n n m O.
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