--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/library_autobatch/Z/plus".
+
+include "auto/Z/z.ma".
+include "auto/nat/minus.ma".
+
+definition Zplus :Z \to Z \to Z \def
+\lambda x,y.
+ match x with
+ [ OZ \Rightarrow y
+ | (pos m) \Rightarrow
+ match y with
+ [ OZ \Rightarrow x
+ | (pos n) \Rightarrow (pos (pred ((S m)+(S n))))
+ | (neg n) \Rightarrow
+ match nat_compare m n with
+ [ LT \Rightarrow (neg (pred (n-m)))
+ | EQ \Rightarrow OZ
+ | GT \Rightarrow (pos (pred (m-n)))] ]
+ | (neg m) \Rightarrow
+ match y with
+ [ OZ \Rightarrow x
+ | (pos n) \Rightarrow
+ match nat_compare m n with
+ [ LT \Rightarrow (pos (pred (n-m)))
+ | EQ \Rightarrow OZ
+ | GT \Rightarrow (neg (pred (m-n)))]
+ | (neg n) \Rightarrow (neg (pred ((S m)+(S n))))] ].
+
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "integer plus" 'plus x y = (cic:/matita/library_autobatch/Z/plus/Zplus.con x y).
+
+theorem Zplus_z_OZ: \forall z:Z. z+OZ = z.
+intro.
+elim z;autobatch.
+ (*simplify;reflexivity.*)
+qed.
+
+(* theorem symmetric_Zplus: symmetric Z Zplus. *)
+
+theorem sym_Zplus : \forall x,y:Z. x+y = y+x.
+intros.
+elim x
+[ autobatch
+ (*rewrite > Zplus_z_OZ.
+ reflexivity*)
+| elim y
+ [ autobatch
+ (*simplify.
+ reflexivity*)
+ | simplify.
+ autobatch
+ (*rewrite < plus_n_Sm.
+ rewrite < plus_n_Sm.
+ rewrite < sym_plus.
+ reflexivity*)
+ | simplify.
+ rewrite > nat_compare_n_m_m_n.
+ simplify.
+ elim nat_compare;autobatch
+ (*[ simplify.
+ reflexivity
+ | simplify.
+ reflexivity
+ | simplify.
+ reflexivity
+ ]*)
+ ]
+| elim y
+ [ autobatch
+ (*simplify.
+ reflexivity*)
+ | simplify.
+ rewrite > nat_compare_n_m_m_n.
+ simplify.
+ elim nat_compare;autobatch
+ (*[ simplify.
+ reflexivity
+ | simplify.
+ reflexivity
+ | simplify.
+ reflexivity
+ ]*)
+ | simplify.
+ autobatch
+ (*rewrite < plus_n_Sm.
+ rewrite < plus_n_Sm.
+ rewrite < sym_plus.
+ reflexivity*)
+ ]
+]
+qed.
+
+theorem Zpred_Zplus_neg_O : \forall z:Z. Zpred z = (neg O)+z.
+intros.
+elim z
+[ autobatch
+ (*simplify.
+ reflexivity*)
+| elim n;autobatch
+ (*[ simplify.
+ reflexivity
+ | simplify.
+ reflexivity
+ ]*)
+| simplify.
+ reflexivity
+]
+qed.
+
+theorem Zsucc_Zplus_pos_O : \forall z:Z. Zsucc z = (pos O)+z.
+intros.
+elim z
+[ autobatch
+ (*simplify.
+ reflexivity*)
+| autobatch
+ (*simplify.
+ reflexivity*)
+| elim n;autobatch
+ (*[ simplify.
+ reflexivity
+ | simplify.
+ reflexivity
+ ]*)
+]
+qed.
+
+theorem Zplus_pos_pos:
+\forall n,m. (pos n)+(pos m) = (Zsucc (pos n))+(Zpred (pos m)).
+intros.
+elim n
+[ elim m;autobatch
+ (*[ simplify.
+ reflexivity
+ | simplify.
+ reflexivity
+ ]*)
+| elim m
+ [ autobatch
+ (*simplify.
+ rewrite < plus_n_Sm.
+ rewrite < plus_n_O.
+ reflexivity*)
+ | simplify.
+ autobatch
+ (*rewrite < plus_n_Sm.
+ rewrite < plus_n_Sm.
+ reflexivity*)
+ ]
+]
+qed.
+
+theorem Zplus_pos_neg:
+\forall n,m. (pos n)+(neg m) = (Zsucc (pos n))+(Zpred (neg m)).
+intros.
+reflexivity.
+qed.
+
+theorem Zplus_neg_pos :
+\forall n,m. (neg n)+(pos m) = (Zsucc (neg n))+(Zpred (pos m)).
+intros.
+elim n
+[ elim m;autobatch
+ (*[ simplify.
+ reflexivity
+ | simplify.
+ reflexivity
+ ]*)
+| elim m;autobatch
+ (*[ simplify.
+ reflexivity
+ | simplify.
+ reflexivity
+ ]*)
+]
+qed.
+
+theorem Zplus_neg_neg:
+\forall n,m. (neg n)+(neg m) = (Zsucc (neg n))+(Zpred (neg m)).
+intros.
+elim n
+[ autobatch
+ (*elim m
+ [ simplify.
+ reflexivity
+ | simplify.
+ reflexivity
+ ]*)
+| elim m
+ [ autobatch
+ (*simplify.
+ rewrite > plus_n_Sm.
+ reflexivity*)
+ | simplify.
+ autobatch
+ (*rewrite > plus_n_Sm.
+ reflexivity*)
+ ]
+]
+qed.
+
+theorem Zplus_Zsucc_Zpred:
+\forall x,y. x+y = (Zsucc x)+(Zpred y).
+intros.
+elim x
+[ autobatch
+ (*elim y
+ [ simplify.
+ reflexivity
+ | rewrite < Zsucc_Zplus_pos_O.
+ rewrite > Zsucc_Zpred.
+ reflexivity
+ | simplify.
+ reflexivity
+ ]*)
+| elim y;autobatch
+ (*[ simplify.
+ reflexivity
+ | apply Zplus_pos_pos
+ | apply Zplus_pos_neg
+ ]*)
+| elim y;autobatch
+ (*[ rewrite < sym_Zplus.
+ rewrite < (sym_Zplus (Zpred OZ)).
+ rewrite < Zpred_Zplus_neg_O.
+ rewrite > Zpred_Zsucc.
+ simplify.
+ reflexivity
+ | apply Zplus_neg_pos
+ | rewrite < Zplus_neg_neg.
+ reflexivity
+ ]*)
+]
+qed.
+
+theorem Zplus_Zsucc_pos_pos :
+\forall n,m. (Zsucc (pos n))+(pos m) = Zsucc ((pos n)+(pos m)).
+intros.
+reflexivity.
+qed.
+
+theorem Zplus_Zsucc_pos_neg:
+\forall n,m. (Zsucc (pos n))+(neg m) = (Zsucc ((pos n)+(neg m))).
+intros.
+apply (nat_elim2
+(\lambda n,m. (Zsucc (pos n))+(neg m) = (Zsucc ((pos n)+(neg m)))))
+[ intro.
+ elim n1;autobatch
+ (*[ simplify.
+ reflexivity
+ | elim n2; simplify; reflexivity
+ ]*)
+| intros.
+ autobatch
+ (*elim n1;simplify;reflexivity*)
+| intros.
+ rewrite < (Zplus_pos_neg ? m1).
+ elim H.
+ reflexivity
+]
+qed.
+
+theorem Zplus_Zsucc_neg_neg :
+\forall n,m. Zsucc (neg n) + neg m = Zsucc (neg n + neg m).
+intros.
+apply (nat_elim2
+(\lambda n,m. Zsucc (neg n) + neg m = Zsucc (neg n + neg m)))
+[ intros.
+ autobatch
+ (*elim n1
+ [ simplify.
+ reflexivity
+ | elim n2;simplify;reflexivity
+ ]*)
+| intros.
+ autobatch
+ (*elim n1;simplify;reflexivity*)
+| intros.
+ autobatch.
+ (*rewrite < (Zplus_neg_neg ? m1).
+ reflexivity*)
+]
+qed.
+
+theorem Zplus_Zsucc_neg_pos:
+\forall n,m. Zsucc (neg n)+(pos m) = Zsucc ((neg n)+(pos m)).
+intros.
+apply (nat_elim2
+(\lambda n,m. Zsucc (neg n) + (pos m) = Zsucc (neg n + pos m)))
+[ intros.
+ autobatch
+ (*elim n1
+ [ simplify.
+ reflexivity
+ | elim n2;simplify;reflexivity
+ ]*)
+| intros.
+ autobatch
+ (*elim n1;simplify;reflexivity*)
+| intros.
+ autobatch
+ (*rewrite < H.
+ rewrite < (Zplus_neg_pos ? (S m1)).
+ reflexivity*)
+]
+qed.
+
+theorem Zplus_Zsucc : \forall x,y:Z. (Zsucc x)+y = Zsucc (x+y).
+intros.
+elim x
+[ autobatch
+ (*elim y
+ [ simplify.
+ reflexivity
+ | simplify.
+ reflexivity
+ | rewrite < Zsucc_Zplus_pos_O.
+ reflexivity
+ ]*)
+| elim y;autobatch
+ (*[ rewrite < (sym_Zplus OZ).
+ reflexivity
+ | apply Zplus_Zsucc_pos_pos
+ | apply Zplus_Zsucc_pos_neg
+ ]*)
+| elim y;autobatch
+ (*[ rewrite < sym_Zplus.
+ rewrite < (sym_Zplus OZ).
+ simplify.
+ reflexivity
+ | apply Zplus_Zsucc_neg_pos
+ | apply Zplus_Zsucc_neg_neg
+ ]*)
+]
+qed.
+
+theorem Zplus_Zpred: \forall x,y:Z. (Zpred x)+y = Zpred (x+y).
+intros.
+cut (Zpred (x+y) = Zpred ((Zsucc (Zpred x))+y));autobatch.
+(*[ rewrite > Hcut.
+ rewrite > Zplus_Zsucc.
+ rewrite > Zpred_Zsucc.
+ reflexivity
+| rewrite > Zsucc_Zpred.
+ reflexivity
+]*)
+qed.
+
+
+theorem associative_Zplus: associative Z Zplus.
+change with (\forall x,y,z:Z. (x + y) + z = x + (y + z)).
+(* simplify. *)
+intros.
+elim x
+[ autobatch
+ (*simplify.
+ reflexivity*)
+| elim n
+ [ rewrite < Zsucc_Zplus_pos_O.
+ autobatch
+ (*rewrite < Zsucc_Zplus_pos_O.
+ rewrite > Zplus_Zsucc.
+ reflexivity*)
+ | rewrite > (Zplus_Zsucc (pos n1)).
+ rewrite > (Zplus_Zsucc (pos n1)).
+ autobatch
+ (*rewrite > (Zplus_Zsucc ((pos n1)+y)).
+ apply eq_f.
+ assumption*)
+ ]
+| elim n
+ [ rewrite < (Zpred_Zplus_neg_O (y+z)).
+ autobatch
+ (*rewrite < (Zpred_Zplus_neg_O y).
+ rewrite < Zplus_Zpred.
+ reflexivity*)
+ | rewrite > (Zplus_Zpred (neg n1)).
+ rewrite > (Zplus_Zpred (neg n1)).
+ autobatch
+ (*rewrite > (Zplus_Zpred ((neg n1)+y)).
+ apply eq_f.
+ assumption*)
+ ]
+]
+qed.
+
+variant assoc_Zplus : \forall x,y,z:Z. (x+y)+z = x+(y+z)
+\def associative_Zplus.
+
+(* Zopp *)
+definition Zopp : Z \to Z \def
+\lambda x:Z. match x with
+[ OZ \Rightarrow OZ
+| (pos n) \Rightarrow (neg n)
+| (neg n) \Rightarrow (pos n) ].
+
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "integer unary minus" 'uminus x = (cic:/matita/library_autobatch/Z/plus/Zopp.con x).
+
+theorem Zopp_Zplus: \forall x,y:Z. -(x+y) = -x + -y.
+intros.
+elim x
+[ elim y;autobatch
+ (*simplify;reflexivity*)
+| elim y
+ [ autobatch
+ (*simplify.
+ reflexivity*)
+ | autobatch
+ (*simplify.
+ reflexivity*)
+ | simplify.
+ apply nat_compare_elim;
+ intro;autobatch (*simplify;reflexivity*)
+ ]
+| elim y
+ [ autobatch
+ (*simplify.
+ reflexivity*)
+ | simplify.
+ apply nat_compare_elim;
+ intro;autobatch
+ (*simplify;reflexivity*)
+ | autobatch
+ (*simplify.
+ reflexivity*)
+ ]
+]
+qed.
+
+theorem Zopp_Zopp: \forall x:Z. --x = x.
+intro.
+elim x;reflexivity.
+qed.
+
+theorem Zplus_Zopp: \forall x:Z. x+ -x = OZ.
+intro.
+elim x
+[ apply refl_eq
+| simplify.
+ rewrite > nat_compare_n_n.
+ autobatch
+ (*simplify.
+ apply refl_eq*)
+| simplify.
+ rewrite > nat_compare_n_n.
+ autobatch
+ (*simplify.
+ apply refl_eq*)
+]
+qed.
+
+(* minus *)
+definition Zminus : Z \to Z \to Z \def \lambda x,y:Z. x + (-y).
+
+interpretation "integer minus" 'minus x y = (cic:/matita/library_autobatch/Z/plus/Zminus.con x y).
projects / helm.git / blobdiff