set "baseuri" "cic:/matita/Z/z".
-include "nat/compare.ma".
-include "nat/minus.ma".
-include "higher_order_defs/functions.ma".
+include "datatypes/bool.ma".
+include "nat/nat.ma".
inductive Z : Set \def
OZ : Z
theorem OZ_test_to_Prop :\forall z:Z.
match OZ_test z with
[true \Rightarrow z=OZ
-|false \Rightarrow \lnot (z=OZ)].
+|false \Rightarrow z \neq OZ].
intros.elim z.
simplify.reflexivity.
-simplify.intros.
-cut match neg n with
-[ OZ \Rightarrow True
-| (pos n) \Rightarrow False
-| (neg n) \Rightarrow False].
-apply Hcut.rewrite > H.simplify.exact I.
-simplify.intros.
-cut match pos n with
-[ OZ \Rightarrow True
-| (pos n) \Rightarrow False
-| (neg n) \Rightarrow False].
-apply Hcut. rewrite > H.simplify.exact I.
+simplify. unfold Not. intros (H).
+discriminate H.
+simplify. unfold Not. intros (H).
+discriminate H.
qed.
+(* discrimination *)
+theorem injective_pos: injective nat Z pos.
+unfold injective.
+intros.
+change with (abs (pos x) = abs (pos y)).
+apply eq_f.assumption.
+qed.
+
+variant inj_pos : \forall n,m:nat. pos n = pos m \to n = m
+\def injective_pos.
+
+theorem injective_neg: injective nat Z neg.
+unfold injective.
+intros.
+change with (abs (neg x) = abs (neg y)).
+apply eq_f.assumption.
+qed.
+
+variant inj_neg : \forall n,m:nat. neg n = neg m \to n = m
+\def injective_neg.
+
+theorem not_eq_OZ_pos: \forall n:nat. OZ \neq pos n.
+unfold Not.intros (n H).
+discriminate H.
+qed.
+
+theorem not_eq_OZ_neg :\forall n:nat. OZ \neq neg n.
+unfold Not.intros (n H).
+discriminate H.
+qed.
+
+theorem not_eq_pos_neg :\forall n,m:nat. pos n \neq neg m.
+unfold Not.intros (n m H).
+discriminate H.
+qed.
+
+theorem decidable_eq_Z : \forall x,y:Z. decidable (x=y).
+intros.unfold decidable.
+elim x.
+(* goal: x=OZ *)
+ elim y.
+ (* goal: x=OZ y=OZ *)
+ left.reflexivity.
+ (* goal: x=OZ 2=2 *)
+ right.apply not_eq_OZ_pos.
+ (* goal: x=OZ 2=3 *)
+ right.apply not_eq_OZ_neg.
+(* goal: x=pos *)
+ elim y.
+ (* goal: x=pos y=OZ *)
+ right.unfold Not.intro.
+ apply (not_eq_OZ_pos n). symmetry. assumption.
+ (* goal: x=pos y=pos *)
+ elim (decidable_eq_nat n n1:((n=n1) \lor ((n=n1) \to False))).
+ left.apply eq_f.assumption.
+ right.unfold Not.intros (H_inj).apply H. injection H_inj. assumption.
+ (* goal: x=pos y=neg *)
+ right.unfold Not.intro.apply (not_eq_pos_neg n n1). assumption.
+(* goal: x=neg *)
+ elim y.
+ (* goal: x=neg y=OZ *)
+ right.unfold Not.intro.
+ apply (not_eq_OZ_neg n). symmetry. assumption.
+ (* goal: x=neg y=pos *)
+ right. unfold Not.intro. apply (not_eq_pos_neg n1 n). symmetry. assumption.
+ (* goal: x=neg y=neg *)
+ elim (decidable_eq_nat n n1:((n=n1) \lor ((n=n1) \to False))).
+ left.apply eq_f.assumption.
+ right.unfold Not.intro.apply H.apply injective_neg.assumption.
+qed.
+
+(* end discrimination *)
+
definition Zsucc \def
\lambda z. match z with
[ OZ \Rightarrow pos O
| (neg n) \Rightarrow neg (S n)].
theorem Zpred_Zsucc: \forall z:Z. Zpred (Zsucc z) = z.
-intros.elim z.reflexivity.
-elim n.reflexivity.
-reflexivity.
-reflexivity.
-qed.
-
-theorem Zsucc_Zpred: \forall z:Z. Zsucc (Zpred z) = z.
-intros.elim z.reflexivity.
-reflexivity.
-elim n.reflexivity.
-reflexivity.
-qed.
-
-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 (S (plus m n)))
- | (neg n) \Rightarrow
- match nat_compare m n with
- [ LT \Rightarrow (neg (pred (minus n m)))
- | EQ \Rightarrow OZ
- | GT \Rightarrow (pos (pred (minus m n)))]]
- | (neg m) \Rightarrow
- match y with
- [ OZ \Rightarrow x
- | (pos n) \Rightarrow
- match nat_compare m n with
- [ LT \Rightarrow (pos (pred (minus n m)))
- | EQ \Rightarrow OZ
- | GT \Rightarrow (neg (pred (minus m n)))]
- | (neg n) \Rightarrow (neg (S (plus m n)))]].
-
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "integer plus" 'plus x y = (cic:/matita/Z/z/Zplus.con x y).
-
-theorem Zplus_z_OZ: \forall z:Z. z+OZ = z.
-intro.elim z.
-simplify.reflexivity.
-simplify.reflexivity.
-simplify.reflexivity.
-qed.
-
-(* theorem symmetric_Zplus: symmetric Z Zplus. *)
-
-theorem sym_Zplus : \forall x,y:Z. x+y = y+x.
-intros.elim x.rewrite > Zplus_z_OZ.reflexivity.
-elim y.simplify.reflexivity.
-simplify.
-rewrite < sym_plus.reflexivity.
-simplify.
-rewrite > nat_compare_n_m_m_n.
-simplify.elim nat_compare ? ?.simplify.reflexivity.
-simplify. reflexivity.
-simplify. reflexivity.
-elim y.simplify.reflexivity.
-simplify.rewrite > nat_compare_n_m_m_n.
-simplify.elim nat_compare ? ?.simplify.reflexivity.
-simplify. reflexivity.
-simplify. reflexivity.
-simplify.rewrite < sym_plus.reflexivity.
-qed.
-
-theorem Zpred_Zplus_neg_O : \forall z:Z. Zpred z = (neg O)+z.
-intros.elim z.
-simplify.reflexivity.
-simplify.reflexivity.
-elim n.simplify.reflexivity.
-simplify.reflexivity.
-qed.
-
-theorem Zsucc_Zplus_pos_O : \forall z:Z. Zsucc z = (pos O)+z.
-intros.elim z.
-simplify.reflexivity.
-elim n.simplify.reflexivity.
-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.
-simplify.reflexivity.
-simplify.reflexivity.
-elim m.
-simplify.
-rewrite < plus_n_O.reflexivity.
-simplify.
-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.
+elim z.
+ reflexivity.
+ reflexivity.
+ elim n.
+ reflexivity.
+ reflexivity.
qed.
-theorem Zplus_neg_pos :
-\forall n,m. (neg n)+(pos m) = (Zsucc (neg n))+(Zpred (pos m)).
-intros.
-elim n.elim m.
-simplify.reflexivity.
-simplify.reflexivity.
-elim m.
-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.elim m.
-simplify.reflexivity.
-simplify.reflexivity.
-elim m.
-simplify.rewrite < plus_n_Sm.reflexivity.
-simplify.rewrite > plus_n_Sm.reflexivity.
-qed.
-
-theorem Zplus_Zsucc_Zpred:
-\forall x,y. x+y = (Zsucc x)+(Zpred y).
-intros.
-elim x. elim y.
-simplify.reflexivity.
-simplify.reflexivity.
-rewrite < Zsucc_Zplus_pos_O.
-rewrite > Zsucc_Zpred.reflexivity.
-elim y.rewrite < sym_Zplus.rewrite < sym_Zplus (Zpred OZ).
-rewrite < Zpred_Zplus_neg_O.
-rewrite > Zpred_Zsucc.
-simplify.reflexivity.
-rewrite < Zplus_neg_neg.reflexivity.
-apply Zplus_neg_pos.
-elim y.simplify.reflexivity.
-apply Zplus_pos_neg.
-apply Zplus_pos_pos.
-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.
-intros.elim n1.
-simplify. reflexivity.
-elim n2.simplify. reflexivity.
-simplify. reflexivity.
-intros. elim n1.
-simplify. reflexivity.
-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))).intro.
-intros.elim n1.
-simplify. reflexivity.
-elim n2.simplify. reflexivity.
-simplify. reflexivity.
-intros. elim n1.
-simplify. reflexivity.
-simplify.reflexivity.
-intros.
-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.elim n1.
-simplify. reflexivity.
-elim n2.simplify. reflexivity.
-simplify. reflexivity.
-intros. elim n1.
-simplify. reflexivity.
-simplify.reflexivity.
-intros.
-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.elim y.
-simplify. reflexivity.
-rewrite < Zsucc_Zplus_pos_O.reflexivity.
-simplify.reflexivity.
-elim y.rewrite < sym_Zplus.rewrite < sym_Zplus OZ.simplify.reflexivity.
-apply Zplus_Zsucc_neg_neg.
-apply Zplus_Zsucc_neg_pos.
-elim y.
-rewrite < sym_Zplus OZ.reflexivity.
-apply Zplus_Zsucc_pos_neg.
-apply Zplus_Zsucc_pos_pos.
-qed.
-
-theorem Zplus_Zpred: \forall x,y:Z. (Zpred x)+y = Zpred (x+y).
+theorem Zsucc_Zpred: \forall z:Z. Zsucc (Zpred z) = z.
intros.
-cut Zpred (x+y) = Zpred ((Zsucc (Zpred x))+y).
-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.simplify.reflexivity.
-elim n.rewrite < (Zpred_Zplus_neg_O (y+z)).
-rewrite < (Zpred_Zplus_neg_O y).
-rewrite < Zplus_Zpred.
-reflexivity.
-rewrite > Zplus_Zpred (neg n1).
-rewrite > Zplus_Zpred (neg n1).
-rewrite > Zplus_Zpred ((neg n1)+y).
-apply eq_f.assumption.
-elim n.rewrite < Zsucc_Zplus_pos_O.
-rewrite < Zsucc_Zplus_pos_O.
-rewrite > Zplus_Zsucc.
-reflexivity.
-rewrite > Zplus_Zsucc (pos n1).
-rewrite > Zplus_Zsucc (pos n1).
-rewrite > Zplus_Zsucc ((pos 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/Z/z/Zopp.con x).
-
-theorem Zplus_Zopp: \forall x:Z. x+ -x = OZ.
-intro.elim x.
-apply refl_eq.
-simplify.
-rewrite > nat_compare_n_n.
-simplify.apply refl_eq.
-simplify.
-rewrite > nat_compare_n_n.
-simplify.apply refl_eq.
+elim z.
+ reflexivity.
+ elim n.
+ reflexivity.
+ reflexivity.
+ reflexivity.
qed.
projects / helm.git / blobdiff