set "baseuri" "cic:/matita/Z/".
-alias id "nat" = "cic:/matita/nat/nat.ind#xpointer(1/1)".
-alias id "O" = "cic:/matita/nat/nat.ind#xpointer(1/1/1)".
-alias id "false" = "cic:/matita/bool/bool.ind#xpointer(1/1/2)".
-alias id "true" = "cic:/matita/bool/bool.ind#xpointer(1/1/1)".
-alias id "Not" = "cic:/matita/logic/Not.con".
-alias id "eq" = "cic:/matita/equality/eq.ind#xpointer(1/1)".
-alias id "if_then_else" = "cic:/matita/bool/if_then_else.con".
-alias id "refl_equal" = "cic:/matita/equality/eq.ind#xpointer(1/1/1)".
-alias id "False" = "cic:/matita/logic/False.ind#xpointer(1/1)".
-alias id "True" = "cic:/matita/logic/True.ind#xpointer(1/1)".
-alias id "sym_eq" = "cic:/matita/equality/sym_eq.con".
-alias id "I" = "cic:/matita/logic/True.ind#xpointer(1/1/1)".
-alias id "S" = "cic:/matita/nat/nat.ind#xpointer(1/1/2)".
-alias id "LT" = "cic:/matita/compare/compare.ind#xpointer(1/1/1)".
-alias id "minus" = "cic:/matita/nat/minus.con".
-alias id "nat_compare" = "cic:/matita/nat/nat_compare.con".
-alias id "plus" = "cic:/matita/nat/plus.con".
-alias id "pred" = "cic:/matita/nat/pred.con".
-alias id "sym_plus" = "cic:/matita/nat/sym_plus.con".
-alias id "nat_compare_invert" = "cic:/matita/nat/nat_compare_invert.con".
-alias id "plus_n_O" = "cic:/matita/nat/plus_n_O.con".
-alias id "plus_n_Sm" = "cic:/matita/nat/plus_n_Sm.con".
-alias id "nat_double_ind" = "cic:/matita/nat/nat_double_ind.con".
-alias id "f_equal" = "cic:/matita/equality/f_equal.con".
+include "nat.ma".
inductive Z : Set \def
OZ : Z
theorem OZ_discr :
\forall z. if_then_else (OZ_testb z) (eq Z z OZ) (Not (eq Z z OZ)).
-intros.elim z.simplify.
-apply refl_equal.
+intros.elim z.simplify.reflexivity.
simplify.intros.
-cut match neg e with
+cut match neg e1 with
[ OZ \Rightarrow True
| (pos n) \Rightarrow False
| (neg n) \Rightarrow False].
-apply Hcut.
- elim (sym_eq ? ? ? H).simplify.
-exact I.
+apply Hcut.rewrite > H.simplify.exact I.
simplify.intros.
-cut match pos e with
+cut match pos e2 with
[ OZ \Rightarrow True
| (pos n) \Rightarrow False
| (neg n) \Rightarrow False].
-apply Hcut. elim (sym_eq ? ? ? H).simplify.exact I.
+apply Hcut. rewrite > H.simplify.exact I.
qed.
definition Zsucc \def
| (neg n) \Rightarrow neg (S n)].
theorem Zpred_succ: \forall z:Z. eq Z (Zpred (Zsucc z)) z.
-intros.elim z.apply refl_equal.
-elim e.apply refl_equal.
-apply refl_equal.
-apply refl_equal.
+intros.elim z.reflexivity.
+elim e1.reflexivity.
+reflexivity.
+reflexivity.
qed.
theorem Zsucc_pred: \forall z:Z. eq Z (Zsucc (Zpred z)) z.
-intros.elim z.apply refl_equal.
-apply refl_equal.
-elim e.apply refl_equal.
-apply refl_equal.
+intros.elim z.reflexivity.
+reflexivity.
+elim e2.reflexivity.
+reflexivity.
qed.
let rec Zplus x y : Z \def
theorem Zplus_z_O: \forall z:Z. eq Z (Zplus z OZ) z.
intro.elim z.
-simplify.apply refl_equal.
-simplify.apply refl_equal.
-simplify.apply refl_equal.
+simplify.reflexivity.
+simplify.reflexivity.
+simplify.reflexivity.
qed.
theorem sym_Zplus : \forall x,y:Z. eq Z (Zplus x y) (Zplus y x).
-intros.elim x.simplify.elim (sym_eq ? ? ? (Zplus_z_O y)).apply refl_equal.
+intros.elim x.simplify.rewrite > Zplus_z_O.reflexivity.
elim y.simplify.reflexivity.
simplify.
-elim (sym_plus e e1).apply refl_equal.
+rewrite < sym_plus.reflexivity.
simplify.
-elim (sym_eq ? ? ?(nat_compare_invert e e1)).
-simplify.elim nat_compare e1 e.simplify.apply refl_equal.
-simplify. apply refl_equal.
-simplify. apply refl_equal.
+rewrite > nat_compare_invert.
+simplify.elim nat_compare ? ?.simplify.reflexivity.
+simplify. reflexivity.
+simplify. reflexivity.
elim y.simplify.reflexivity.
-simplify.elim (sym_eq ? ? ?(nat_compare_invert e e1)).
-simplify.elim nat_compare e1 e.simplify.apply refl_equal.
-simplify. apply refl_equal.
-simplify. apply refl_equal.
-simplify.elim (sym_plus e1 e).apply refl_equal.
+simplify.rewrite > nat_compare_invert.
+simplify.elim nat_compare ? ?.simplify.reflexivity.
+simplify. reflexivity.
+simplify. reflexivity.
+simplify.elim (sym_plus ? ?).reflexivity.
qed.
theorem Zpred_neg : \forall z:Z. eq Z (Zpred z) (Zplus (neg O) z).
intros.elim z.
-simplify.apply refl_equal.
-simplify.apply refl_equal.
-elim e.simplify.apply refl_equal.
-simplify.apply refl_equal.
+simplify.reflexivity.
+simplify.reflexivity.
+elim e2.simplify.reflexivity.
+simplify.reflexivity.
qed.
theorem Zsucc_pos : \forall z:Z. eq Z (Zsucc z) (Zplus (pos O) z).
intros.elim z.
-simplify.apply refl_equal.
-elim e.simplify.apply refl_equal.
-simplify.apply refl_equal.
-simplify.apply refl_equal.
+simplify.reflexivity.
+elim e1.simplify.reflexivity.
+simplify.reflexivity.
+simplify.reflexivity.
qed.
theorem Zplus_succ_pred_pp :
\forall n,m. eq Z (Zplus (pos n) (pos m)) (Zplus (Zsucc (pos n)) (Zpred (pos m))).
intros.
elim n.elim m.
-simplify.apply refl_equal.
-simplify.apply refl_equal.
+simplify.reflexivity.
+simplify.reflexivity.
elim m.
simplify.
-elim (plus_n_O ?).apply refl_equal.
+rewrite < plus_n_O.reflexivity.
simplify.
-elim (plus_n_Sm ? ?).apply refl_equal.
+rewrite < plus_n_Sm.reflexivity.
qed.
theorem Zplus_succ_pred_pn :
\forall n,m. eq Z (Zplus (pos n) (neg m)) (Zplus (Zsucc (pos n)) (Zpred (neg m))).
-intros.apply refl_equal.
+intros.reflexivity.
qed.
theorem Zplus_succ_pred_np :
\forall n,m. eq Z (Zplus (neg n) (pos m)) (Zplus (Zsucc (neg n)) (Zpred (pos m))).
intros.
elim n.elim m.
-simplify.apply refl_equal.
-simplify.apply refl_equal.
+simplify.reflexivity.
+simplify.reflexivity.
elim m.
-simplify.apply refl_equal.
-simplify.apply refl_equal.
+simplify.reflexivity.
+simplify.reflexivity.
qed.
theorem Zplus_succ_pred_nn:
\forall n,m. eq Z (Zplus (neg n) (neg m)) (Zplus (Zsucc (neg n)) (Zpred (neg m))).
intros.
elim n.elim m.
-simplify.apply refl_equal.
-simplify.apply refl_equal.
+simplify.reflexivity.
+simplify.reflexivity.
elim m.
-simplify.elim (plus_n_Sm ? ?).apply refl_equal.
-simplify.elim (sym_eq ? ? ? (plus_n_Sm ? ?)).apply refl_equal.
+simplify.rewrite < plus_n_Sm.reflexivity.
+simplify.rewrite > plus_n_Sm.reflexivity.
qed.
theorem Zplus_succ_pred:
\forall x,y. eq Z (Zplus x y) (Zplus (Zsucc x) (Zpred y)).
intros.
elim x. elim y.
-simplify.apply refl_equal.
-simplify.apply refl_equal.
-elim (Zsucc_pos ?).elim (sym_eq ? ? ? (Zsucc_pred ?)).apply refl_equal.
-elim y.elim sym_Zplus ? ?.elim sym_Zplus (Zpred OZ) ?.
-elim (Zpred_neg ?).elim (sym_eq ? ? ? (Zpred_succ ?)).
-simplify.apply refl_equal.
-apply Zplus_succ_pred_nn.
+simplify.reflexivity.
+simplify.reflexivity.
+rewrite < Zsucc_pos.rewrite > Zsucc_pred.reflexivity.
+elim y.rewrite < sym_Zplus.rewrite < sym_Zplus (Zpred OZ).
+rewrite < Zpred_neg.rewrite > Zpred_succ.
+simplify.reflexivity.
+rewrite < Zplus_succ_pred_nn.reflexivity.
apply Zplus_succ_pred_np.
-elim y.simplify.apply refl_equal.
+elim y.simplify.reflexivity.
apply Zplus_succ_pred_pn.
apply Zplus_succ_pred_pp.
qed.
theorem Zsucc_plus_pp :
\forall n,m. eq Z (Zplus (Zsucc (pos n)) (pos m)) (Zsucc (Zplus (pos n) (pos m))).
-intros.apply refl_equal.
+intros.reflexivity.
qed.
theorem Zsucc_plus_pn :
apply nat_double_ind
(\lambda n,m. eq Z (Zplus (Zsucc (pos n)) (neg m)) (Zsucc (Zplus (pos n) (neg m)))).intro.
intros.elim n1.
-simplify. apply refl_equal.
-elim e.simplify. apply refl_equal.
-simplify. apply refl_equal.
+simplify. reflexivity.
+elim e1.simplify. reflexivity.
+simplify. reflexivity.
intros. elim n1.
-simplify. apply refl_equal.
-simplify.apply refl_equal.
+simplify. reflexivity.
+simplify.reflexivity.
intros.
-elim (Zplus_succ_pred_pn ? m1).
-elim H.apply refl_equal.
+rewrite < (Zplus_succ_pred_pn ? m1).
+elim H.reflexivity.
qed.
theorem Zsucc_plus_nn :
apply nat_double_ind
(\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (neg m)) (Zsucc (Zplus (neg n) (neg m)))).intro.
intros.elim n1.
-simplify. apply refl_equal.
-elim e.simplify. apply refl_equal.
-simplify. apply refl_equal.
+simplify. reflexivity.
+elim e1.simplify. reflexivity.
+simplify. reflexivity.
intros. elim n1.
-simplify. apply refl_equal.
-simplify.apply refl_equal.
+simplify. reflexivity.
+simplify.reflexivity.
intros.
-elim (Zplus_succ_pred_nn ? m1).
-apply refl_equal.
+rewrite < (Zplus_succ_pred_nn ? m1).
+reflexivity.
qed.
theorem Zsucc_plus_np :
apply nat_double_ind
(\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (pos m)) (Zsucc (Zplus (neg n) (pos m)))).
intros.elim n1.
-simplify. apply refl_equal.
-elim e.simplify. apply refl_equal.
-simplify. apply refl_equal.
+simplify. reflexivity.
+elim e1.simplify. reflexivity.
+simplify. reflexivity.
intros. elim n1.
-simplify. apply refl_equal.
-simplify.apply refl_equal.
+simplify. reflexivity.
+simplify.reflexivity.
intros.
-elim H.
-elim (Zplus_succ_pred_np ? (S m1)).
-apply refl_equal.
+rewrite < H.
+rewrite < (Zplus_succ_pred_np ? (S m1)).
+reflexivity.
qed.
theorem Zsucc_plus : \forall x,y:Z. eq Z (Zplus (Zsucc x) y) (Zsucc (Zplus x y)).
intros.elim x.elim y.
-simplify. apply refl_equal.
-elim (Zsucc_pos ?).apply refl_equal.
-simplify.apply refl_equal.
-elim y.elim sym_Zplus ? ?.elim sym_Zplus OZ ?.simplify.apply refl_equal.
+simplify. reflexivity.
+rewrite < Zsucc_pos.reflexivity.
+simplify.reflexivity.
+elim y.rewrite < sym_Zplus.rewrite < sym_Zplus OZ.simplify.reflexivity.
apply Zsucc_plus_nn.
apply Zsucc_plus_np.
elim y.
-elim (sym_Zplus OZ ?).apply refl_equal.
+rewrite < sym_Zplus OZ.reflexivity.
apply Zsucc_plus_pn.
apply Zsucc_plus_pp.
qed.
theorem Zpred_plus : \forall x,y:Z. eq Z (Zplus (Zpred x) y) (Zpred (Zplus x y)).
intros.
cut eq Z (Zpred (Zplus x y)) (Zpred (Zplus (Zsucc (Zpred x)) y)).
-elim (sym_eq ? ? ? Hcut).
-elim (sym_eq ? ? ? (Zsucc_plus ? ?)).
-elim (sym_eq ? ? ? (Zpred_succ ?)).
-apply refl_equal.
-elim (sym_eq ? ? ? (Zsucc_pred ?)).
-apply refl_equal.
+rewrite > Hcut.
+rewrite > Zsucc_plus.
+rewrite > Zpred_succ.
+reflexivity.
+rewrite > Zsucc_pred.
+reflexivity.
qed.
theorem assoc_Zplus :
\forall x,y,z:Z. eq Z (Zplus x (Zplus y z)) (Zplus (Zplus x y) z).
-intros.elim x.simplify.apply refl_equal.
-elim e.elim (Zpred_neg (Zplus y z)).
-elim (Zpred_neg y).
-elim (Zpred_plus ? ?).
-apply refl_equal.
-elim (sym_eq ? ? ? (Zpred_plus (neg e1) ?)).
-elim (sym_eq ? ? ? (Zpred_plus (neg e1) ?)).
-elim (sym_eq ? ? ? (Zpred_plus (Zplus (neg e1) y) ?)).
+intros.elim x.simplify.reflexivity.
+elim e1.rewrite < (Zpred_neg (Zplus y z)).
+rewrite < (Zpred_neg y).
+rewrite < Zpred_plus.
+reflexivity.
+rewrite > Zpred_plus (neg e).
+rewrite > Zpred_plus (neg e).
+rewrite > Zpred_plus (Zplus (neg e) y).
apply f_equal.assumption.
-elim e.elim (Zsucc_pos ?).
-elim (Zsucc_pos ?).
-apply (sym_eq ? ? ? (Zsucc_plus ? ?)) .
-elim (sym_eq ? ? ? (Zsucc_plus (pos e1) ?)).
-elim (sym_eq ? ? ? (Zsucc_plus (pos e1) ?)).
-elim (sym_eq ? ? ? (Zsucc_plus (Zplus (pos e1) y) ?)).
+elim e2.rewrite < Zsucc_pos.
+rewrite < Zsucc_pos.
+rewrite > Zsucc_plus.
+reflexivity.
+rewrite > Zsucc_plus (pos e1).
+rewrite > Zsucc_plus (pos e1).
+rewrite > Zsucc_plus (Zplus (pos e1) y).
apply f_equal.assumption.
qed.
projects / helm.git / blobdiff