simplify;
apply f_eq_extensional;
intro;
- auto.
+ autobatch.
qed.
lemma Fmult_zero_f: ∀f:R→R.0·f=0.
simplify;
apply f_eq_extensional;
intro;
- auto.
+ autobatch.
qed.
lemma Fmult_commutative: symmetric ? Fmult.
unfold Fmult;
apply f_eq_extensional;
intros;
- auto.
+ autobatch.
qed.
lemma Fmult_associative: associative ? Fmult.
unfold Fmult;
apply f_eq_extensional;
intros;
- auto.
+ autobatch.
qed.
lemma Fmult_Fplus_distr: distributive ? Fmult Fplus.
apply f_eq_extensional;
intros;
simplify;
- auto.
+ autobatch.
qed.
lemma monomio_product:
[ simplify;
apply f_eq_extensional;
intro;
- auto
+ autobatch
| simplify;
apply f_eq_extensional;
intro;
cut (x\sup (n1+m) = x \sup n1 · x \sup m);
[ rewrite > Hcut;
- auto
+ autobatch
| change in ⊢ (? ? % ?) with ((λx:R.x\sup(n1+m)) x);
rewrite > H;
reflexivity
intros;
elim n;
[ simplify;
- auto
+ autobatch
| simplify;
clear x;
clear H;
[ simplify;
elim m;
[ simplify;
- auto
+ autobatch
| simplify;
rewrite < H;
- auto
+ autobatch
]
| simplify;
rewrite < H;
clear H;
elim n;
[ simplify;
- auto
+ autobatch
| simplify;
- auto
+ autobatch
]
]
].
theorem derivative_power: ∀n:nat. D[x \sup n] = n·x \sup (pred n).
assume n:nat.
- we proceed by induction on n to prove
- (D[x \sup n] = n · x \sup (pred n)).
+ (*we proceed by induction on n to prove
+ (D[x \sup n] = n · x \sup (pred n)).*)
+ elim n 0.
case O.
the thesis becomes (D[x \sup 0] = 0·x \sup (pred 0)).
by _
theorem derivative_power': ∀n:nat. D[x \sup (1+n)] = (1+n) · x \sup n.
assume n:nat.
- we proceed by induction on n to prove
- (D[x \sup (1+n)] = (1+n) · x \sup n).
+ (*we proceed by induction on n to prove
+ (D[x \sup (1+n)] = (1+n) · x \sup n).*) elim n 0.
case O.
the thesis becomes (D[x \sup 1] = 1 · x \sup 0).
by _