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
]
]
].
suppose (0=m) (m_zero). by _ done.
conclude
(D[x \sup (1+m)])
- = (D[x · x \sup m]) by _.
- = (D[x] · x \sup m + x · D[x \sup m]) by _.
- = (x \sup m + x · (m · x \sup (pred m))) by _.
-clear H.
- = (x \sup m + m · (x \sup (1 + pred m))) by _.
- = (x \sup m + m · x \sup m) by _.
- = ((1+m) · x \sup m) by _ (timeout=30)
+ = (D[x · x \sup m]).
+ = (D[x] · x \sup m + x · D[x \sup m]).
+ = (x \sup m + x · (m · x \sup (pred m))).
+ = (x \sup m + m · (x \sup (1 + pred m))).
+ = (x \sup m + m · x \sup m).
+ = ((1+m) · x \sup m) by Fmult_one_f Fmult_commutative Fmult_Fplus_distr costante_sum
done.
qed.
(D[x \sup (2+m)] = (2+m) · x \sup (1+m)).
conclude
(D[x \sup (2+m)])
- = (D[x · x \sup (1+m)]) by _.
- = (D[x] · x \sup (1+m) + x · D[x \sup (1+m)]) by _.
- = (x \sup (1+m) + x · (costante (1+m) · x \sup m)) by _.
-clear H.
- = (x \sup (1+m) + costante (1+m) · x \sup (1+m)) by _.
- = (x \sup (1+m) · (costante (2 + m))) by _
+ = (D[x · x \sup (1+m)]).
+ = (D[x] · x \sup (1+m) + x · D[x \sup (1+m)]).
+ = (x \sup (1+m) + x · (costante (1+m) · x \sup m)).
+ = (x \sup (1+m) + costante (1+m) · x \sup (1+m)).
+ = ((2+m) · x \sup (1+m)) by Fmult_one_f Fmult_commutative
+ Fmult_Fplus_distr assoc_plus plus_n_SO costante_sum
done.
qed.
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