interpretation "Rderivative" 'derivative f = (derivative f).
-notation "hvbox('x' \sup n)"
+(* FG: we definitely do not want 'x' as a keyward!
+ * Any file that includes this one can not use 'x' as an identifier
+ *)
+notation "hvbox('X' \sup n)"
non associative with precedence 60
for @{ 'monomio $n }.
-notation "hvbox('x')"
+notation "hvbox('X')"
non associative with precedence 60
for @{ 'monomio 1 }.
interpretation "Rmonomio" 'monomio n = (monomio n).
-axiom derivative_x0: D[x \sup 0] = 0.
-axiom derivative_x1: D[x] = 1.
+axiom derivative_x0: D[X \sup 0] = 0.
+axiom derivative_x1: D[X] = 1.
+
axiom derivative_mult: ∀f,g:R→R. D[f·g] = D[f]·g + f·D[g].
alias symbol "middot" = "Fmult".
-theorem derivative_power: ∀n:nat. D[x \sup n] = n·x \sup (pred n).
+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)).*)
+ (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)).
+ the thesis becomes (D[X \sup 0] = 0·X \sup (pred 0)).
done.
case S (m:nat).
by induction hypothesis we know
- (D[x \sup m] = m·x \sup (pred m)) (H).
+ (D[X \sup m] = m·X \sup (pred m)) (H).
the thesis becomes
- (D[x \sup (1+m)] = (1+m) · x \sup m).
+ (D[X \sup (1+m)] = (1+m) · X \sup m).
we need to prove
- (m · (x \sup (1+ pred m)) = m · x \sup m) (Ppred).
+ (m · (X \sup (1+ pred m)) = m · X \sup m) (Ppred).
+ lapply depth=0 le_n;
we proved (0 < m ∨ 0=m) (cases).
we proceed by induction on cases
- to prove (m · (x \sup (1+ pred m)) = m · x \sup m).
+ to prove (m · (X \sup (1+ pred m)) = m · X \sup m).
case left.
suppose (0 < m) (m_pos).
using (S_pred ? m_pos) we proved (m = 1 + pred m) (H1).
- by H1 done.
+ by H1 done.
case right.
suppose (0=m) (m_zero).
by m_zero, Fmult_zero_f done.
conclude
- (D[x \sup (1+m)])
- = (D[x · x \sup m]).
- = (D[x] · x \sup m + x · D[x \sup m]).
- = (x \sup m + x · (m · x \sup (pred m))) timeout=30.
- = (x \sup m + m · (x \sup (1 + pred m))).
- = (x \sup m + m · x \sup m).
- = ((1+m) · x \sup m) timeout=30 by Fmult_one_f, Fmult_commutative, Fmult_Fplus_distr, costante_sum
+ (D[X \sup (1+m)])
+ = (D[X · X \sup m]).
+ = (D[X] · X \sup m + X · D[X \sup m]).
+ = (X \sup m + X · (m · X \sup (pred m))).
+ lapply depth=0 Fmult_associative;
+ lapply depth=0 Fmult_commutative;
+ = (X \sup m + m · (X · X \sup (pred m))) by Fmult_associative, Fmult_commutative.
+ = (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.
interpretation "Rderivative" 'derivative \eta.f = (derivative f).
*)
-notation "hvbox(\frac 'd' ('d' 'x') break p)" with precedence 90
+notation "hvbox(\frac 'd' ('d' 'X') break p)" with precedence 90
for @{ 'derivative $p}.
interpretation "Rderivative" 'derivative f = (derivative f).
-theorem derivative_power': ∀n:nat. D[x \sup (1+n)] = (1+n) · x \sup n.
+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).*) elim n 0.
+ (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).
+ the thesis becomes (D[X \sup 1] = 1 · X \sup 0).
done.
case S (m:nat).
by induction hypothesis we know
- (D[x \sup (1+m)] = (1+m) · x \sup m) (H).
+ (D[X \sup (1+m)] = (1+m) · X \sup m) (H).
the thesis becomes
- (D[x \sup (2+m)] = (2+m) · x \sup (1+m)).
+ (D[X \sup (2+m)] = (2+m) · X \sup (1+m)).
conclude
- (D[x \sup (2+m)])
- = (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)) timeout=30 by Fmult_one_f, Fmult_commutative,
+ (D[X \sup (2+m)])
+ = (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)) timeout=30 by Fmult_one_f, Fmult_commutative,
Fmult_Fplus_distr, assoc_plus, plus_n_SO, costante_sum
done.
qed.