(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/demo/power_derivative".
-
include "nat/plus.ma".
include "nat/orders.ma".
include "nat/compare.ma".
interpretation "Rplus" 'plus x y =
(cic:/matita/demo/power_derivative/Rplus.con x y).
-notation "hvbox(a break \middot b)"
- left associative with precedence 55
-for @{ 'times $a $b }.
-
-interpretation "Rmult" 'times x y =
+interpretation "Rmult" 'middot x y =
(cic:/matita/demo/power_derivative/Rmult.con x y).
definition Fplus ≝
interpretation "Fplus" 'plus x y =
(cic:/matita/demo/power_derivative/Fplus.con x y).
-interpretation "Fmult" 'times x y =
+interpretation "Fmult" 'middot x y =
(cic:/matita/demo/power_derivative/Fmult.con x y).
notation "2" with precedence 89
]
].
-coercion cic:/matita/demo/power_derivative/inj.con.
+coercion inj.
axiom Rplus_Rzero_x: ∀x:R.0+x=x.
axiom Rplus_comm: symmetric ? Rplus.
axiom Rplus_assoc: associative ? Rplus.
-axiom Rmult_Rone_x: ∀x:R.1*x=x.
-axiom Rmult_Rzero_x: ∀x:R.0*x=0.
+axiom Rmult_Rone_x: ∀x:R.1 · x=x.
+axiom Rmult_Rzero_x: ∀x:R.0 · x=0.
axiom Rmult_assoc: associative ? Rmult.
axiom Rmult_comm: symmetric ? Rmult.
axiom Rmult_Rplus_distr: distributive ? Rmult Rplus.
-alias symbol "times" = "Rmult".
+alias symbol "middot" = "Rmult".
alias symbol "plus" = "natural plus".
definition monomio ≝
definition costante : nat → R → R ≝
λa:nat.λx:R.inj a.
-coercion cic:/matita/demo/power_derivative/costante.con 1.
+coercion costante with 1.
axiom f_eq_extensional:
∀f,g:R→R.(∀x:R.f x = g x) → f=g.
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
]
]
].
axiom derivative_x1: D[x] = 1.
axiom derivative_mult: ∀f,g:R→R. D[f·g] = D[f]·g + f·D[g].
-alias symbol "times" = "Fmult".
+alias symbol "middot" = "Fmult".
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 _
done.
case S (m:nat).
by induction hypothesis we know
(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).
- by _ we proved (0 < m ∨ 0=m) (cases).
+ 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).
case left.
suppose (0 < m) (m_pos).
- by (S_pred m m_pos) we proved (m = 1 + pred m) (H1).
- by _
+ using (S_pred ? m_pos) we proved (m = 1 + pred m) (H1).
done.
case right.
- suppose (0=m) (m_zero). by _ done.
+ suppose (0=m) (m_zero).
+ 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))) 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
done.
qed.
(cic:/matita/demo/power_derivative/derivative.con f).
*)
-notation "hvbox(\frac 'd' ('d' 'x') break p)"
- right associative with precedence 90
+notation "hvbox(\frac 'd' ('d' 'x') break p)" with precedence 90
for @{ 'derivative $p}.
interpretation "Rderivative" 'derivative f =
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 _
done.
case S (m:nat).
by induction hypothesis we know
(D[x \sup (2+m)] = (2+m) · x \sup (1+m)).
conclude
(D[x \sup (2+m)])
- = (D[x \sup 1 · x \sup (1+m)]) by _.
- = (D[x \sup 1] · 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)) timeout=30 by Fmult_one_f, Fmult_commutative,
+ Fmult_Fplus_distr, assoc_plus, plus_n_SO, costante_sum
done.
-qed.
\ No newline at end of file
+qed.