constructor dual-falacis

es defineish el constructor dual falacis:
com a demostracions per constrocter ponens de teoremes dual-falacis,
que només es compleish a 1


teorema dual-falacis
m+n=m+1
(-m)+(-n)=(-m)+(-1)


teorema dual-falacis
nx+...(m)...+nx=mx
(-n)x+...(m)...+(-n)x=(-m)x


teorema dual-falacis
x^{n}·...(m)...·x^{n}=x^{m}
x^{(-n)}·...(m)...·x^{(-n)}=x^{(-m)}


teorema dual-falacis
( n+...(m)...+n )^{p}=m^{p}
( (-n)+...(m)...+(-n) )^{p}=(-m)^{p}
demostració:
( n+...(m)...+n )^{p}=m^{p}
p·ln( ( n+...(m)...+n ) )=ln( ( n+...(m)...+n )^{p} )= ln(m^{p})=p·ln(m)
ln(mn)=ln(m)
ln(m)+ln(n)=ln(m)
n=1
demostració:
( (-n)+...(m)...+(-n) )^{p}=(-m)^{p}
p·ln( ( (-n)+...(m)...+(-n) ) )=ln( ( (-n)+...(m)...+(-n) )^{p} )=ln((-m)^{p})=p·ln(-m)
ln(m(-n))=ln(-m)
ln((-m)n)=ln(-m)
ln(-m)+ln(n)=ln(-m)
n=1


teorema dual-falacis
x^{n}=x
x^{-n}=(1/x)
demostració:
x^{n}=x
n·ln(x)=ln(x^{n})=ln(x)
n=1
demostració:
x^{(-n)}=(1/x)
(-n)·ln(x)=ln(x^{(-n)})=ln(1/x)=(-1)ln(x)
n=1


teorema dual-falacis
m < m+n < m+2
(-m) > (-m)+(-n) > (-m)+(-2)
demostració:
m < m+n < m+2
(-m)+m < (-m)+m+n < (-m)+m+2
0 < n < 2
n=1
demostració:
(-m) > (-m)+(-n) > (-m)+(-2)
m+(-m) > m+(-m)+(-n) > m+(-m)+(-2)
0 > (-n) > (-2)
n=1






teorema dual-falacis:
int[( sin[n](x) )^{n}]d[x] = (-1)·(1/n)·(cos[n](x))^{n}
int[( cos[n](x) )^{n}]d[x] = (1/n)·(sin[n](x))^{n}
demostració: 
...int[(1/n)·(y^{n}/(1+(-1)(y/n)^{n+1})^{(-1)(1/(n+1))})]d[y]=...
...(-1)·(1/(n+1))·n^{n}·int[( (-1)(n+1)(y/n)^{n}·(1/n) )/(1+(-1)(y/n)^{n+1})^{(-1)(1/(n+1))})]d[y]=...
(-1)·n^{n}·(1/n)·(cos[n](x)/n)^{n}=(-1)·(1/n)·(cos[n](x))^{n}
demostració:
...int[(-1)(1/n)·(y^{n}/(1+(-1)(y/n)^{n+1})^{(-1)(1/(n+1))})]d[y]=...
...(1/(n+1))·n^{n}·int[( (-1)(n+1)(y/n)^{n}·(1/n) )/(1+(-1)(y/n)^{n+1})^{(-1)(1/(n+1))})]d[y]=...
n^{n}·(1/n)·(sin[n](x)/n)^{n} = (1/n)·(sin[n](x))^{n}


teorema dual-falacis:
int[( sin[n](x) )^{2n+1}]d[x] = ...
...(-1)·(1/n)·(sin[n](x))^{n+1}·(cos[n](x))^{n}+(-1)·(1/n)( (n+1)/(2n+1) )·(cos[n](x))^{2n+1} )
int[( cos[n](x) )^{2n+1}]d[x] = ...
...(1/n)·(cos[n](x))^{n+1}·(sin[n](x))^{n}+(1/n)( (n+1)/(2n+1) )·(sin[n](x))^{2n+1} )
demostració:
...int[y^{n+1}·(1/n)·(y^{n}/(1+(-1)(y/n)^{n+1})^{(-1)(1/(n+1))})]d[y]=...
...(-1)·int[(y^{n+1}/(n+1))·n^{n}·((-1)(n+1)(y/n)^{n}·(1/n)/(1+(-1)(y/n)^{n+1})^{(-1)(1/(n+1))})]d[y]=...
...(-1)n^{n}( (y^{n+1}/(n+1))·((n+1)/n)(1+(-1)(y/n)^{n+1})^{(n/(n+1))}+...
...(-1)int[y^{n}·((n+1)/n)·(1+(-1)(y/n)^{n+1})^{(n/(n+1))}]d[y] )=...
...(-1)n^{n}( (y^{n+1}/n)(1+(-1)(y/n)^{n+1})^{(n/(n+1))}+...
...n^{n}int[(-1)(n+1)(y/n)^{n}·(1/n)·(1+(-1)(y/n)^{n+1})^{(n/(n+1))}]d[y] )=...
...(-1)n^{n}(y^{n+1}/n)(1+(-1)(y/n)^{n+1})^{(n/(n+1))}+...
...(-1)n^{2n}((n+1)/(2n+1))·(1+(-1)(y/n)^{n+1})^{((2n+1)/(n+1))}=...
...(-1)·(1/n)·(sin[n](x))^{n+1}·(cos[n](x))^{n}+(-1)·(1/n)( (n+1)/(2n+1) )·(cos[n](x))^{2n+1} )


demostració:
...int[y^{n+1}·(1/n)·((-1)·y^{n}/(1+(-1)(y/n)^{n+1})^{(-1)(1/(n+1))})]d[y]=...
...(1/n)·(cos[n](x))^{n+1}·(sin[n](x))^{n}+(1/n)( (n+1)/(2n+1) )·(sin[n](x))^{2n+1} )