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This question is linked to the question https://ask.sagemath.org/question/68450/access-to-all-the-coefficients-of-an-expression-seems-to-fail/

Is there a bug in substitution of a function in SageMath? In asking the former question I have seen an error in the result but I was thinking it was my fact. After analysing the code and the result I have discovered an error in the substitution.

I rewrite only the necessary part of the code

def use_prime(expr):
    """
    Return the expression with one-variable function derivatives "prime
    """
    op = expr.operator()
    if op:
        args = expr.operands()
        aargs = (use_prime(a) for a in args)
        opp = op
        if (isinstance(op, sage.symbolic.operators.FDerivativeOperator)and len(args) == 1):
            name = op.function().name()
            primes = "’" * len(op.parameter_set())
            opp = function(f"{name}{primes}")
        return opp(*aargs)
    else:
        return expr


##########################

var('x y p w_0 θ I')
assume(p>=0)
assume(p<=1)
assume(θ>=1)
U = function('U')#déclare U comme le nom d'une fonction
EU(x,y,p) = p * U(x) + (1-p) * U(y)
w_1 = function('w_1')
w_1(w_0, π) = (w_0-π)
w_2 = function('w_2')
w_2(w_0, π, x, I, p) = (w_0 - π) - x + I 
π = function('π')
π(p,I,θ)=θ*(1-p)*I
U = function('U')
EU(x, y, p) = p*U(x) + (1 - p)*U(y)
x, y, p, w_0, θ, I, = SR.var('x y p w_0 θ I')
f = EU(w_1(w_0, π(p,I,θ)),w_2(w_0, π, x, I, π(p,I,θ)) )
h = diff(f, I)
hh = use_prime(h)
k = diff(f, I,2)
kk = use_prime(k)
show(LatexExpr(r'''\text{L'assuré cherche à résoudre le programme suivant :} '''))
show(LatexExpr(r'''I^\star=  \textrm{argmax}_{\{I\}}\left\{pU(w_1)+(1-p) U(w_2)| w_1=w_0 - \pi, w_2=w_0 - \pi -x + I, \pi = \theta(1-p)I \right\} '''))     
show(LatexExpr(r'''\text{ou encore : } I^\star = \textrm{argmax}_{\{I\}}\{'''), f,LatexExpr(r'\}'))     
show(LatexExpr(r'''\text{La condition d'optimalité du premier ordre est } d\mathbb{E}U = 0 \text{, soit : } ''' ))
show(LatexExpr(r'\,\,\,\,\,\,\,\,\,\,\,\,\,\, d\mathbb{E}U = '),hh, ' = 0')

The result is

notice that $\pi = \theta (1-p)I$ which is rewriten directly by Sagemath $\pi = - \theta (p-1) I$ we must add the * of course. But if you look to the definition of $I^\star$ after the substitution the negative signe has disapeared inside $U()$.

Why ?

This question is linked to the a previous question:

• Ask Sage question https://ask.sagemath.org/question/68450/access-to-all-the-coefficients-of-an-expression-seems-to-fail/
  68450: Access to all the coefficients of an expression seems to fail

Is there a bug in substitution of a function in SageMathSageMath??

In asking the former question I have seen saw an error in the result but I was thinking thought it was might be my fact. fault.

After analysing the code and the result I have discovered an error in the substitution.

I rewrite only the necessary part of the codecode:

def use_prime(expr):
    """
    Return the expression with one-variable function derivatives "prime
in prime notation.
    """
    op = expr.operator()
    if op:
        args = expr.operands()
        aargs = (use_prime(a) for a in args)
        opp = op
        if (isinstance(op, sage.symbolic.operators.FDerivativeOperator)and sage.symbolic.operators.FDerivativeOperator)
                and len(args) == 1):
            name = op.function().name()
            primes = "’" * len(op.parameter_set())
            opp = function(f"{name}{primes}")
        return opp(*aargs)
    else:
        return expr


##########################

var('x y p w_0 θ I')
assume(p>=0)
assume(p<=1)
assume(θ>=1)
assume(p >= 0)
assume(p <= 1)
assume(θ >= 1)
U = function('U')#déclare function('U')  # déclare U comme le nom d'une fonction
EU(x,y,p) EU(x, y, p) = p * U(x) + (1-p) * U(y)
w_1 = function('w_1')
w_1(w_0, π) = (w_0-π)
(w_0 - π)
w_2 = function('w_2')
w_2(w_0, π, x, I, p) = (w_0 - π) - x + I 
π = function('π')
π(p,I,θ)=θ*(1-p)*I
π(p, I, θ) = θ * (1 - p) * I
U = function('U')
EU(x, y, p) = p*U(x) p * U(x) + (1 - p)*U(y)
p) * U(y)
x, y, p, w_0, θ, I, = SR.var('x y p w_0 θ I')
f = EU(w_1(w_0, π(p,I,θ)),w_2(w_0, π(p, I, θ)), w_2(w_0, π, x, I, π(p,I,θ)) )
π(p, I, θ)))
h = diff(f, I)
hh = use_prime(h)
k = diff(f, I,2)
I, 2)
kk = use_prime(k)
show(LatexExpr(r'''\text{L'assuré cherche à résoudre le programme suivant :} '''))
show(LatexExpr(r'''I^\star=  show(LatexExpr(r'''I^\star = \textrm{argmax}_{\{I\}}\left\{pU(w_1)+(1-p) U(w_2)| w_1=w_0 w_1'''
               r''' = w_0 - \pi, w_2=w_0 - \pi -x + I, \pi = \theta(1-p)I \right\} '''))     
'''))
show(LatexExpr(r'''\text{ou encore : } I^\star = \textrm{argmax}_{\{I\}}\{'''), f,LatexExpr(r'\}')) \textrm{argmax}_{\{I\}}\{'''),
     f, LatexExpr(r'\}'))     
show(LatexExpr(r'''\text{La condition d'optimalité du premier ordre est } d\mathbb{E}U '''
               r'''d\mathbb{E}U = 0 \text{, soit : } ''' ))
}'''))
show(LatexExpr(r'\,\,\,\,\,\,\,\,\,\,\,\,\,\, d\mathbb{E}U = '),hh, '), hh, ' = 0')

The result is

notice Notice that $\pi = \theta (1-p)I$ which is rewriten rewritten directly by Sagemath SageMath as $\pi = - \theta (p-1) I$ we must add the * * of course. But if you look to at the definition of $I^\star$ after the substitution substitution, the negative signe sign has disapeared disappeared inside $U()$.

Why ?Why?

This question is linked to a previous question:

• Ask Sage question 68450: Access to all the coefficients of an expression seems to fail

Is there a bug in substitution of a function in SageMath?

In asking the former question I saw an error in the result but I thought it might be my fault.

After analysing the code and the result I have discovered an error in the substitution.

I rewrite only the necessary part of the code:

def use_prime(expr):
    """
    Return the expression with one-variable function derivatives in prime notation.
    """
    op = expr.operator()
    if op:
        args = expr.operands()
        aargs = (use_prime(a) for a in args)
        opp = op
        if (isinstance(op, sage.symbolic.operators.FDerivativeOperator)
                and len(args) == 1):
            name = op.function().name()
            primes = "’" * len(op.parameter_set())
            opp = function(f"{name}{primes}")
        return opp(*aargs)
    else:
        return expr

var('x y p w_0 θ I')
assume(p >= 0)
assume(p <= 1)
assume(θ >= 1)
U = function('U')  # déclare U comme le nom d'une fonction
EU(x, y, p) = p * U(x) + (1-p) * U(y)
w_1 = function('w_1')
w_1(w_0, π) = (w_0 - π)
w_2 = function('w_2')
w_2(w_0, π, x, I, p) = (w_0 - π) - x + I 
π = function('π')
π(p, I, θ) = θ * (1 - p) * I
U = function('U')
EU(x, y, p) = p * U(x) + (1 - p) * U(y)
x, y, p, w_0, θ, I, = SR.var('x y p w_0 θ I')
f = EU(w_1(w_0, π(p, I, θ)), w_2(w_0, π, x, I, π(p, I, θ)))
h = diff(f, I)
hh = use_prime(h)
k = diff(f, I, 2)
kk = use_prime(k)
show(LatexExpr(r'''\text{L'assuré show(LatexExpr(r'''\text{L’assuré cherche à résoudre le programme suivant :} '''))
show(LatexExpr(r'''I^\star = \textrm{argmax}_{\{I\}}\left\{pU(w_1)+(1-p) U(w_2)| w_1'''
               r''' = w_0 - \pi, w_2=w_0 - \pi -x + I, \pi = \theta(1-p)I \right\} '''))
show(LatexExpr(r'''\text{ou encore : } I^\star = \textrm{argmax}_{\{I\}}\{'''),
     f, LatexExpr(r'\}'))     
show(LatexExpr(r'''\text{La condition d'optimalité d’optimalité du premier ordre est } '''
               r'''d\mathbb{E}U = 0 \text{, soit : }'''))
show(LatexExpr(r'\,\,\,\,\,\,\,\,\,\,\,\,\,\, d\mathbb{E}U = '), hh, ' = 0')

The result is

Notice that $\pi = \theta (1-p)I$ which is rewritten directly by SageMath as $\pi = - \theta (p-1) I$ we must add the * of course. But if you look at the definition of $I^\star$ after the substitution, the negative sign has disappeared inside $U()$.

Why?

This question is linked to a previous question:

• Ask Sage question 68450: Access to all the coefficients of an expression seems to fail

Is there a bug in substitution of a function in SageMath?

In asking the former question I saw an error in the result but I thought it might be my fault.

After analysing the code and the result I have discovered an error in the substitution.

I rewrite only the necessary part of the code:

def use_prime(expr):
    """
    Return the expression with one-variable function derivatives in prime notation.
    """
    op = expr.operator()
    if op:
        args = expr.operands()
        aargs = (use_prime(a) for a in args)
        opp = op
        if (isinstance(op, sage.symbolic.operators.FDerivativeOperator)
                and len(args) == 1):
            name = op.function().name()
            primes = "’" * len(op.parameter_set())
            opp = function(f"{name}{primes}")
        return opp(*aargs)
    else:
        return expr

var('x y p w_0 θ I')
assume(p >= 0)
assume(p <= 1)
assume(θ >= 1)
U = function('U')  # déclare U comme le nom d'une fonction
EU(x, y, p) = p * U(x) + (1-p) * U(y)
w_1 = function('w_1')
w_1(w_0, π) = (w_0 - π)
w_2 = function('w_2')
w_2(w_0, π, x, I, p) = (w_0 - π) - x + I 
π = function('π')
π(p, I, θ) = θ * (1 - p) * I
U = function('U')
EU(x, y, p) = p * U(x) + (1 - p) * U(y)
x, y, p, w_0, θ, I, = SR.var('x y p w_0 θ I')
f = EU(w_1(w_0, π(p, I, θ)), w_2(w_0, π, x, I, π(p, I, θ)))
h = diff(f, I)
hh = use_prime(h)
k = diff(f, I, 2)
kk = use_prime(k)
show(LatexExpr(r'''\text{L’assuré cherche à résoudre le programme suivant :} '''))
show(LatexExpr(r'''I^\star = \textrm{argmax}_{\{I\}}\left\{pU(w_1)+(1-p) U(w_2)| w_1'''
               r''' = w_0 - \pi, w_2=w_0 - \pi -x + I, \pi = \theta(1-p)I \right\} '''))
show(LatexExpr(r'''\text{ou encore : } I^\star = \textrm{argmax}_{\{I\}}\{'''),
     f, LatexExpr(r'\}'))     
show(LatexExpr(r'''\text{La condition d’optimalité du premier ordre est } '''
               r'''d\mathbb{E}U = 0 \text{, soit : }'''))
show(LatexExpr(r'\,\,\,\,\,\,\,\,\,\,\,\,\,\, d\mathbb{E}U = '), hh, ' = 0')

The result is

Notice that $\pi = \theta (1-p)I$ which is rewritten directly by SageMath as $\pi = - \theta (p-1) I$ we must add the * of course. But if you look at the definition of $I^\star$ after the substitution, the negative sign has disappeared inside $U()$.

Why?

This question is linked related to a previous question:

• Ask Sage question 68450: Access to all the coefficients of an expression seems to fail

Is there a bug in substitution of a function in SageMath?

In asking the former question I saw an error in the result but I thought it might be my fault.

After analysing the code and the result I have discovered an error in the substitution.

I rewrite only the necessary part of the code:

def use_prime(expr):
    """
    Return the expression with one-variable function derivatives in prime notation.
    """
    op = expr.operator()
    if op:
        args = expr.operands()
        aargs = (use_prime(a) for a in args)
        opp = op
        if (isinstance(op, sage.symbolic.operators.FDerivativeOperator)
                and len(args) == 1):
            name = op.function().name()
            primes = "’" * len(op.parameter_set())
            opp = function(f"{name}{primes}")
        return opp(*aargs)
    else:
        return expr

var('x y p w_0 θ I')
assume(p >= 0)
assume(p <= 1)
assume(θ >= 1)
U = function('U')  # déclare U comme le nom d'une fonction
EU(x, y, p) = p * U(x) + (1-p) * U(y)
w_1 = function('w_1')
w_1(w_0, π) = (w_0 - π)
w_2 = function('w_2')
w_2(w_0, π, x, I, p) = (w_0 - π) - x + I 
π = function('π')
π(p, I, θ) = θ * (1 - p) * I
U = function('U')
EU(x, y, p) = p * U(x) + (1 - p) * U(y)
x, y, p, w_0, θ, I, = SR.var('x y p w_0 θ I')
f = EU(w_1(w_0, π(p, I, θ)), w_2(w_0, π, x, I, π(p, I, θ)))
h = diff(f, I)
hh = use_prime(h)
k = diff(f, I, 2)
kk = use_prime(k)
show(LatexExpr(r'''\text{L’assuré cherche à résoudre le programme suivant :} '''))
show(LatexExpr(r'''I^\star = \textrm{argmax}_{\{I\}}\left\{pU(w_1)+(1-p) U(w_2)| w_1'''
               r''' = w_0 - \pi, w_2=w_0 - \pi -x + I, \pi = \theta(1-p)I \right\} '''))
show(LatexExpr(r'''\text{ou encore : } I^\star = \textrm{argmax}_{\{I\}}\{'''),
     f, LatexExpr(r'\}'))     
show(LatexExpr(r'''\text{La condition d’optimalité du premier ordre est } '''
               r'''d\mathbb{E}U = 0 \text{, soit : }'''))
show(LatexExpr(r'\,\,\,\,\,\,\,\,\,\,\,\,\,\, d\mathbb{E}U = '), hh, ' = 0')

The result is

Notice that $\pi = \theta (1-p)I$ which is rewritten directly by SageMath as $\pi = - \theta (p-1) I$ we must add the * of course. But if you look at the definition of $I^\star$ after the substitution, the negative sign has disappeared inside $U()$.

Why?