# from the theory of proveit.linear_algebra.inner_products¶

In [1]:
import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
# import Expression classes needed to build the expression
from proveit import Conditional, Py, X, x, y
from proveit.linear_algebra import Hspace, InnerProd, VecZero
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import zero

In [2]:
# build up the expression from sub-expressions
expr = Conditional(Implies(Forall(instance_param_or_params = [x], instance_expr = Equals(InnerProd(x, y), zero), domain = X), Equals(Py, VecZero(Hspace))), InSet(y, Hspace))

expr:
In [3]:
# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

In [4]:
# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left\{\left[\forall_{x \in X}~\left(\left \langle x, y\right \rangle = 0\right)\right] \Rightarrow \left(P\left(y\right) = \vec{0}\left(\mathcal{H}\right)\right) \textrm{ if } y \in \mathcal{H}\right..

In [5]:
stored_expr.style_options()

namedescriptiondefaultcurrent valuerelated methods
condition_delimiter'comma' or 'and'commacomma('with_comma_delimiter', 'with_conjunction_delimiter')
In [6]:
# display the expression information
stored_expr.expr_info()

core typesub-expressionsexpression
0Conditionalvalue: 1
condition: 2
1Operationoperator: 3
operands: 4
2Operationoperator: 25
operands: 5
3Literal
4ExprTuple6, 7
5ExprTuple33, 22
6Operationoperator: 8
operand: 11
7Operationoperator: 23
operands: 10
8Literal
9ExprTuple11
10ExprTuple12, 13
11Lambdaparameter: 32
body: 15
12Operationoperator: 16
operand: 33
13Operationoperator: 18
operand: 22
14ExprTuple32
15Conditionalvalue: 20
condition: 21
16Variable
17ExprTuple33
18Literal
19ExprTuple22
20Operationoperator: 23
operands: 24
21Operationoperator: 25
operands: 26
22Variable
23Literal
24ExprTuple27, 28
25Literal
26ExprTuple32, 29
27Operationoperator: 30
operands: 31
28Literal
29Variable
30Literal
31ExprTuple32, 33
32Variable
33Variable