# 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 ExprTuple, Py, X, x, y
from proveit.linear_algebra import Hspace, InnerProd, VecZero
from proveit.logic import Equals, Forall
from proveit.numbers import zero

In [2]:
# build up the expression from sub-expressions
expr = ExprTuple(Forall(instance_param_or_params = [x], instance_expr = Equals(InnerProd(x, y), zero), domain = X), Equals(Py, VecZero(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(\forall_{x \in X}~\left(\left \langle x, y\right \rangle = 0\right), P\left(y\right) = \vec{0}\left(\mathcal{H}\right)\right)

In [5]:
stored_expr.style_options()

namedescriptiondefaultcurrent valuerelated methods
wrap_positionsposition(s) at which wrapping is to occur; 'n' is after the nth comma.()()('with_wrapping_at',)
justificationif any wrap positions are set, justify to the 'left', 'center', or 'right'leftleft('with_justification',)
In [6]:
# display the expression information
stored_expr.expr_info()

core typesub-expressionsexpression
0ExprTuple1, 2
1Operationoperator: 3
operand: 6
2Operationoperator: 18
operands: 5
3Literal
4ExprTuple6
5ExprTuple7, 8
6Lambdaparameter: 27
body: 10
7Operationoperator: 11
operand: 28
8Operationoperator: 13
operand: 17
9ExprTuple27
10Conditionalvalue: 15
condition: 16
11Variable
12ExprTuple28
13Literal
14ExprTuple17
15Operationoperator: 18
operands: 19
16Operationoperator: 20
operands: 21
17Variable
18Literal
19ExprTuple22, 23
20Literal
21ExprTuple27, 24
22Operationoperator: 25
operands: 26
23Literal
24Variable
25Literal
26ExprTuple27, 28
27Variable
28Variable