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

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

\forall_{y \in \mathcal{H}}~\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)\right)

In [5]:
stored_expr.style_options()

namedescriptiondefaultcurrent valuerelated methods
with_wrappingIf 'True', wrap the Expression after the parametersNoneNone/False('with_wrapping',)
condition_wrappingWrap 'before' or 'after' the condition (or None).NoneNone/False('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_paramsIf 'True', wraps every two parameters AND wraps the Expression after the parametersNoneNone/False('with_params',)
justificationjustify to the 'left', 'center', or 'right' in the array cellscentercenter('with_justification',)
In [6]:
# display the expression information
stored_expr.expr_info()

core typesub-expressionsexpression
0Operationoperator: 11
operand: 2
1ExprTuple2
2Lambdaparameter: 36
body: 3
3Conditionalvalue: 4
condition: 5
4Operationoperator: 6
operands: 7
5Operationoperator: 28
operands: 8
6Literal
7ExprTuple9, 10
8ExprTuple36, 25
9Operationoperator: 11
operand: 14
10Operationoperator: 26
operands: 13
11Literal
12ExprTuple14
13ExprTuple15, 16
14Lambdaparameter: 35
body: 18
15Operationoperator: 19
operand: 36
16Operationoperator: 21
operand: 25
17ExprTuple35
18Conditionalvalue: 23
condition: 24
19Variable
20ExprTuple36
21Literal
22ExprTuple25
23Operationoperator: 26
operands: 27
24Operationoperator: 28
operands: 29
25Variable
26Literal
27ExprTuple30, 31
28Literal
29ExprTuple35, 32
30Operationoperator: 33
operands: 34
31Literal
32Variable
33Literal
34ExprTuple35, 36
35Variable
36Variable