# 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 H, K, x
from proveit.linear_algebra import InnerProd, InnerProdSpaces, Norm
from proveit.logic import Equals, Forall
from proveit.numbers import sqrt

In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [K], instance_expr = Forall(instance_param_or_params = [H], instance_expr = Forall(instance_param_or_params = [x], instance_expr = Equals(Norm(x), sqrt(InnerProd(x, x))), domain = H), domain = InnerProdSpaces(K)))

Out[2]:
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_{K}~\left[\forall_{H \underset{{\scriptscriptstyle c}}{\in} \textrm{InnerProdSpaces}\left(K\right)}~\left[\forall_{x \in H}~\left(\left \|x\right \| = \sqrt{\left \langle x, x\right \rangle}\right)\right]\right]

In [5]:
stored_expr.style_options()

Out[5]:
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()

Out[6]:
core typesub-expressionsexpression
0Operationoperator: 10
operand: 2
1ExprTuple2
2Lambdaparameter: 21
body: 3
3Operationoperator: 10
operand: 5
4ExprTuple5
5Lambdaparameter: 28
body: 7
6ExprTuple28
7Conditionalvalue: 8
condition: 9
8Operationoperator: 10
operand: 14
9Operationoperator: 12
operands: 13
10Literal
11ExprTuple14
12Literal
13ExprTuple28, 15
14Lambdaparameter: 39
body: 16
15Operationoperator: 17
operand: 21
16Conditionalvalue: 19
condition: 20
17Literal
18ExprTuple21
19Operationoperator: 22
operands: 23
20Operationoperator: 24
operands: 25
21Variable
22Literal
23ExprTuple26, 27
24Literal
25ExprTuple39, 28
26Operationoperator: 29
operand: 39
27Operationoperator: 31
operands: 32
28Variable
29Literal
30ExprTuple39
31Literal
32ExprTuple33, 34
33Operationoperator: 35
operands: 36
34Operationoperator: 37
operands: 38
35Literal
36ExprTuple39, 39
37Literal
38ExprTuple40, 41
39Variable
40Literal
41Literal