# 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, y
from proveit.linear_algebra import InnerProd, InnerProdSpaces, VecSpaces
from proveit.logic import And, Equals, Forall, InClass, InSet

In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [K, H], instance_expr = Equals(InClass(H, InnerProdSpaces(K)), And(InClass(H, VecSpaces(K)), Forall(instance_param_or_params = [x, y], instance_expr = InSet(InnerProd(x, y), K)))))

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, H}~\left(\left(H \underset{{\scriptscriptstyle c}}{\in} \textrm{InnerProdSpaces}\left(K\right)\right) = \left(\left(H \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)\right) \land \left[\forall_{x, y}~\left(\left \langle x, y\right \rangle \in K\right)\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: 18
operand: 2
1ExprTuple2
2Lambdaparameters: 3
body: 4
3ExprTuple29, 20
4Operationoperator: 5
operands: 6
5Literal
6ExprTuple7, 8
7Operationoperator: 16
operands: 9
8Operationoperator: 10
operands: 11
9ExprTuple20, 12
10Literal
11ExprTuple13, 14
12Operationoperator: 15
operand: 29
13Operationoperator: 16
operands: 17
14Operationoperator: 18
operand: 22
15Literal
16Literal
17ExprTuple20, 21
18Literal
19ExprTuple22
20Variable
21Operationoperator: 23
operand: 29
22Lambdaparameters: 31
body: 25
23Literal
24ExprTuple29
25Operationoperator: 26
operands: 27
26Literal
27ExprTuple28, 29
28Operationoperator: 30
operands: 31
29Variable
30Literal
31ExprTuple32, 33
32Variable
33Variable