# from the theory of proveit.physics.quantum.algebra¶

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 A, B, Composition, X, Y
from proveit.linear_algebra import HilbertSpaces, Hspace, LinMap
from proveit.logic import Equals, Forall, InSet
from proveit.physics.quantum import Qmult

In [2]:
# build up the expression from sub-expressions
sub_expr1 = Qmult(A)
sub_expr2 = Qmult(B)
expr = Forall(instance_param_or_params = [Hspace, X, Y], instance_expr = Forall(instance_param_or_params = [A, B], instance_expr = Equals(Qmult(A, B), Composition(sub_expr1, sub_expr2)), conditions = [InSet(sub_expr1, LinMap(X, Y)), InSet(sub_expr2, LinMap(Hspace, X))]), domain = HilbertSpaces)

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_{\mathcal{H}, X, Y \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces}}~\left[\forall_{A, B~|~\left[A\right] \in \mathcal{L}\left(X, Y\right), \left[B\right] \in \mathcal{L}\left(\mathcal{H}, X\right)}~\left(\left(A \thinspace B\right) = \left(\left[A\right] \circ \left[B\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: 7
operand: 2
1ExprTuple2
2Lambdaparameters: 3
body: 4
3ExprTuple49, 50, 47
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 10
6Operationoperator: 24
operands: 9
7Literal
8ExprTuple10
9ExprTuple11, 12, 13
10Lambdaparameters: 30
body: 14
11Operationoperator: 17
operands: 15
12Operationoperator: 17
operands: 16
13Operationoperator: 17
operands: 18
14Conditionalvalue: 19
condition: 20
15ExprTuple49, 21
16ExprTuple50, 21
17Literal
18ExprTuple47, 21
19Operationoperator: 22
operands: 23
20Operationoperator: 24
operands: 25
21Literal
22Literal
23ExprTuple26, 27
24Literal
25ExprTuple28, 29
26Operationoperator: 42
operands: 30
27Operationoperator: 31
operands: 32
28Operationoperator: 34
operands: 33
29Operationoperator: 34
operands: 35
30ExprTuple46, 48
31Literal
32ExprTuple36, 38
33ExprTuple36, 37
34Literal
35ExprTuple38, 39
36Operationoperator: 42
operand: 46
37Operationoperator: 44
operands: 41
38Operationoperator: 42
operand: 48
39Operationoperator: 44
operands: 45
40ExprTuple46
41ExprTuple50, 47
42Literal
43ExprTuple48
44Literal
45ExprTuple49, 50
46Variable
47Variable
48Variable
49Variable
50Variable