# 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 M, m, n
from proveit.linear_algebra import LinMap, MatrixSpace
from proveit.logic import CartExp, Forall, InSet
from proveit.numbers import Complex, NaturalPos
from proveit.physics.quantum import Qmult

In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [m, n], instance_expr = Forall(instance_param_or_params = [M], instance_expr = InSet(Qmult(M), LinMap(CartExp(Complex, n), CartExp(Complex, m))), domain = MatrixSpace(field = Complex, rows = m, columns = n)), domain = NaturalPos)

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_{m, n \in \mathbb{N}^+}~\left[\forall_{M \in \mathbb{C}^{m \times n}}~\left(\left[M\right] \in \mathcal{L}\left(\mathbb{C}^{n}, \mathbb{C}^{m}\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
3ExprTuple40, 38
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 11
6Operationoperator: 9
operands: 10
7Literal
8ExprTuple11
9Literal
10ExprTuple12, 13
11Lambdaparameter: 32
body: 14
12Operationoperator: 21
operands: 15
13Operationoperator: 21
operands: 16
14Conditionalvalue: 17
condition: 18
15ExprTuple40, 19
16ExprTuple38, 19
17Operationoperator: 21
operands: 20
18Operationoperator: 21
operands: 22
19Literal
20ExprTuple23, 24
21Literal
22ExprTuple32, 25
23Operationoperator: 26
operand: 32
24Operationoperator: 28
operands: 29
25Operationoperator: 30
operands: 31
26Literal
27ExprTuple32
28Literal
29ExprTuple33, 34
30Literal
31NamedExprsfield: 39
rows: 40
columns: 38
32Variable
33Operationoperator: 36
operands: 35
34Operationoperator: 36
operands: 37
35ExprTuple39, 38
36Literal
37ExprTuple39, 40
38Variable
39Literal
40Variable