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Expression of type Equals

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.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import ExprRange, Function, IndexedVar, Q, Variable, alpha, beta, c, delta, f, gamma, j
from proveit.linear_algebra import ScalarMult, VecSum
from proveit.logic import Equals
from proveit.numbers import Mult, one
from proveit.physics.quantum import Qmult, bra_varphi, ket_psi
In [2]:
# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = [ExprRange(sub_expr1, IndexedVar(c, sub_expr1), one, j)]
sub_expr3 = VecSum(index_or_indices = sub_expr2, summand = Function(f, sub_expr2), condition = Function(Q, sub_expr2))
expr = Equals(Qmult(alpha, beta, gamma, delta, bra_varphi, sub_expr3, ket_psi, bra_varphi), ScalarMult(Mult(alpha, beta, gamma, delta), Qmult(bra_varphi, sub_expr3, ket_psi, bra_varphi))).with_wrapping_at(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())
\begin{array}{c} \begin{array}{l} \left(\alpha \thinspace \beta \thinspace \gamma \thinspace \delta \thinspace \langle \varphi \rvert \thinspace \left[\sum_{c_{1}, c_{2}, \ldots, c_{j}~|~Q\left(c_{1}, c_{2}, \ldots, c_{j}\right)}~f\left(c_{1}, c_{2}, \ldots, c_{j}\right)\right] \thinspace \lvert \psi \rangle \thinspace \langle \varphi \rvert\right) =  \\ \left(\left(\alpha \cdot \beta \cdot \gamma \cdot \delta\right) \cdot \left(\langle \varphi \rvert \thinspace \left[\sum_{c_{1}, c_{2}, \ldots, c_{j}~|~Q\left(c_{1}, c_{2}, \ldots, c_{j}\right)}~f\left(c_{1}, c_{2}, \ldots, c_{j}\right)\right] \thinspace \lvert \psi \rangle \thinspace \langle \varphi \rvert\right)\right) \end{array} \end{array}
In [5]:
stored_expr.style_options()
namedescriptiondefaultcurrent valuerelated methods
operation'infix' or 'function' style formattinginfixinfix
wrap_positionsposition(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.()(2)('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justificationif any wrap positions are set, justify to the 'left', 'center', or 'right'centercenter('with_justification',)
directionDirection of the relation (normal or reversed)normalnormal('with_direction_reversed', 'is_reversed')
In [6]:
# display the expression information
stored_expr.expr_info()
 core typesub-expressionsexpression
0Operationoperator: 1
operands: 2
1Literal
2ExprTuple3, 4
3Operationoperator: 12
operands: 5
4Operationoperator: 6
operands: 7
5ExprTuple14, 15, 16, 17, 20, 18, 19, 20
6Literal
7ExprTuple8, 9
8Operationoperator: 10
operands: 11
9Operationoperator: 12
operands: 13
10Literal
11ExprTuple14, 15, 16, 17
12Literal
13ExprTuple20, 18, 19, 20
14Variable
15Variable
16Variable
17Variable
18Operationoperator: 21
operand: 27
19Operationoperator: 23
operand: 28
20Operationoperator: 25
operand: 29
21Literal
22ExprTuple27
23Literal
24ExprTuple28
25Literal
26ExprTuple29
27Lambdaparameters: 35
body: 30
28Variable
29Variable
30Conditionalvalue: 31
condition: 32
31Operationoperator: 33
operands: 35
32Operationoperator: 34
operands: 35
33Variable
34Variable
35ExprTuple36
36ExprRangelambda_map: 37
start_index: 38
end_index: 39
37Lambdaparameter: 43
body: 40
38Literal
39Variable
40IndexedVarvariable: 41
index: 43
41Variable
42ExprTuple43
43Variable