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Expression of type Equals

from the theory of proveit.physics.quantum.QPE

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 Conditional, k, m
from proveit.logic import Equals, InSet
from proveit.numbers import Exp, Mult, Neg, e, frac, i, one, pi, two
from proveit.physics.quantum.QPE import _m_domain, _phase, _t, _two_pow_t
In [2]:
# build up the expression from sub-expressions
sub_expr1 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr2 = frac(one, Exp(two, frac(_t, two)))
sub_expr3 = Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t)))
sub_expr4 = InSet(k, _m_domain)
expr = Equals(Conditional(Mult(sub_expr1, Mult(sub_expr2, sub_expr3)), sub_expr4), Conditional(Mult(sub_expr1, sub_expr2, sub_expr3), sub_expr4)).with_wrapping_at(1)
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\{\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \left(\frac{1}{2^{\frac{t}{2}}} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right) \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right.. \\  = \left\{\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \frac{1}{2^{\frac{t}{2}}} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}} \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\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.()(1)('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
3Conditionalvalue: 5
condition: 7
4Conditionalvalue: 6
condition: 7
5Operationoperator: 46
operands: 8
6Operationoperator: 46
operands: 9
7Operationoperator: 10
operands: 11
8ExprTuple13, 12
9ExprTuple13, 19, 20
10Literal
11ExprTuple52, 14
12Operationoperator: 46
operands: 15
13Operationoperator: 48
operands: 16
14Operationoperator: 17
operands: 18
15ExprTuple19, 20
16ExprTuple30, 21
17Literal
18ExprTuple22, 23
19Operationoperator: 42
operands: 24
20Operationoperator: 48
operands: 25
21Operationoperator: 46
operands: 26
22Literal
23Operationoperator: 27
operands: 28
24ExprTuple40, 29
25ExprTuple30, 31
26ExprTuple54, 50, 51, 32, 52
27Literal
28ExprTuple45, 33
29Operationoperator: 48
operands: 34
30Literal
31Operationoperator: 36
operand: 39
32Literal
33Operationoperator: 36
operand: 40
34ExprTuple54, 38
35ExprTuple39
36Literal
37ExprTuple40
38Operationoperator: 42
operands: 41
39Operationoperator: 42
operands: 43
40Literal
41ExprTuple55, 54
42Literal
43ExprTuple44, 45
44Operationoperator: 46
operands: 47
45Operationoperator: 48
operands: 49
46Literal
47ExprTuple54, 50, 51, 52, 53
48Literal
49ExprTuple54, 55
50Literal
51Literal
52Variable
53Variable
54Literal
55Literal