Expression of type Equals

from the theory of proveit.physics.quantum.QPE

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 k
from proveit.linear_algebra import ScalarMult, VecSum
from proveit.logic import Equals
from proveit.numbers import Exp, Mult, e, frac, i, one, pi, two
from proveit.physics.quantum import NumKet, Qmult
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import _m_domain, _phase, _t

# build up the expression from sub-expressions
sub_expr1 = InverseFourierTransform(_t)
sub_expr2 = frac(one, Exp(two, frac(_t, two)))
sub_expr3 = VecSum(index_or_indices = [k], summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, _t)), domain = _m_domain)
expr = Equals(Qmult(sub_expr1, sub_expr2, sub_expr3), Qmult(sub_expr2, sub_expr1, sub_expr3)).with_wrapping_at(2)

expr:

# 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

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\begin{array}{c} \begin{array}{l} \left({\mathrm {FT}}^{\dag}_{t} \thinspace \frac{1}{2^{\frac{t}{2}}} \thinspace \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right)\right) =  \\ \left(\frac{1}{2^{\frac{t}{2}}} \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right)\right) \end{array} \end{array}

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(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')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 6 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	9, 8, 10
6	Literal
7	ExprTuple	8, 9, 10
8	Operation	operator: 24 operands: 11
9	Operation	operator: 12 operand: 57
10	Operation	operator: 14 operand: 17
11	ExprTuple	58, 16
12	Literal
13	ExprTuple	57
14	Literal
15	ExprTuple	17
16	Operation	operator: 52 operands: 18
17	Lambda	parameter: 49 body: 20
18	ExprTuple	56, 21
19	ExprTuple	49
20	Conditional	value: 22 condition: 23
21	Operation	operator: 24 operands: 25
22	Operation	operator: 26 operands: 27
23	Operation	operator: 28 operands: 29
24	Literal
25	ExprTuple	57, 56
26	Literal
27	ExprTuple	30, 31
28	Literal
29	ExprTuple	49, 32
30	Operation	operator: 52 operands: 33
31	Operation	operator: 34 operands: 35
32	Operation	operator: 36 operands: 37
33	ExprTuple	38, 39
34	Literal
35	ExprTuple	49, 57
36	Literal
37	ExprTuple	40, 41
38	Literal
39	Operation	operator: 42 operands: 43
40	Literal
41	Operation	operator: 44 operands: 45
42	Literal
43	ExprTuple	56, 46, 47, 48, 49
44	Literal
45	ExprTuple	50, 51
46	Literal
47	Literal
48	Literal
49	Variable
50	Operation	operator: 52 operands: 53
51	Operation	operator: 54 operand: 58
52	Literal
53	ExprTuple	56, 57
54	Literal
55	ExprTuple	58
56	Literal
57	Literal
58	Literal