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, m
from proveit.linear_algebra import ScalarMult, VecSum
from proveit.logic import Equals
from proveit.numbers import Exp, Mult, Sum, e, i, pi, two
from proveit.physics.quantum import NumBra, 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 = [k]
sub_expr2 = NumBra(m, _t)
sub_expr3 = InverseFourierTransform(_t)
sub_expr4 = NumKet(k, _t)
sub_expr5 = Exp(e, Mult(two, pi, i, _phase, k))
expr = Equals(Qmult(sub_expr2, sub_expr3, VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr5, sub_expr4), domain = _m_domain)), Sum(index_or_indices = sub_expr1, summand = Mult(sub_expr5, Qmult(sub_expr2, sub_expr3, sub_expr4)), domain = _m_domain)).with_wrapping_at(1)

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({_{t}}\langle m \rvert \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) \\  = \left(\sum_{k = 0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \left({_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \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.	()	(1)	('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: 25 operands: 5
4	Operation	operator: 6 operand: 9
5	ExprTuple	29, 30, 8
6	Literal
7	ExprTuple	9
8	Operation	operator: 10 operand: 13
9	Lambda	parameter: 53 body: 12
10	Literal
11	ExprTuple	13
12	Conditional	value: 14 condition: 19
13	Lambda	parameter: 53 body: 16
14	Operation	operator: 46 operands: 17
15	ExprTuple	53
16	Conditional	value: 18 condition: 19
17	ExprTuple	27, 20
18	Operation	operator: 21 operands: 22
19	Operation	operator: 23 operands: 24
20	Operation	operator: 25 operands: 26
21	Literal
22	ExprTuple	27, 31
23	Literal
24	ExprTuple	53, 28
25	Literal
26	ExprTuple	29, 30, 31
27	Operation	operator: 56 operands: 32
28	Operation	operator: 33 operands: 34
29	Operation	operator: 35 operands: 36
30	Operation	operator: 37 operand: 61
31	Operation	operator: 39 operands: 40
32	ExprTuple	41, 42
33	Literal
34	ExprTuple	43, 44
35	Literal
36	ExprTuple	45, 61
37	Literal
38	ExprTuple	61
39	Literal
40	ExprTuple	53, 61
41	Literal
42	Operation	operator: 46 operands: 47
43	Literal
44	Operation	operator: 48 operands: 49
45	Variable
46	Literal
47	ExprTuple	60, 50, 51, 52, 53
48	Literal
49	ExprTuple	54, 55
50	Literal
51	Literal
52	Literal
53	Variable
54	Operation	operator: 56 operands: 57
55	Operation	operator: 58 operand: 62
56	Literal
57	ExprTuple	60, 61
58	Literal
59	ExprTuple	62
60	Literal
61	Literal
62	Literal