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(VecSum(index_or_indices = sub_expr1, summand = Qmult(sub_expr2, sub_expr3, 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))

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())

\left(\sum_{k=0}^{2^{t} - 1} \left({_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \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)

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.	()	()	('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: 5 operand: 9
4	Operation	operator: 7 operand: 10
5	Literal
6	ExprTuple	9
7	Literal
8	ExprTuple	10
9	Lambda	parameter: 59 body: 11
10	Lambda	parameter: 59 body: 13
11	Conditional	value: 14 condition: 16
12	ExprTuple	59
13	Conditional	value: 15 condition: 16
14	Operation	operator: 26 operands: 17
15	Operation	operator: 50 operands: 18
16	Operation	operator: 19 operands: 20
17	ExprTuple	31, 32, 21
18	ExprTuple	30, 22
19	Literal
20	ExprTuple	59, 23
21	Operation	operator: 24 operands: 25
22	Operation	operator: 26 operands: 27
23	Operation	operator: 28 operands: 29
24	Literal
25	ExprTuple	30, 33
26	Literal
27	ExprTuple	31, 32, 33
28	Literal
29	ExprTuple	34, 35
30	Operation	operator: 52 operands: 36
31	Operation	operator: 37 operands: 38
32	Operation	operator: 39 operand: 61
33	Operation	operator: 41 operands: 42
34	Literal
35	Operation	operator: 43 operands: 44
36	ExprTuple	45, 46
37	Literal
38	ExprTuple	47, 61
39	Literal
40	ExprTuple	61
41	Literal
42	ExprTuple	59, 61
43	Literal
44	ExprTuple	48, 49
45	Literal
46	Operation	operator: 50 operands: 51
47	Variable
48	Operation	operator: 52 operands: 53
49	Operation	operator: 54 operand: 62
50	Literal
51	ExprTuple	60, 56, 57, 58, 59
52	Literal
53	ExprTuple	60, 61
54	Literal
55	ExprTuple	62
56	Literal
57	Literal
58	Literal
59	Variable
60	Literal
61	Literal
62	Literal