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, e, frac, i, one, 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 = NumBra(m, _t)
sub_expr2 = InverseFourierTransform(_t)
sub_expr3 = frac(one, Exp(two, frac(_t, two)))
sub_expr4 = 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, ScalarMult(sub_expr3, Qmult(sub_expr2, sub_expr4))), Qmult(sub_expr1, sub_expr3, sub_expr2, sub_expr4))

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({_{t}}\langle m \rvert \thinspace \left(\frac{1}{2^{\frac{t}{2}}} \cdot \left({\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)\right)\right) = \left({_{t}}\langle m \rvert \thinspace \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)

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: 16 operands: 5
4	Operation	operator: 16 operands: 6
5	ExprTuple	8, 7
6	ExprTuple	8, 12, 19, 20
7	Operation	operator: 34 operands: 9
8	Operation	operator: 10 operands: 11
9	ExprTuple	12, 13
10	Literal
11	ExprTuple	14, 65
12	Operation	operator: 28 operands: 15
13	Operation	operator: 16 operands: 17
14	Variable
15	ExprTuple	66, 18
16	Literal
17	ExprTuple	19, 20
18	Operation	operator: 60 operands: 21
19	Operation	operator: 22 operand: 65
20	Operation	operator: 24 operand: 27
21	ExprTuple	64, 26
22	Literal
23	ExprTuple	65
24	Literal
25	ExprTuple	27
26	Operation	operator: 28 operands: 29
27	Lambda	parameter: 57 body: 31
28	Literal
29	ExprTuple	65, 64
30	ExprTuple	57
31	Conditional	value: 32 condition: 33
32	Operation	operator: 34 operands: 35
33	Operation	operator: 36 operands: 37
34	Literal
35	ExprTuple	38, 39
36	Literal
37	ExprTuple	57, 40
38	Operation	operator: 60 operands: 41
39	Operation	operator: 42 operands: 43
40	Operation	operator: 44 operands: 45
41	ExprTuple	46, 47
42	Literal
43	ExprTuple	57, 65
44	Literal
45	ExprTuple	48, 49
46	Literal
47	Operation	operator: 50 operands: 51
48	Literal
49	Operation	operator: 52 operands: 53
50	Literal
51	ExprTuple	64, 54, 55, 56, 57
52	Literal
53	ExprTuple	58, 59
54	Literal
55	Literal
56	Literal
57	Variable
58	Operation	operator: 60 operands: 61
59	Operation	operator: 62 operand: 66
60	Literal
61	ExprTuple	64, 65
62	Literal
63	ExprTuple	66
64	Literal
65	Literal
66	Literal