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))), Mult(sub_expr3, Qmult(sub_expr1, 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(\frac{1}{2^{\frac{t}{2}}} \cdot \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)\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: 15 operands: 5
4	Operation	operator: 52 operands: 6
5	ExprTuple	13, 7
6	ExprTuple	11, 8
7	Operation	operator: 36 operands: 9
8	Operation	operator: 15 operands: 10
9	ExprTuple	11, 12
10	ExprTuple	13, 20, 21
11	Operation	operator: 30 operands: 14
12	Operation	operator: 15 operands: 16
13	Operation	operator: 17 operands: 18
14	ExprTuple	68, 19
15	Literal
16	ExprTuple	20, 21
17	Literal
18	ExprTuple	22, 67
19	Operation	operator: 62 operands: 23
20	Operation	operator: 24 operand: 67
21	Operation	operator: 26 operand: 29
22	Variable
23	ExprTuple	66, 28
24	Literal
25	ExprTuple	67
26	Literal
27	ExprTuple	29
28	Operation	operator: 30 operands: 31
29	Lambda	parameter: 59 body: 33
30	Literal
31	ExprTuple	67, 66
32	ExprTuple	59
33	Conditional	value: 34 condition: 35
34	Operation	operator: 36 operands: 37
35	Operation	operator: 38 operands: 39
36	Literal
37	ExprTuple	40, 41
38	Literal
39	ExprTuple	59, 42
40	Operation	operator: 62 operands: 43
41	Operation	operator: 44 operands: 45
42	Operation	operator: 46 operands: 47
43	ExprTuple	48, 49
44	Literal
45	ExprTuple	59, 67
46	Literal
47	ExprTuple	50, 51
48	Literal
49	Operation	operator: 52 operands: 53
50	Literal
51	Operation	operator: 54 operands: 55
52	Literal
53	ExprTuple	66, 56, 57, 58, 59
54	Literal
55	ExprTuple	60, 61
56	Literal
57	Literal
58	Literal
59	Variable
60	Operation	operator: 62 operands: 63
61	Operation	operator: 64 operand: 68
62	Literal
63	ExprTuple	66, 67
64	Literal
65	ExprTuple	68
66	Literal
67	Literal
68	Literal