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_expr3, sub_expr1, sub_expr2, sub_expr4), Mult(sub_expr3, Qmult(sub_expr1, sub_expr2, sub_expr4))).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(\frac{1}{2^{\frac{t}{2}}} \thinspace {_{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(\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) \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: 10 operands: 5
4	Operation	operator: 48 operands: 6
5	ExprTuple	7, 13, 14, 15
6	ExprTuple	7, 8
7	Operation	operator: 26 operands: 9
8	Operation	operator: 10 operands: 11
9	ExprTuple	64, 12
10	Literal
11	ExprTuple	13, 14, 15
12	Operation	operator: 58 operands: 16
13	Operation	operator: 17 operands: 18
14	Operation	operator: 19 operand: 63
15	Operation	operator: 21 operand: 25
16	ExprTuple	62, 23
17	Literal
18	ExprTuple	24, 63
19	Literal
20	ExprTuple	63
21	Literal
22	ExprTuple	25
23	Operation	operator: 26 operands: 27
24	Variable
25	Lambda	parameter: 55 body: 29
26	Literal
27	ExprTuple	63, 62
28	ExprTuple	55
29	Conditional	value: 30 condition: 31
30	Operation	operator: 32 operands: 33
31	Operation	operator: 34 operands: 35
32	Literal
33	ExprTuple	36, 37
34	Literal
35	ExprTuple	55, 38
36	Operation	operator: 58 operands: 39
37	Operation	operator: 40 operands: 41
38	Operation	operator: 42 operands: 43
39	ExprTuple	44, 45
40	Literal
41	ExprTuple	55, 63
42	Literal
43	ExprTuple	46, 47
44	Literal
45	Operation	operator: 48 operands: 49
46	Literal
47	Operation	operator: 50 operands: 51
48	Literal
49	ExprTuple	62, 52, 53, 54, 55
50	Literal
51	ExprTuple	56, 57
52	Literal
53	Literal
54	Literal
55	Variable
56	Operation	operator: 58 operands: 59
57	Operation	operator: 60 operand: 64
58	Literal
59	ExprTuple	62, 63
60	Literal
61	ExprTuple	64
62	Literal
63	Literal
64	Literal