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 Variable, k, l
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Mult, Sum, e, frac, i, one, pi, subtract, two
from proveit.physics.quantum.QPE import SubIndexed, _alpha, _b_floor, _m_domain, _phase, _two_pow_t

# build up the expression from sub-expressions
expr = Equals(SubIndexed(_alpha, [Variable("_a", latex_format = r"{_{-}a}")]), Mult(frac(one, _two_pow_t), Sum(index_or_indices = [k], summand = Exp(Exp(e, Mult(two, pi, i, subtract(_phase, frac(Add(_b_floor, l), _two_pow_t)))), k), 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())

\alpha_{{_{-}a}} = \left(\frac{1}{2^{t}} \cdot \left(\sum_{k = 0}^{2^{t} - 1} (\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(\varphi - \frac{b_{\textit{f}} + l}{2^{t}}\right)})^{k}\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: 8
4	Operation	operator: 32 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Variable
9	Operation	operator: 47 operands: 11
10	Operation	operator: 12 operand: 14
11	ExprTuple	43, 50
12	Literal
13	ExprTuple	14
14	Lambda	parameter: 23 body: 16
15	ExprTuple	23
16	Conditional	value: 17 condition: 18
17	Operation	operator: 53 operands: 19
18	Operation	operator: 20 operands: 21
19	ExprTuple	22, 23
20	Literal
21	ExprTuple	23, 24
22	Operation	operator: 53 operands: 25
23	Variable
24	Operation	operator: 26 operands: 27
25	ExprTuple	28, 29
26	Literal
27	ExprTuple	30, 31
28	Literal
29	Operation	operator: 32 operands: 33
30	Literal
31	Operation	operator: 51 operands: 34
32	Literal
33	ExprTuple	57, 35, 36, 37
34	ExprTuple	50, 38
35	Literal
36	Literal
37	Operation	operator: 51 operands: 39
38	Operation	operator: 44 operand: 43
39	ExprTuple	41, 42
40	ExprTuple	43
41	Literal
42	Operation	operator: 44 operand: 46
43	Literal
44	Literal
45	ExprTuple	46
46	Operation	operator: 47 operands: 48
47	Literal
48	ExprTuple	49, 50
49	Operation	operator: 51 operands: 52
50	Operation	operator: 53 operands: 54
51	Literal
52	ExprTuple	55, 56
53	Literal
54	ExprTuple	57, 58
55	Literal
56	Variable
57	Literal
58	Literal