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 e, l
from proveit.logic import Equals
from proveit.numbers import Abs, Add, Exp, Interval, Neg, Sum, one, subtract, two
from proveit.physics.quantum.QPE import _neg_domain, _pos_domain, _rel_indexed_alpha, _two_pow__t_minus_one

# build up the expression from sub-expressions
sub_expr1 = [l]
sub_expr2 = Exp(Abs(_rel_indexed_alpha), two)
sub_expr3 = Sum(index_or_indices = sub_expr1, summand = sub_expr2, domain = _pos_domain)
expr = Equals(Add(Sum(index_or_indices = sub_expr1, summand = sub_expr2, domain = _neg_domain), sub_expr3), Add(Sum(index_or_indices = sub_expr1, summand = sub_expr2, domain = Interval(Add(Neg(_two_pow__t_minus_one), one), subtract(Neg(e), one))), sub_expr3))

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(\left(\sum_{l = -2^{t - 1} + 1}^{-\left(e + 1\right)} \left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2}\right) + \left(\sum_{l = e + 1}^{2^{t - 1}} \left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2}\right)\right) = \left(\left(\sum_{l = -2^{t - 1} + 1}^{-e - 1} \left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2}\right) + \left(\sum_{l = e + 1}^{2^{t - 1}} \left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2}\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: 66 operands: 5
4	Operation	operator: 66 operands: 6
5	ExprTuple	7, 9
6	ExprTuple	8, 9
7	Operation	operator: 12 operand: 14
8	Operation	operator: 12 operand: 15
9	Operation	operator: 12 operand: 16
10	ExprTuple	14
11	ExprTuple	15
12	Literal
13	ExprTuple	16
14	Lambda	parameter: 61 body: 17
15	Lambda	parameter: 61 body: 18
16	Lambda	parameter: 61 body: 20
17	Conditional	value: 23 condition: 21
18	Conditional	value: 23 condition: 22
19	ExprTuple	61
20	Conditional	value: 23 condition: 24
21	Operation	operator: 28 operands: 25
22	Operation	operator: 28 operands: 26
23	Operation	operator: 62 operands: 27
24	Operation	operator: 28 operands: 29
25	ExprTuple	61, 30
26	ExprTuple	61, 31
27	ExprTuple	32, 64
28	Literal
29	ExprTuple	61, 33
30	Operation	operator: 38 operands: 34
31	Operation	operator: 38 operands: 35
32	Operation	operator: 36 operand: 43
33	Operation	operator: 38 operands: 39
34	ExprTuple	41, 40
35	ExprTuple	41, 42
36	Literal
37	ExprTuple	43
38	Literal
39	ExprTuple	49, 58
40	Operation	operator: 70 operand: 49
41	Operation	operator: 66 operands: 45
42	Operation	operator: 66 operands: 46
43	Operation	operator: 47 operand: 52
44	ExprTuple	49
45	ExprTuple	50, 72
46	ExprTuple	51, 69
47	Literal
48	ExprTuple	52
49	Operation	operator: 66 operands: 53
50	Operation	operator: 70 operand: 58
51	Operation	operator: 70 operand: 59
52	Operation	operator: 56 operands: 57
53	ExprTuple	59, 72
54	ExprTuple	58
55	ExprTuple	59
56	Literal
57	ExprTuple	60, 61
58	Operation	operator: 62 operands: 63
59	Variable
60	Literal
61	Variable
62	Literal
63	ExprTuple	64, 65
64	Literal
65	Operation	operator: 66 operands: 67
66	Literal
67	ExprTuple	68, 69
68	Literal
69	Operation	operator: 70 operand: 72
70	Literal
71	ExprTuple	72
72	Literal