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 Function, Variable, l
from proveit.logic import Equals
from proveit.numbers import Abs, Add, Exp, Sum, two
from proveit.physics.quantum.QPE import _neg_domain, _pos_domain, _rel_indexed_alpha
from proveit.statistics import ProbOfAll

# build up the expression from sub-expressions
sub_expr1 = [l]
sub_expr2 = Exp(Abs(_rel_indexed_alpha), two)
expr = Equals(ProbOfAll(instance_param_or_params = sub_expr1, instance_element = Function(Variable("_b", latex_format = r"{_{-}b}"), sub_expr1)), Add(Sum(index_or_indices = sub_expr1, summand = sub_expr2, domain = _neg_domain), Sum(index_or_indices = sub_expr1, summand = sub_expr2, domain = _pos_domain))).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[\textrm{Prob}_{l}~{_{-}b}\left(l\right)\right] =  \\ \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) \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: 5 operand: 8
4	Operation	operator: 58 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameter: 53 body: 11
9	Operation	operator: 13 operand: 16
10	Operation	operator: 13 operand: 17
11	Operation	operator: 15 operand: 53
12	ExprTuple	16
13	Literal
14	ExprTuple	17
15	Variable
16	Lambda	parameter: 53 body: 18
17	Lambda	parameter: 53 body: 20
18	Conditional	value: 22 condition: 21
19	ExprTuple	53
20	Conditional	value: 22 condition: 23
21	Operation	operator: 26 operands: 24
22	Operation	operator: 54 operands: 25
23	Operation	operator: 26 operands: 27
24	ExprTuple	53, 28
25	ExprTuple	29, 56
26	Literal
27	ExprTuple	53, 30
28	Operation	operator: 34 operands: 31
29	Operation	operator: 32 operand: 38
30	Operation	operator: 34 operands: 35
31	ExprTuple	36, 37
32	Literal
33	ExprTuple	38
34	Literal
35	ExprTuple	44, 50
36	Operation	operator: 58 operands: 39
37	Operation	operator: 62 operand: 44
38	Operation	operator: 41 operand: 45
39	ExprTuple	43, 64
40	ExprTuple	44
41	Literal
42	ExprTuple	45
43	Operation	operator: 62 operand: 50
44	Operation	operator: 58 operands: 47
45	Operation	operator: 48 operands: 49
46	ExprTuple	50
47	ExprTuple	51, 64
48	Literal
49	ExprTuple	52, 53
50	Operation	operator: 54 operands: 55
51	Variable
52	Literal
53	Variable
54	Literal
55	ExprTuple	56, 57
56	Literal
57	Operation	operator: 58 operands: 59
58	Literal
59	ExprTuple	60, 61
60	Literal
61	Operation	operator: 62 operand: 64
62	Literal
63	ExprTuple	64
64	Literal