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, Sum, two
from proveit.physics.quantum.QPE import Pfail, _neg_domain, _pos_domain, _rel_indexed_alpha

# build up the expression from sub-expressions
sub_expr1 = [l]
sub_expr2 = Exp(Abs(_rel_indexed_alpha), two)
expr = Equals(Pfail(e), 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)))

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[P_{\rm fail}\right]\left(e\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)

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: 48
4	Operation	operator: 55 operands: 7
5	Literal
6	ExprTuple	48
7	ExprTuple	8, 9
8	Operation	operator: 11 operand: 13
9	Operation	operator: 11 operand: 14
10	ExprTuple	13
11	Literal
12	ExprTuple	14
13	Lambda	parameter: 50 body: 15
14	Lambda	parameter: 50 body: 17
15	Conditional	value: 19 condition: 18
16	ExprTuple	50
17	Conditional	value: 19 condition: 20
18	Operation	operator: 23 operands: 21
19	Operation	operator: 51 operands: 22
20	Operation	operator: 23 operands: 24
21	ExprTuple	50, 25
22	ExprTuple	26, 53
23	Literal
24	ExprTuple	50, 27
25	Operation	operator: 31 operands: 28
26	Operation	operator: 29 operand: 35
27	Operation	operator: 31 operands: 32
28	ExprTuple	33, 34
29	Literal
30	ExprTuple	35
31	Literal
32	ExprTuple	41, 47
33	Operation	operator: 55 operands: 36
34	Operation	operator: 59 operand: 41
35	Operation	operator: 38 operand: 42
36	ExprTuple	40, 61
37	ExprTuple	41
38	Literal
39	ExprTuple	42
40	Operation	operator: 59 operand: 47
41	Operation	operator: 55 operands: 44
42	Operation	operator: 45 operands: 46
43	ExprTuple	47
44	ExprTuple	48, 61
45	Literal
46	ExprTuple	49, 50
47	Operation	operator: 51 operands: 52
48	Variable
49	Literal
50	Variable
51	Literal
52	ExprTuple	53, 54
53	Literal
54	Operation	operator: 55 operands: 56
55	Literal
56	ExprTuple	57, 58
57	Literal
58	Operation	operator: 59 operand: 61
59	Literal
60	ExprTuple	61
61	Literal