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 Pfail, _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)
expr = Equals(Pfail(e), Add(Sum(index_or_indices = sub_expr1, summand = sub_expr2, domain = Interval(Add(Neg(_two_pow__t_minus_one), one), subtract(Neg(e), one))), 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}^{-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: 5 operand: 49
4	Operation	operator: 56 operands: 6
5	Literal
6	ExprTuple	7, 8
7	Operation	operator: 10 operand: 12
8	Operation	operator: 10 operand: 13
9	ExprTuple	12
10	Literal
11	ExprTuple	13
12	Lambda	parameter: 51 body: 14
13	Lambda	parameter: 51 body: 16
14	Conditional	value: 18 condition: 17
15	ExprTuple	51
16	Conditional	value: 18 condition: 19
17	Operation	operator: 22 operands: 20
18	Operation	operator: 52 operands: 21
19	Operation	operator: 22 operands: 23
20	ExprTuple	51, 24
21	ExprTuple	25, 54
22	Literal
23	ExprTuple	51, 26
24	Operation	operator: 30 operands: 27
25	Operation	operator: 28 operand: 34
26	Operation	operator: 30 operands: 31
27	ExprTuple	32, 33
28	Literal
29	ExprTuple	34
30	Literal
31	ExprTuple	35, 48
32	Operation	operator: 56 operands: 36
33	Operation	operator: 56 operands: 37
34	Operation	operator: 38 operand: 43
35	Operation	operator: 56 operands: 40
36	ExprTuple	41, 62
37	ExprTuple	42, 59
38	Literal
39	ExprTuple	43
40	ExprTuple	49, 62
41	Operation	operator: 60 operand: 48
42	Operation	operator: 60 operand: 49
43	Operation	operator: 46 operands: 47
44	ExprTuple	48
45	ExprTuple	49
46	Literal
47	ExprTuple	50, 51
48	Operation	operator: 52 operands: 53
49	Variable
50	Literal
51	Variable
52	Literal
53	ExprTuple	54, 55
54	Literal
55	Operation	operator: 56 operands: 57
56	Literal
57	ExprTuple	58, 59
58	Literal
59	Operation	operator: 60 operand: 62
60	Literal
61	ExprTuple	62
62	Literal