Expression of type LessEq

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 l
from proveit.numbers import Abs, Add, Exp, LessEq, Mult, Sum, four, frac, one, two
from proveit.physics.quantum.QPE import _diff_l_scaled_delta_floor, _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)
sub_expr3 = frac(one, Mult(four, Exp(_diff_l_scaled_delta_floor, two)))
expr = LessEq(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)), Add(Sum(index_or_indices = sub_expr1, summand = sub_expr3, domain = _neg_domain), Sum(index_or_indices = sub_expr1, summand = sub_expr3, 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(\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) \leq \left(\left(\sum_{l = -2^{t - 1} + 1}^{-\left(e + 1\right)} \frac{1}{4 \cdot \left(l - \left(2^{t} \cdot \delta_{b_{\textit{f}}}\right)\right)^{2}}\right) + \left(\sum_{l = e + 1}^{2^{t - 1}} \frac{1}{4 \cdot \left(l - \left(2^{t} \cdot \delta_{b_{\textit{f}}}\right)\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: 72 operands: 5
4	Operation	operator: 72 operands: 6
5	ExprTuple	7, 8
6	ExprTuple	9, 10
7	Operation	operator: 14 operand: 16
8	Operation	operator: 14 operand: 17
9	Operation	operator: 14 operand: 18
10	Operation	operator: 14 operand: 19
11	ExprTuple	16
12	ExprTuple	17
13	ExprTuple	18
14	Literal
15	ExprTuple	19
16	Lambda	parameter: 66 body: 20
17	Lambda	parameter: 66 body: 21
18	Lambda	parameter: 66 body: 22
19	Lambda	parameter: 66 body: 24
20	Conditional	value: 25 condition: 26
21	Conditional	value: 25 condition: 28
22	Conditional	value: 27 condition: 26
23	ExprTuple	66
24	Conditional	value: 27 condition: 28
25	Operation	operator: 81 operands: 29
26	Operation	operator: 33 operands: 30
27	Operation	operator: 31 operands: 32
28	Operation	operator: 33 operands: 34
29	ExprTuple	35, 86
30	ExprTuple	66, 36
31	Literal
32	ExprTuple	85, 37
33	Literal
34	ExprTuple	66, 38
35	Operation	operator: 39 operand: 45
36	Operation	operator: 43 operands: 41
37	Operation	operator: 74 operands: 42
38	Operation	operator: 43 operands: 44
39	Literal
40	ExprTuple	45
41	ExprTuple	46, 47
42	ExprTuple	48, 49
43	Literal
44	ExprTuple	57, 64
45	Operation	operator: 50 operand: 55
46	Operation	operator: 72 operands: 52
47	Operation	operator: 79 operand: 57
48	Literal
49	Operation	operator: 81 operands: 54
50	Literal
51	ExprTuple	55
52	ExprTuple	56, 85
53	ExprTuple	57
54	ExprTuple	58, 86
55	Operation	operator: 59 operands: 60
56	Operation	operator: 79 operand: 64
57	Operation	operator: 72 operands: 62
58	Operation	operator: 72 operands: 63
59	Literal
60	ExprTuple	88, 66
61	ExprTuple	64
62	ExprTuple	65, 85
63	ExprTuple	66, 67
64	Operation	operator: 81 operands: 68
65	Variable
66	Variable
67	Operation	operator: 79 operand: 71
68	ExprTuple	86, 70
69	ExprTuple	71
70	Operation	operator: 72 operands: 73
71	Operation	operator: 74 operands: 75
72	Literal
73	ExprTuple	87, 76
74	Literal
75	ExprTuple	77, 78
76	Operation	operator: 79 operand: 85
77	Operation	operator: 81 operands: 82
78	Operation	operator: 83 operand: 88
79	Literal
80	ExprTuple	85
81	Literal
82	ExprTuple	86, 87
83	Literal
84	ExprTuple	88
85	Literal
86	Literal
87	Literal
88	Literal