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 e, l
from proveit.numbers import Add, Exp, LessEq, Mult, Sum, four, frac, one, two
from proveit.physics.quantum.QPE import Pfail, _diff_l_scaled_delta_floor, _neg_domain, _pos_domain

# build up the expression from sub-expressions
sub_expr1 = [l]
sub_expr2 = frac(one, Mult(four, Exp(_diff_l_scaled_delta_floor, two)))
expr = LessEq(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) \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: 5 operand: 47
4	Operation	operator: 54 operands: 7
5	Literal
6	ExprTuple	47
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: 48 body: 15
14	Lambda	parameter: 48 body: 17
15	Conditional	value: 19 condition: 18
16	ExprTuple	48
17	Conditional	value: 19 condition: 20
18	Operation	operator: 24 operands: 21
19	Operation	operator: 22 operands: 23
20	Operation	operator: 24 operands: 25
21	ExprTuple	48, 26
22	Literal
23	ExprTuple	67, 27
24	Literal
25	ExprTuple	48, 28
26	Operation	operator: 31 operands: 29
27	Operation	operator: 56 operands: 30
28	Operation	operator: 31 operands: 32
29	ExprTuple	33, 34
30	ExprTuple	35, 36
31	Literal
32	ExprTuple	41, 46
33	Operation	operator: 54 operands: 37
34	Operation	operator: 61 operand: 41
35	Literal
36	Operation	operator: 63 operands: 39
37	ExprTuple	40, 67
38	ExprTuple	41
39	ExprTuple	42, 68
40	Operation	operator: 61 operand: 46
41	Operation	operator: 54 operands: 44
42	Operation	operator: 54 operands: 45
43	ExprTuple	46
44	ExprTuple	47, 67
45	ExprTuple	48, 49
46	Operation	operator: 63 operands: 50
47	Variable
48	Variable
49	Operation	operator: 61 operand: 53
50	ExprTuple	68, 52
51	ExprTuple	53
52	Operation	operator: 54 operands: 55
53	Operation	operator: 56 operands: 57
54	Literal
55	ExprTuple	69, 58
56	Literal
57	ExprTuple	59, 60
58	Operation	operator: 61 operand: 67
59	Operation	operator: 63 operands: 64
60	Operation	operator: 65 operand: 70
61	Literal
62	ExprTuple	67
63	Literal
64	ExprTuple	68, 69
65	Literal
66	ExprTuple	70
67	Literal
68	Literal
69	Literal
70	Literal