Expression of type Lambda

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 Conditional, Lambda, e, l
from proveit.logic import InSet
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, _e_domain, _neg_domain, _pos_domain

# build up the expression from sub-expressions
sub_expr1 = [l]
sub_expr2 = frac(one, Exp(_diff_l_scaled_delta_floor, two))
expr = Lambda(e, Conditional(LessEq(Pfail(e), Mult(frac(one, four), 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)))), InSet(e, _e_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())

e \mapsto \left\{\left[P_{\rm fail}\right]\left(e\right) \leq \left(\frac{1}{4} \cdot \left(\left(\sum_{l = -2^{t - 1} + 1}^{-\left(e + 1\right)} \frac{1}{\left(l - \left(2^{t} \cdot \delta_{b_{\textit{f}}}\right)\right)^{2}}\right) + \left(\sum_{l = e + 1}^{2^{t - 1}} \frac{1}{\left(l - \left(2^{t} \cdot \delta_{b_{\textit{f}}}\right)\right)^{2}}\right)\right)\right) \textrm{ if } e \in \{1~\ldotp \ldotp~2^{t - 1} - 2\}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 62 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 4 operands: 5
3	Operation	operator: 39 operands: 6
4	Literal
5	ExprTuple	7, 8
6	ExprTuple	62, 9
7	Operation	operator: 10 operand: 62
8	Operation	operator: 65 operands: 12
9	Operation	operator: 46 operands: 13
10	Literal
11	ExprTuple	62
12	ExprTuple	14, 15
13	ExprTuple	82, 16
14	Operation	operator: 37 operands: 17
15	Operation	operator: 70 operands: 18
16	Operation	operator: 70 operands: 19
17	ExprTuple	82, 20
18	ExprTuple	21, 22
19	ExprTuple	61, 23
20	Literal
21	Operation	operator: 25 operand: 28
22	Operation	operator: 25 operand: 29
23	Operation	operator: 80 operand: 77
24	ExprTuple	28
25	Literal
26	ExprTuple	29
27	ExprTuple	77
28	Lambda	parameter: 56 body: 30
29	Lambda	parameter: 56 body: 32
30	Conditional	value: 34 condition: 33
31	ExprTuple	56
32	Conditional	value: 34 condition: 35
33	Operation	operator: 39 operands: 36
34	Operation	operator: 37 operands: 38
35	Operation	operator: 39 operands: 40
36	ExprTuple	56, 41
37	Literal
38	ExprTuple	82, 42
39	Literal
40	ExprTuple	56, 43
41	Operation	operator: 46 operands: 44
42	Operation	operator: 72 operands: 45
43	Operation	operator: 46 operands: 47
44	ExprTuple	48, 49
45	ExprTuple	50, 77
46	Literal
47	ExprTuple	55, 61
48	Operation	operator: 70 operands: 51
49	Operation	operator: 80 operand: 55
50	Operation	operator: 70 operands: 53
51	ExprTuple	54, 82
52	ExprTuple	55
53	ExprTuple	56, 57
54	Operation	operator: 80 operand: 61
55	Operation	operator: 70 operands: 59
56	Variable
57	Operation	operator: 80 operand: 63
58	ExprTuple	61
59	ExprTuple	62, 82
60	ExprTuple	63
61	Operation	operator: 72 operands: 64
62	Variable
63	Operation	operator: 65 operands: 66
64	ExprTuple	77, 67
65	Literal
66	ExprTuple	68, 69
67	Operation	operator: 70 operands: 71
68	Operation	operator: 72 operands: 73
69	Operation	operator: 74 operand: 79
70	Literal
71	ExprTuple	78, 76
72	Literal
73	ExprTuple	77, 78
74	Literal
75	ExprTuple	79
76	Operation	operator: 80 operand: 82
77	Literal
78	Literal
79	Literal
80	Literal
81	ExprTuple	82
82	Literal