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 Equals, InSet
from proveit.numbers import Abs, Add, Exp, Sum, two
from proveit.physics.quantum.QPE import Pfail, _e_domain, _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 = Lambda(e, Conditional(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))), 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) = \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) \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: 58 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 4 operands: 5
3	Operation	operator: 33 operands: 6
4	Literal
5	ExprTuple	7, 8
6	ExprTuple	58, 9
7	Operation	operator: 10 operand: 58
8	Operation	operator: 65 operands: 12
9	Operation	operator: 41 operands: 13
10	Literal
11	ExprTuple	58
12	ExprTuple	14, 15
13	ExprTuple	71, 16
14	Operation	operator: 18 operand: 21
15	Operation	operator: 18 operand: 22
16	Operation	operator: 65 operands: 20
17	ExprTuple	21
18	Literal
19	ExprTuple	22
20	ExprTuple	57, 23
21	Lambda	parameter: 60 body: 24
22	Lambda	parameter: 60 body: 26
23	Operation	operator: 69 operand: 63
24	Conditional	value: 29 condition: 28
25	ExprTuple	60
26	Conditional	value: 29 condition: 30
27	ExprTuple	63
28	Operation	operator: 33 operands: 31
29	Operation	operator: 61 operands: 32
30	Operation	operator: 33 operands: 34
31	ExprTuple	60, 35
32	ExprTuple	36, 63
33	Literal
34	ExprTuple	60, 37
35	Operation	operator: 41 operands: 38
36	Operation	operator: 39 operand: 45
37	Operation	operator: 41 operands: 42
38	ExprTuple	43, 44
39	Literal
40	ExprTuple	45
41	Literal
42	ExprTuple	51, 57
43	Operation	operator: 65 operands: 46
44	Operation	operator: 69 operand: 51
45	Operation	operator: 48 operand: 52
46	ExprTuple	50, 71
47	ExprTuple	51
48	Literal
49	ExprTuple	52
50	Operation	operator: 69 operand: 57
51	Operation	operator: 65 operands: 54
52	Operation	operator: 55 operands: 56
53	ExprTuple	57
54	ExprTuple	58, 71
55	Literal
56	ExprTuple	59, 60
57	Operation	operator: 61 operands: 62
58	Variable
59	Literal
60	Variable
61	Literal
62	ExprTuple	63, 64
63	Literal
64	Operation	operator: 65 operands: 66
65	Literal
66	ExprTuple	67, 68
67	Literal
68	Operation	operator: 69 operand: 71
69	Literal
70	ExprTuple	71
71	Literal