Expression of type Forall

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, Forall
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 = Forall(instance_param_or_params = [e], instance_expr = 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))), domain = _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())

\forall_{e \in \{1~\ldotp \ldotp~2^{t - 1} - 2\}}~\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)\right)

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

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