Theorem _failure_upper_bound_lemma of type Forall ¶

from the theory of proveit.physics.quantum.QPE ¶

In [1]:

import proveit
# Automation is not needed when only building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_theorem_expr # Load the stored theorem expression as 'stored_expr'
# import the special expression
from proveit.physics.quantum.QPE import _failure_upper_bound_lemma

In [2]:

# check that the built expression is the same as the stored expression
assert _failure_upper_bound_lemma.expr == stored_expr
assert _failure_upper_bound_lemma.expr._style_id == stored_expr._style_id
print("Passed sanity check: _failure_upper_bound_lemma matches stored_expr")

Passed sanity check: _failure_upper_bound_lemma matches stored_expr

In [3]:

# 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) \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)\right)

In [4]:

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',)

In [5]:

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions
0	Operation	operator: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameter: 65 body: 4
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operands: 8
6	Operation	operator: 42 operands: 9
7	Literal
8	ExprTuple	10, 11
9	ExprTuple	65, 12
10	Operation	operator: 13 operand: 65
11	Operation	operator: 68 operands: 15
12	Operation	operator: 49 operands: 16
13	Literal
14	ExprTuple	65
15	ExprTuple	17, 18
16	ExprTuple	85, 19
17	Operation	operator: 40 operands: 20
18	Operation	operator: 73 operands: 21
19	Operation	operator: 73 operands: 22
20	ExprTuple	85, 23
21	ExprTuple	24, 25
22	ExprTuple	64, 26
23	Literal
24	Operation	operator: 28 operand: 31
25	Operation	operator: 28 operand: 32
26	Operation	operator: 83 operand: 80
27	ExprTuple	31
28	Literal
29	ExprTuple	32
30	ExprTuple	80
31	Lambda	parameter: 59 body: 33
32	Lambda	parameter: 59 body: 35
33	Conditional	value: 37 condition: 36
34	ExprTuple	59
35	Conditional	value: 37 condition: 38
36	Operation	operator: 42 operands: 39
37	Operation	operator: 40 operands: 41
38	Operation	operator: 42 operands: 43
39	ExprTuple	59, 44
40	Literal
41	ExprTuple	85, 45
42	Literal
43	ExprTuple	59, 46
44	Operation	operator: 49 operands: 47
45	Operation	operator: 75 operands: 48
46	Operation	operator: 49 operands: 50
47	ExprTuple	51, 52
48	ExprTuple	53, 80
49	Literal
50	ExprTuple	58, 64
51	Operation	operator: 73 operands: 54
52	Operation	operator: 83 operand: 58
53	Operation	operator: 73 operands: 56
54	ExprTuple	57, 85
55	ExprTuple	58
56	ExprTuple	59, 60
57	Operation	operator: 83 operand: 64
58	Operation	operator: 73 operands: 62
59	Variable
60	Operation	operator: 83 operand: 66
61	ExprTuple	64
62	ExprTuple	65, 85
63	ExprTuple	66
64	Operation	operator: 75 operands: 67
65	Variable
66	Operation	operator: 68 operands: 69
67	ExprTuple	80, 70
68	Literal
69	ExprTuple	71, 72
70	Operation	operator: 73 operands: 74
71	Operation	operator: 75 operands: 76
72	Operation	operator: 77 operand: 82
73	Literal
74	ExprTuple	81, 79
75	Literal
76	ExprTuple	80, 81
77	Literal
78	ExprTuple	82
79	Operation	operator: 83 operand: 85
80	Literal
81	Literal
82	Literal
83	Literal
84	ExprTuple	85
85	Literal