Theorem complementary_event_prob_of_all of type Forall

from the theory of proveit.statistics

see dependencies

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.statistics import complementary_event_prob_of_all

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

Passed sanity check: complementary_event_prob_of_all matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\forall_{\Omega \underset{{\scriptscriptstyle c}}{\in} \textrm{SampleSpaces}}~\left[\forall_{X}~\left[\forall_{f \in \left[X \xrightarrow[\text{onto}]{\text{1-to-1}} \Omega\right]}~\left[\forall_{Q, R}~\left(\begin{array}{c} \begin{array}{l} \left[\forall_{x \in X}~\left(Q\left(x\right) = (\lnot R\left(x\right))\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[\textrm{Prob}_{x \in X~|~Q\left(x\right)}~f\left(x\right)\right] =  \\ \left(1 - \left[\textrm{Prob}_{y \in X~|~R\left(y\right)}~f\left(y\right)\right]\right) \end{array} \end{array}\right) \end{array} \end{array}\right)\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: 33 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 30 body: 4
3	ExprTuple	30
4	Conditional	value: 5 condition: 6
5	Operation	operator: 33 operand: 10
6	Operation	operator: 8 operands: 9
7	ExprTuple	10
8	Literal
9	ExprTuple	30, 11
10	Lambda	parameter: 81 body: 13
11	Literal
12	ExprTuple	81
13	Operation	operator: 33 operand: 15
14	ExprTuple	15
15	Lambda	parameter: 72 body: 17
16	ExprTuple	72
17	Conditional	value: 18 condition: 19
18	Operation	operator: 33 operand: 22
19	Operation	operator: 77 operands: 21
20	ExprTuple	22
21	ExprTuple	72, 23
22	Lambda	parameters: 24 body: 25
23	Operation	operator: 26 operands: 27
24	ExprTuple	66, 79
25	Operation	operator: 28 operands: 29
26	Literal
27	ExprTuple	81, 30
28	Literal
29	ExprTuple	31, 32
30	Variable
31	Operation	operator: 33 operand: 36
32	Operation	operator: 47 operands: 35
33	Literal
34	ExprTuple	36
35	ExprTuple	37, 38
36	Lambda	parameter: 69 body: 39
37	Operation	operator: 59 operand: 44
38	Operation	operator: 41 operands: 42
39	Conditional	value: 43 condition: 62
40	ExprTuple	44
41	Literal
42	ExprTuple	45, 46
43	Operation	operator: 47 operands: 48
44	Lambda	parameter: 69 body: 49
45	Literal
46	Operation	operator: 50 operand: 55
47	Literal
48	ExprTuple	63, 52
49	Conditional	value: 53 condition: 54
50	Literal
51	ExprTuple	55
52	Operation	operator: 56 operand: 61
53	Operation	operator: 72 operand: 69
54	Operation	operator: 73 operands: 58
55	Operation	operator: 59 operand: 64
56	Literal
57	ExprTuple	61
58	ExprTuple	62, 63
59	Literal
60	ExprTuple	64
61	Operation	operator: 79 operand: 69
62	Operation	operator: 77 operands: 65
63	Operation	operator: 66 operand: 69
64	Lambda	parameter: 82 body: 68
65	ExprTuple	69, 81
66	Variable
67	ExprTuple	69
68	Conditional	value: 70 condition: 71
69	Variable
70	Operation	operator: 72 operand: 82
71	Operation	operator: 73 operands: 74
72	Variable
73	Literal
74	ExprTuple	75, 76
75	Operation	operator: 77 operands: 78
76	Operation	operator: 79 operand: 82
77	Literal
78	ExprTuple	82, 81
79	Variable
80	ExprTuple	82
81	Variable
82	Variable