Axiom prob_of_all_def 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_axiom_expr # Load the stored axiom expression as 'stored_expr'
# import the special expression
from proveit.statistics import prob_of_all_def

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

Passed sanity check: prob_of_all_def 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 \rightarrow \Omega\right]}~\left[\forall_{Q}~\left(\begin{array}{c} \begin{array}{l} \left[\textrm{Prob}_{x \in X~|~Q\left(x\right)}~f\left(x\right)\right] =  \\ \textrm{Pr}\left(\left\{f\left(x\right)~|~Q\left(x\right)\right\}_{x \in X}\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: 20 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 31 body: 4
3	ExprTuple	31
4	Conditional	value: 5 condition: 6
5	Operation	operator: 20 operand: 10
6	Operation	operator: 8 operands: 9
7	ExprTuple	10
8	Literal
9	ExprTuple	31, 11
10	Lambda	parameter: 53 body: 13
11	Literal
12	ExprTuple	53
13	Operation	operator: 20 operand: 15
14	ExprTuple	15
15	Lambda	parameter: 44 body: 17
16	ExprTuple	44
17	Conditional	value: 18 condition: 19
18	Operation	operator: 20 operand: 23
19	Operation	operator: 49 operands: 22
20	Literal
21	ExprTuple	23
22	ExprTuple	44, 24
23	Lambda	parameter: 51 body: 26
24	Operation	operator: 27 operands: 28
25	ExprTuple	51
26	Operation	operator: 29 operands: 30
27	Literal
28	ExprTuple	53, 31
29	Literal
30	ExprTuple	32, 33
31	Variable
32	Operation	operator: 34 operand: 40
33	Operation	operator: 35 operand: 37
34	Literal
35	Literal
36	ExprTuple	37
37	Operation	operator: 38 operand: 40
38	Literal
39	ExprTuple	40
40	Lambda	parameter: 54 body: 41
41	Conditional	value: 42 condition: 43
42	Operation	operator: 44 operand: 54
43	Operation	operator: 45 operands: 46
44	Variable
45	Literal
46	ExprTuple	47, 48
47	Operation	operator: 49 operands: 50
48	Operation	operator: 51 operand: 54
49	Literal
50	ExprTuple	54, 53
51	Variable
52	ExprTuple	54
53	Variable
54	Variable