Theorem prob_of_all_events_transformation 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 prob_of_all_events_transformation

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

Passed sanity check: prob_of_all_events_transformation 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_{A, B, Q}~\left[\begin{array}{l}\forall_{f \in \left[A \xrightarrow[]{\text{1-to-1}} \Omega\right], g \in \left[B \xrightarrow[\text{onto}]{\text{1-to-1}} A\right]}~\\
\left(\left[\textrm{Prob}_{x \in A~|~Q\left(x\right)}~f\left(x\right)\right] = \left[\textrm{Prob}_{y \in B~|~Q\left(g\left(y\right)\right)}~f\left(g\left(y\right)\right)\right]\right)\end{array}\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: 14 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 47 body: 4
3	ExprTuple	47
4	Conditional	value: 5 condition: 6
5	Operation	operator: 14 operand: 10
6	Operation	operator: 8 operands: 9
7	ExprTuple	10
8	Literal
9	ExprTuple	47, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	62, 64, 60
13	Operation	operator: 14 operand: 16
14	Literal
15	ExprTuple	16
16	Lambda	parameters: 17 body: 18
17	ExprTuple	49, 66
18	Conditional	value: 19 condition: 20
19	Operation	operator: 21 operands: 22
20	Operation	operator: 50 operands: 23
21	Literal
22	ExprTuple	24, 25
23	ExprTuple	26, 27
24	Operation	operator: 29 operand: 33
25	Operation	operator: 29 operand: 34
26	Operation	operator: 58 operands: 31
27	Operation	operator: 58 operands: 32
28	ExprTuple	33
29	Literal
30	ExprTuple	34
31	ExprTuple	49, 35
32	ExprTuple	66, 36
33	Lambda	parameter: 63 body: 37
34	Lambda	parameter: 68 body: 38
35	Operation	operator: 39 operands: 40
36	Operation	operator: 41 operands: 42
37	Conditional	value: 43 condition: 44
38	Conditional	value: 45 condition: 46
39	Literal
40	ExprTuple	62, 47
41	Literal
42	ExprTuple	64, 62
43	Operation	operator: 49 operand: 63
44	Operation	operator: 50 operands: 48
45	Operation	operator: 49 operand: 65
46	Operation	operator: 50 operands: 51
47	Variable
48	ExprTuple	52, 53
49	Variable
50	Literal
51	ExprTuple	54, 55
52	Operation	operator: 58 operands: 56
53	Operation	operator: 60 operand: 63
54	Operation	operator: 58 operands: 59
55	Operation	operator: 60 operand: 65
56	ExprTuple	63, 62
57	ExprTuple	63
58	Literal
59	ExprTuple	68, 64
60	Variable
61	ExprTuple	65
62	Variable
63	Variable
64	Variable
65	Operation	operator: 66 operand: 68
66	Variable
67	ExprTuple	68
68	Variable