Theorem sample_space_via_bijections 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 sample_space_via_bijections

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

Passed sanity check: sample_space_via_bijections 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, Y}~\left[\forall_{f \in \left[X \xrightarrow[\text{onto}]{\text{1-to-1}} \Omega\right], g \in \left[X \xrightarrow[\text{onto}]{\text{1-to-1}} Y\right]}~\left(\begin{array}{c} \begin{array}{l} \left[\forall_{x \in X}~\left(\textrm{Pr}\left(f\left(x\right)\right) = \textrm{Pr}\left(g\left(x\right)\right)\right)\right] \\  \Rightarrow \left(Y \underset{{\scriptscriptstyle c}}{\in} \textrm{SampleSpaces}\right) \end{array} \end{array}\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: 25 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 41 body: 4
3	ExprTuple	41
4	Conditional	value: 5 condition: 6
5	Operation	operator: 25 operand: 9
6	Operation	operator: 27 operands: 8
7	ExprTuple	9
8	ExprTuple	41, 32
9	Lambda	parameters: 38 body: 10
10	Operation	operator: 25 operand: 12
11	ExprTuple	12
12	Lambda	parameters: 13 body: 14
13	ExprTuple	55, 56
14	Conditional	value: 15 condition: 16
15	Operation	operator: 17 operands: 18
16	Operation	operator: 19 operands: 20
17	Literal
18	ExprTuple	21, 22
19	Literal
20	ExprTuple	23, 24
21	Operation	operator: 25 operand: 31
22	Operation	operator: 27 operands: 28
23	Operation	operator: 45 operands: 29
24	Operation	operator: 45 operands: 30
25	Literal
26	ExprTuple	31
27	Literal
28	ExprTuple	42, 32
29	ExprTuple	55, 33
30	ExprTuple	56, 34
31	Lambda	parameter: 58 body: 35
32	Literal
33	Operation	operator: 37 operands: 36
34	Operation	operator: 37 operands: 38
35	Conditional	value: 39 condition: 40
36	ExprTuple	49, 41
37	Literal
38	ExprTuple	49, 42
39	Operation	operator: 43 operands: 44
40	Operation	operator: 45 operands: 46
41	Variable
42	Variable
43	Literal
44	ExprTuple	47, 48
45	Literal
46	ExprTuple	58, 49
47	Operation	operator: 51 operand: 53
48	Operation	operator: 51 operand: 54
49	Variable
50	ExprTuple	53
51	Literal
52	ExprTuple	54
53	Operation	operator: 55 operand: 58
54	Operation	operator: 56 operand: 58
55	Variable
56	Variable
57	ExprTuple	58
58	Variable