Theorem prob_of_all_as_sum 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_as_sum

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

Passed sanity check: prob_of_all_as_sum 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{1-to-1}} \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] =  \\ \left[\sum_{x \in X~|~Q\left(x\right)}~\textrm{Pr}\left(f\left(x\right)\right)\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: 56 body: 13
11	Literal
12	ExprTuple	56
13	Operation	operator: 20 operand: 15
14	ExprTuple	15
15	Lambda	parameter: 51 body: 17
16	ExprTuple	51
17	Conditional	value: 18 condition: 19
18	Operation	operator: 20 operand: 23
19	Operation	operator: 52 operands: 22
20	Literal
21	ExprTuple	23
22	ExprTuple	51, 24
23	Lambda	parameter: 54 body: 26
24	Operation	operator: 27 operands: 28
25	ExprTuple	54
26	Operation	operator: 29 operands: 30
27	Literal
28	ExprTuple	56, 31
29	Literal
30	ExprTuple	32, 33
31	Variable
32	Operation	operator: 34 operand: 38
33	Operation	operator: 36 operand: 39
34	Literal
35	ExprTuple	38
36	Literal
37	ExprTuple	39
38	Lambda	parameter: 57 body: 40
39	Lambda	parameter: 57 body: 41
40	Conditional	value: 48 condition: 43
41	Conditional	value: 42 condition: 43
42	Operation	operator: 44 operand: 48
43	Operation	operator: 46 operands: 47
44	Literal
45	ExprTuple	48
46	Literal
47	ExprTuple	49, 50
48	Operation	operator: 51 operand: 57
49	Operation	operator: 52 operands: 53
50	Operation	operator: 54 operand: 57
51	Variable
52	Literal
53	ExprTuple	57, 56
54	Variable
55	ExprTuple	57
56	Variable
57	Variable