Theorem prob_of_disjoint_events_is_prob_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_disjoint_events_is_prob_sum

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

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