Expression of type Forall ¶

from the theory of proveit.statistics ¶

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import A, B, C, Omega, Q, Qx, X, f, fx, x
from proveit.logic import Disjoint, Equals, Forall, Injections, SubsetEq, Union
from proveit.numbers import Add
from proveit.statistics import ProbOfAll, SampleSpaces

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Forall(instance_param_or_params = [Omega], instance_expr = Forall(instance_param_or_params = [A, B, C, X, Q], instance_expr = Forall(instance_param_or_params = [f], instance_expr = Equals(ProbOfAll(instance_param_or_params = sub_expr1, instance_element = fx, domain = C, condition = Qx), Add(ProbOfAll(instance_param_or_params = sub_expr1, instance_element = fx, domain = A, condition = Qx), ProbOfAll(instance_param_or_params = sub_expr1, instance_element = fx, domain = B, condition = Qx))), domain = Injections(X, Omega)), conditions = [SubsetEq(C, X), Equals(C, Union(A, B)), Disjoint(A, B)]).with_wrapping(), domain = SampleSpaces)

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

Passed sanity check: expr 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()

# display the expression information
stored_expr.expr_info()

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',)

	core type	sub-expressions
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

Expression of type Forall¶

from the theory of proveit.statistics¶

Expression of type Forall ¶

from the theory of proveit.statistics ¶