Expression of type Conditional ¶

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 Conditional, Omega, Q, Qx, R, Rx, Ry, X, f, fx, fy, x, y
from proveit.logic import Bijections, Equals, Forall, Implies, InClass, Not
from proveit.numbers import one, subtract
from proveit.statistics import ProbOfAll, SampleSpaces

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Conditional(Forall(instance_param_or_params = [X], instance_expr = Forall(instance_param_or_params = [f], instance_expr = Forall(instance_param_or_params = [Q, R], instance_expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = Equals(Qx, Not(Rx)), domain = X), Equals(ProbOfAll(instance_param_or_params = sub_expr1, instance_element = fx, domain = X, condition = Qx), subtract(one, ProbOfAll(instance_param_or_params = [y], instance_element = fy, domain = X, condition = Ry))).with_wrapping_at(2)).with_wrapping_at(2)), domain = Bijections(X, Omega))), InClass(Omega, 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())

\left\{\forall_{X}~\left[\forall_{f \in \left[X \xrightarrow[\text{onto}]{\text{1-to-1}} \Omega\right]}~\left[\forall_{Q, R}~\left(\begin{array}{c} \begin{array}{l} \left[\forall_{x \in X}~\left(Q\left(x\right) = (\lnot R\left(x\right))\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[\textrm{Prob}_{x \in X~|~Q\left(x\right)}~f\left(x\right)\right] =  \\ \left(1 - \left[\textrm{Prob}_{y \in X~|~R\left(y\right)}~f\left(y\right)\right]\right) \end{array} \end{array}\right) \end{array} \end{array}\right)\right]\right] \textrm{ if } \Omega \underset{{\scriptscriptstyle c}}{\in} \textrm{SampleSpaces}\right..

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions
0	Conditional	value: 1 condition: 2
1	Operation	operator: 29 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	26, 7
6	Lambda	parameter: 77 body: 9
7	Literal
8	ExprTuple	77
9	Operation	operator: 29 operand: 11
10	ExprTuple	11
11	Lambda	parameter: 68 body: 13
12	ExprTuple	68
13	Conditional	value: 14 condition: 15
14	Operation	operator: 29 operand: 18
15	Operation	operator: 73 operands: 17
16	ExprTuple	18
17	ExprTuple	68, 19
18	Lambda	parameters: 20 body: 21
19	Operation	operator: 22 operands: 23
20	ExprTuple	62, 75
21	Operation	operator: 24 operands: 25
22	Literal
23	ExprTuple	77, 26
24	Literal
25	ExprTuple	27, 28
26	Variable
27	Operation	operator: 29 operand: 32
28	Operation	operator: 43 operands: 31
29	Literal
30	ExprTuple	32
31	ExprTuple	33, 34
32	Lambda	parameter: 65 body: 35
33	Operation	operator: 55 operand: 40
34	Operation	operator: 37 operands: 38
35	Conditional	value: 39 condition: 58
36	ExprTuple	40
37	Literal
38	ExprTuple	41, 42
39	Operation	operator: 43 operands: 44
40	Lambda	parameter: 65 body: 45
41	Literal
42	Operation	operator: 46 operand: 51
43	Literal
44	ExprTuple	59, 48
45	Conditional	value: 49 condition: 50
46	Literal
47	ExprTuple	51
48	Operation	operator: 52 operand: 57
49	Operation	operator: 68 operand: 65
50	Operation	operator: 69 operands: 54
51	Operation	operator: 55 operand: 60
52	Literal
53	ExprTuple	57
54	ExprTuple	58, 59
55	Literal
56	ExprTuple	60
57	Operation	operator: 75 operand: 65
58	Operation	operator: 73 operands: 61
59	Operation	operator: 62 operand: 65
60	Lambda	parameter: 78 body: 64
61	ExprTuple	65, 77
62	Variable
63	ExprTuple	65
64	Conditional	value: 66 condition: 67
65	Variable
66	Operation	operator: 68 operand: 78
67	Operation	operator: 69 operands: 70
68	Variable
69	Literal
70	ExprTuple	71, 72
71	Operation	operator: 73 operands: 74
72	Operation	operator: 75 operand: 78
73	Literal
74	ExprTuple	78, 77
75	Variable
76	ExprTuple	78
77	Variable
78	Variable

Expression of type Conditional¶

from the theory of proveit.statistics¶

Expression of type Conditional ¶

from the theory of proveit.statistics ¶