Expression of type Lambda ¶

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, Lambda, 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 = Lambda(Omega, 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())

\Omega \mapsto \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	Lambda	parameter: 28 body: 2
1	ExprTuple	28
2	Conditional	value: 3 condition: 4
3	Operation	operator: 31 operand: 8
4	Operation	operator: 6 operands: 7
5	ExprTuple	8
6	Literal
7	ExprTuple	28, 9
8	Lambda	parameter: 79 body: 11
9	Literal
10	ExprTuple	79
11	Operation	operator: 31 operand: 13
12	ExprTuple	13
13	Lambda	parameter: 70 body: 15
14	ExprTuple	70
15	Conditional	value: 16 condition: 17
16	Operation	operator: 31 operand: 20
17	Operation	operator: 75 operands: 19
18	ExprTuple	20
19	ExprTuple	70, 21
20	Lambda	parameters: 22 body: 23
21	Operation	operator: 24 operands: 25
22	ExprTuple	64, 77
23	Operation	operator: 26 operands: 27
24	Literal
25	ExprTuple	79, 28
26	Literal
27	ExprTuple	29, 30
28	Variable
29	Operation	operator: 31 operand: 34
30	Operation	operator: 45 operands: 33
31	Literal
32	ExprTuple	34
33	ExprTuple	35, 36
34	Lambda	parameter: 67 body: 37
35	Operation	operator: 57 operand: 42
36	Operation	operator: 39 operands: 40
37	Conditional	value: 41 condition: 60
38	ExprTuple	42
39	Literal
40	ExprTuple	43, 44
41	Operation	operator: 45 operands: 46
42	Lambda	parameter: 67 body: 47
43	Literal
44	Operation	operator: 48 operand: 53
45	Literal
46	ExprTuple	61, 50
47	Conditional	value: 51 condition: 52
48	Literal
49	ExprTuple	53
50	Operation	operator: 54 operand: 59
51	Operation	operator: 70 operand: 67
52	Operation	operator: 71 operands: 56
53	Operation	operator: 57 operand: 62
54	Literal
55	ExprTuple	59
56	ExprTuple	60, 61
57	Literal
58	ExprTuple	62
59	Operation	operator: 77 operand: 67
60	Operation	operator: 75 operands: 63
61	Operation	operator: 64 operand: 67
62	Lambda	parameter: 80 body: 66
63	ExprTuple	67, 79
64	Variable
65	ExprTuple	67
66	Conditional	value: 68 condition: 69
67	Variable
68	Operation	operator: 70 operand: 80
69	Operation	operator: 71 operands: 72
70	Variable
71	Literal
72	ExprTuple	73, 74
73	Operation	operator: 75 operands: 76
74	Operation	operator: 77 operand: 80
75	Literal
76	ExprTuple	80, 79
77	Variable
78	ExprTuple	80
79	Variable
80	Variable

Expression of type Lambda¶

from the theory of proveit.statistics¶

Expression of type Lambda ¶

from the theory of proveit.statistics ¶