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 A, B, Conditional, Function, Omega, Q, Qx, f, fx, g, gy, x, y
from proveit.logic import Bijections, Equals, Forall, InClass, Injections
from proveit.statistics import ProbOfAll, SampleSpaces

# build up the expression from sub-expressions
sub_expr1 = [gy]
expr = Conditional(Forall(instance_param_or_params = [A, B, Q], instance_expr = Forall(instance_param_or_params = [f, g], instance_expr = Equals(ProbOfAll(instance_param_or_params = [x], instance_element = fx, domain = A, condition = Qx), ProbOfAll(instance_param_or_params = [y], instance_element = Function(f, sub_expr1), domain = B, condition = Function(Q, sub_expr1))), domains = [Injections(A, Omega), Bijections(B, A)]).with_wrapping()), 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_{A, B, Q}~\left[\begin{array}{l}\forall_{f \in \left[A \xrightarrow[]{\text{1-to-1}} \Omega\right], g \in \left[B \xrightarrow[\text{onto}]{\text{1-to-1}} A\right]}~\\
\left(\left[\textrm{Prob}_{x \in A~|~Q\left(x\right)}~f\left(x\right)\right] = \left[\textrm{Prob}_{y \in B~|~Q\left(g\left(y\right)\right)}~f\left(g\left(y\right)\right)\right]\right)\end{array}\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: 10 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	43, 7
6	Lambda	parameters: 8 body: 9
7	Literal
8	ExprTuple	58, 60, 56
9	Operation	operator: 10 operand: 12
10	Literal
11	ExprTuple	12
12	Lambda	parameters: 13 body: 14
13	ExprTuple	45, 62
14	Conditional	value: 15 condition: 16
15	Operation	operator: 17 operands: 18
16	Operation	operator: 46 operands: 19
17	Literal
18	ExprTuple	20, 21
19	ExprTuple	22, 23
20	Operation	operator: 25 operand: 29
21	Operation	operator: 25 operand: 30
22	Operation	operator: 54 operands: 27
23	Operation	operator: 54 operands: 28
24	ExprTuple	29
25	Literal
26	ExprTuple	30
27	ExprTuple	45, 31
28	ExprTuple	62, 32
29	Lambda	parameter: 59 body: 33
30	Lambda	parameter: 64 body: 34
31	Operation	operator: 35 operands: 36
32	Operation	operator: 37 operands: 38
33	Conditional	value: 39 condition: 40
34	Conditional	value: 41 condition: 42
35	Literal
36	ExprTuple	58, 43
37	Literal
38	ExprTuple	60, 58
39	Operation	operator: 45 operand: 59
40	Operation	operator: 46 operands: 44
41	Operation	operator: 45 operand: 61
42	Operation	operator: 46 operands: 47
43	Variable
44	ExprTuple	48, 49
45	Variable
46	Literal
47	ExprTuple	50, 51
48	Operation	operator: 54 operands: 52
49	Operation	operator: 56 operand: 59
50	Operation	operator: 54 operands: 55
51	Operation	operator: 56 operand: 61
52	ExprTuple	59, 58
53	ExprTuple	59
54	Literal
55	ExprTuple	64, 60
56	Variable
57	ExprTuple	61
58	Variable
59	Variable
60	Variable
61	Operation	operator: 62 operand: 64
62	Variable
63	ExprTuple	64
64	Variable

Expression of type Conditional¶

from the theory of proveit.statistics¶

Expression of type Conditional ¶

from the theory of proveit.statistics ¶