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, X, Y, f, fx, g, gx, x
from proveit.logic import And, Bijections, Equals, Forall, Implies, InClass, InSet
from proveit.statistics import Prob, SampleSpaces

# build up the expression from sub-expressions
expr = Conditional(Implies(Forall(instance_param_or_params = [x], instance_expr = Equals(Prob(fx), Prob(gx)), domain = X), InClass(Y, SampleSpaces)).with_wrapping_at(1), And(InSet(f, Bijections(X, Omega)), InSet(g, Bijections(X, Y))))

expr:

# 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\{\begin{array}{c} \begin{array}{l} \left[\forall_{x \in X}~\left(\textrm{Pr}\left(f\left(x\right)\right) = \textrm{Pr}\left(g\left(x\right)\right)\right)\right] \\  \Rightarrow \left(Y \underset{{\scriptscriptstyle c}}{\in} \textrm{SampleSpaces}\right) \end{array} \end{array} \textrm{ if } f \in \left[X \xrightarrow[\text{onto}]{\text{1-to-1}} \Omega\right] ,  g \in \left[X \xrightarrow[\text{onto}]{\text{1-to-1}} Y\right]\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operands: 4
2	Operation	operator: 5 operands: 6
3	Literal
4	ExprTuple	7, 8
5	Literal
6	ExprTuple	9, 10
7	Operation	operator: 11 operand: 17
8	Operation	operator: 13 operands: 14
9	Operation	operator: 31 operands: 15
10	Operation	operator: 31 operands: 16
11	Literal
12	ExprTuple	17
13	Literal
14	ExprTuple	28, 18
15	ExprTuple	41, 19
16	ExprTuple	42, 20
17	Lambda	parameter: 44 body: 21
18	Literal
19	Operation	operator: 23 operands: 22
20	Operation	operator: 23 operands: 24
21	Conditional	value: 25 condition: 26
22	ExprTuple	35, 27
23	Literal
24	ExprTuple	35, 28
25	Operation	operator: 29 operands: 30
26	Operation	operator: 31 operands: 32
27	Variable
28	Variable
29	Literal
30	ExprTuple	33, 34
31	Literal
32	ExprTuple	44, 35
33	Operation	operator: 37 operand: 39
34	Operation	operator: 37 operand: 40
35	Variable
36	ExprTuple	39
37	Literal
38	ExprTuple	40
39	Operation	operator: 41 operand: 44
40	Operation	operator: 42 operand: 44
41	Variable
42	Variable
43	ExprTuple	44
44	Variable