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

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

\forall_{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]}~\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}\right)

stored_expr.style_options()

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',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 15 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	45, 46
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operands: 8
6	Operation	operator: 9 operands: 10
7	Literal
8	ExprTuple	11, 12
9	Literal
10	ExprTuple	13, 14
11	Operation	operator: 15 operand: 21
12	Operation	operator: 17 operands: 18
13	Operation	operator: 35 operands: 19
14	Operation	operator: 35 operands: 20
15	Literal
16	ExprTuple	21
17	Literal
18	ExprTuple	32, 22
19	ExprTuple	45, 23
20	ExprTuple	46, 24
21	Lambda	parameter: 48 body: 25
22	Literal
23	Operation	operator: 27 operands: 26
24	Operation	operator: 27 operands: 28
25	Conditional	value: 29 condition: 30
26	ExprTuple	39, 31
27	Literal
28	ExprTuple	39, 32
29	Operation	operator: 33 operands: 34
30	Operation	operator: 35 operands: 36
31	Variable
32	Variable
33	Literal
34	ExprTuple	37, 38
35	Literal
36	ExprTuple	48, 39
37	Operation	operator: 41 operand: 43
38	Operation	operator: 41 operand: 44
39	Variable
40	ExprTuple	43
41	Literal
42	ExprTuple	44
43	Operation	operator: 45 operand: 48
44	Operation	operator: 46 operand: 48
45	Variable
46	Variable
47	ExprTuple	48
48	Variable