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, 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 = Lambda([f, g], 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(f, g\right) \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

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