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 Bijections, Equals, Forall, Implies, InClass
from proveit.statistics import Prob, SampleSpaces

# build up the expression from sub-expressions
expr = Lambda(Omega, Conditional(Forall(instance_param_or_params = [X, Y], instance_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)])), InClass(Omega, SampleSpaces)))

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())

\Omega \mapsto \left\{\forall_{X, Y}~\left[\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)\right] \textrm{ if } \Omega \underset{{\scriptscriptstyle c}}{\in} \textrm{SampleSpaces}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 39 body: 2
1	ExprTuple	39
2	Conditional	value: 3 condition: 4
3	Operation	operator: 23 operand: 7
4	Operation	operator: 25 operands: 6
5	ExprTuple	7
6	ExprTuple	39, 30
7	Lambda	parameters: 36 body: 8
8	Operation	operator: 23 operand: 10
9	ExprTuple	10
10	Lambda	parameters: 11 body: 12
11	ExprTuple	53, 54
12	Conditional	value: 13 condition: 14
13	Operation	operator: 15 operands: 16
14	Operation	operator: 17 operands: 18
15	Literal
16	ExprTuple	19, 20
17	Literal
18	ExprTuple	21, 22
19	Operation	operator: 23 operand: 29
20	Operation	operator: 25 operands: 26
21	Operation	operator: 43 operands: 27
22	Operation	operator: 43 operands: 28
23	Literal
24	ExprTuple	29
25	Literal
26	ExprTuple	40, 30
27	ExprTuple	53, 31
28	ExprTuple	54, 32
29	Lambda	parameter: 56 body: 33
30	Literal
31	Operation	operator: 35 operands: 34
32	Operation	operator: 35 operands: 36
33	Conditional	value: 37 condition: 38
34	ExprTuple	47, 39
35	Literal
36	ExprTuple	47, 40
37	Operation	operator: 41 operands: 42
38	Operation	operator: 43 operands: 44
39	Variable
40	Variable
41	Literal
42	ExprTuple	45, 46
43	Literal
44	ExprTuple	56, 47
45	Operation	operator: 49 operand: 51
46	Operation	operator: 49 operand: 52
47	Variable
48	ExprTuple	51
49	Literal
50	ExprTuple	52
51	Operation	operator: 53 operand: 56
52	Operation	operator: 54 operand: 56
53	Variable
54	Variable
55	ExprTuple	56
56	Variable