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 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([X, Y], 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())

\left(X, Y\right) \mapsto \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]

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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