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, Q, Qx, X, f, fx, x
from proveit.logic import Equals, Forall, InClass, Injections
from proveit.numbers import Sum
from proveit.statistics import Prob, ProbOfAll, SampleSpaces

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Lambda(Omega, Conditional(Forall(instance_param_or_params = [X], instance_expr = Forall(instance_param_or_params = [f], instance_expr = Forall(instance_param_or_params = [Q], instance_expr = Equals(ProbOfAll(instance_param_or_params = sub_expr1, instance_element = fx, domain = X, condition = Qx), Sum(index_or_indices = sub_expr1, summand = Prob(fx), domain = X, condition = Qx)).with_wrapping_at(2)), domain = Injections(X, Omega))), 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}~\left[\forall_{f \in \left[X \xrightarrow[]{\text{1-to-1}} \Omega\right]}~\left[\forall_{Q}~\left(\begin{array}{c} \begin{array}{l} \left[\textrm{Prob}_{x \in X~|~Q\left(x\right)}~f\left(x\right)\right] =  \\ \left[\sum_{x \in X~|~Q\left(x\right)}~\textrm{Pr}\left(f\left(x\right)\right)\right] \end{array} \end{array}\right)\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: 29 body: 2
1	ExprTuple	29
2	Conditional	value: 3 condition: 4
3	Operation	operator: 18 operand: 8
4	Operation	operator: 6 operands: 7
5	ExprTuple	8
6	Literal
7	ExprTuple	29, 9
8	Lambda	parameter: 54 body: 11
9	Literal
10	ExprTuple	54
11	Operation	operator: 18 operand: 13
12	ExprTuple	13
13	Lambda	parameter: 49 body: 15
14	ExprTuple	49
15	Conditional	value: 16 condition: 17
16	Operation	operator: 18 operand: 21
17	Operation	operator: 50 operands: 20
18	Literal
19	ExprTuple	21
20	ExprTuple	49, 22
21	Lambda	parameter: 52 body: 24
22	Operation	operator: 25 operands: 26
23	ExprTuple	52
24	Operation	operator: 27 operands: 28
25	Literal
26	ExprTuple	54, 29
27	Literal
28	ExprTuple	30, 31
29	Variable
30	Operation	operator: 32 operand: 36
31	Operation	operator: 34 operand: 37
32	Literal
33	ExprTuple	36
34	Literal
35	ExprTuple	37
36	Lambda	parameter: 55 body: 38
37	Lambda	parameter: 55 body: 39
38	Conditional	value: 46 condition: 41
39	Conditional	value: 40 condition: 41
40	Operation	operator: 42 operand: 46
41	Operation	operator: 44 operands: 45
42	Literal
43	ExprTuple	46
44	Literal
45	ExprTuple	47, 48
46	Operation	operator: 49 operand: 55
47	Operation	operator: 50 operands: 51
48	Operation	operator: 52 operand: 55
49	Variable
50	Literal
51	ExprTuple	55, 54
52	Variable
53	ExprTuple	55
54	Variable
55	Variable