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

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Lambda(X, 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)))

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())

X \mapsto \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]

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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