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, InSet, Injections
from proveit.numbers import Sum
from proveit.statistics import Prob, ProbOfAll

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

f \mapsto \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) \textrm{ if } f \in \left[X \xrightarrow[]{\text{1-to-1}} \Omega\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: 36 body: 2
1	ExprTuple	36
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 37 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	36, 9
8	Lambda	parameter: 39 body: 11
9	Operation	operator: 12 operands: 13
10	ExprTuple	39
11	Operation	operator: 14 operands: 15
12	Literal
13	ExprTuple	41, 16
14	Literal
15	ExprTuple	17, 18
16	Variable
17	Operation	operator: 19 operand: 23
18	Operation	operator: 21 operand: 24
19	Literal
20	ExprTuple	23
21	Literal
22	ExprTuple	24
23	Lambda	parameter: 42 body: 25
24	Lambda	parameter: 42 body: 26
25	Conditional	value: 33 condition: 28
26	Conditional	value: 27 condition: 28
27	Operation	operator: 29 operand: 33
28	Operation	operator: 31 operands: 32
29	Literal
30	ExprTuple	33
31	Literal
32	ExprTuple	34, 35
33	Operation	operator: 36 operand: 42
34	Operation	operator: 37 operands: 38
35	Operation	operator: 39 operand: 42
36	Variable
37	Literal
38	ExprTuple	42, 41
39	Variable
40	ExprTuple	42
41	Variable
42	Variable