Expression of type Forall

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 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 = 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())

\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]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

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