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 = [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)))

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_{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]

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