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, R, Rx, X, f, fx, x
from proveit.logic import Forall, Functions, Implies
from proveit.numbers import LessEq
from proveit.statistics import ProbOfAll, SampleSpaces

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Forall(instance_param_or_params = [Omega], instance_expr = Forall(instance_param_or_params = [X], instance_expr = Forall(instance_param_or_params = [f], instance_expr = Forall(instance_param_or_params = [Q, R], instance_expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = Rx, domain = X, condition = Qx), LessEq(ProbOfAll(instance_param_or_params = sub_expr1, instance_element = fx, domain = X, condition = Qx), ProbOfAll(instance_param_or_params = sub_expr1, instance_element = fx, domain = X, condition = Rx)).with_wrapping_at(2)).with_wrapping_at(2)), domain = Functions(X, Omega)).with_wrapping()), domain = 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())

\forall_{\Omega \underset{{\scriptscriptstyle c}}{\in} \textrm{SampleSpaces}}~\left[\forall_{X}~\left[\begin{array}{l}\forall_{f \in \left[X \rightarrow \Omega\right]}~\\
\left[\forall_{Q, R}~\left(\begin{array}{c} \begin{array}{l} \left[\forall_{x \in X~|~Q\left(x\right)}~R\left(x\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[\textrm{Prob}_{x \in X~|~Q\left(x\right)}~f\left(x\right)\right] \leq  \\ \left[\textrm{Prob}_{x \in X~|~R\left(x\right)}~f\left(x\right)\right] \end{array} \end{array}\right) \end{array} \end{array}\right)\right]\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: 33 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 30 body: 4
3	ExprTuple	30
4	Conditional	value: 5 condition: 6
5	Operation	operator: 33 operand: 10
6	Operation	operator: 8 operands: 9
7	ExprTuple	10
8	Literal
9	ExprTuple	30, 11
10	Lambda	parameter: 63 body: 13
11	Literal
12	ExprTuple	63
13	Operation	operator: 33 operand: 15
14	ExprTuple	15
15	Lambda	parameter: 52 body: 17
16	ExprTuple	52
17	Conditional	value: 18 condition: 19
18	Operation	operator: 33 operand: 22
19	Operation	operator: 59 operands: 21
20	ExprTuple	22
21	ExprTuple	52, 23
22	Lambda	parameters: 24 body: 25
23	Operation	operator: 26 operands: 27
24	ExprTuple	58, 61
25	Operation	operator: 28 operands: 29
26	Literal
27	ExprTuple	63, 30
28	Literal
29	ExprTuple	31, 32
30	Variable
31	Operation	operator: 33 operand: 37
32	Operation	operator: 35 operands: 36
33	Literal
34	ExprTuple	37
35	Literal
36	ExprTuple	38, 39
37	Lambda	parameter: 64 body: 40
38	Operation	operator: 42 operand: 44
39	Operation	operator: 42 operand: 45
40	Conditional	value: 57 condition: 48
41	ExprTuple	44
42	Literal
43	ExprTuple	45
44	Lambda	parameter: 64 body: 46
45	Lambda	parameter: 64 body: 47
46	Conditional	value: 49 condition: 48
47	Conditional	value: 49 condition: 50
48	Operation	operator: 53 operands: 51
49	Operation	operator: 52 operand: 64
50	Operation	operator: 53 operands: 54
51	ExprTuple	56, 55
52	Variable
53	Literal
54	ExprTuple	56, 57
55	Operation	operator: 58 operand: 64
56	Operation	operator: 59 operands: 60
57	Operation	operator: 61 operand: 64
58	Variable
59	Literal
60	ExprTuple	64, 63
61	Variable
62	ExprTuple	64
63	Variable
64	Variable