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

# 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, 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())

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[\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]

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: 24 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 54 body: 4
3	ExprTuple	54
4	Operation	operator: 24 operand: 6
5	ExprTuple	6
6	Lambda	parameter: 43 body: 8
7	ExprTuple	43
8	Conditional	value: 9 condition: 10
9	Operation	operator: 24 operand: 13
10	Operation	operator: 50 operands: 12
11	ExprTuple	13
12	ExprTuple	43, 14
13	Lambda	parameters: 15 body: 16
14	Operation	operator: 17 operands: 18
15	ExprTuple	49, 52
16	Operation	operator: 19 operands: 20
17	Literal
18	ExprTuple	54, 21
19	Literal
20	ExprTuple	22, 23
21	Variable
22	Operation	operator: 24 operand: 28
23	Operation	operator: 26 operands: 27
24	Literal
25	ExprTuple	28
26	Literal
27	ExprTuple	29, 30
28	Lambda	parameter: 55 body: 31
29	Operation	operator: 33 operand: 35
30	Operation	operator: 33 operand: 36
31	Conditional	value: 48 condition: 39
32	ExprTuple	35
33	Literal
34	ExprTuple	36
35	Lambda	parameter: 55 body: 37
36	Lambda	parameter: 55 body: 38
37	Conditional	value: 40 condition: 39
38	Conditional	value: 40 condition: 41
39	Operation	operator: 44 operands: 42
40	Operation	operator: 43 operand: 55
41	Operation	operator: 44 operands: 45
42	ExprTuple	47, 46
43	Variable
44	Literal
45	ExprTuple	47, 48
46	Operation	operator: 49 operand: 55
47	Operation	operator: 50 operands: 51
48	Operation	operator: 52 operand: 55
49	Variable
50	Literal
51	ExprTuple	55, 54
52	Variable
53	ExprTuple	55
54	Variable
55	Variable