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 = [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())

\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}

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	True	('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: 20 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 39 body: 4
3	ExprTuple	39
4	Conditional	value: 5 condition: 6
5	Operation	operator: 20 operand: 9
6	Operation	operator: 46 operands: 8
7	ExprTuple	9
8	ExprTuple	39, 10
9	Lambda	parameters: 11 body: 12
10	Operation	operator: 13 operands: 14
11	ExprTuple	45, 48
12	Operation	operator: 15 operands: 16
13	Literal
14	ExprTuple	50, 17
15	Literal
16	ExprTuple	18, 19
17	Variable
18	Operation	operator: 20 operand: 24
19	Operation	operator: 22 operands: 23
20	Literal
21	ExprTuple	24
22	Literal
23	ExprTuple	25, 26
24	Lambda	parameter: 51 body: 27
25	Operation	operator: 29 operand: 31
26	Operation	operator: 29 operand: 32
27	Conditional	value: 44 condition: 35
28	ExprTuple	31
29	Literal
30	ExprTuple	32
31	Lambda	parameter: 51 body: 33
32	Lambda	parameter: 51 body: 34
33	Conditional	value: 36 condition: 35
34	Conditional	value: 36 condition: 37
35	Operation	operator: 40 operands: 38
36	Operation	operator: 39 operand: 51
37	Operation	operator: 40 operands: 41
38	ExprTuple	43, 42
39	Variable
40	Literal
41	ExprTuple	43, 44
42	Operation	operator: 45 operand: 51
43	Operation	operator: 46 operands: 47
44	Operation	operator: 48 operand: 51
45	Variable
46	Literal
47	ExprTuple	51, 50
48	Variable
49	ExprTuple	51
50	Variable
51	Variable