Expression of type Conditional

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 Conditional, Omega, Q, Qx, R, Rx, X, f, fx, x
from proveit.logic import Forall, Functions, Implies, InSet
from proveit.numbers import LessEq
from proveit.statistics import ProbOfAll

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Conditional(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)), InSet(f, Functions(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())

\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) \textrm{ if } f \in \left[X \rightarrow \Omega\right]\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

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