Expression of type Implies

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 Qx, Rx, X, fx, x
from proveit.logic import Forall, Implies
from proveit.numbers import LessEq
from proveit.statistics import ProbOfAll

# build up the expression from sub-expressions
sub_expr1 = [x]
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)

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}{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}

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	(2)	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

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