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 Q, Qx, R, Rx, Ry, X, fx, fy, x, y
from proveit.logic import Equals, Forall, Implies, Not
from proveit.numbers import one, subtract
from proveit.statistics import ProbOfAll

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Forall(instance_param_or_params = [Q, R], instance_expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = Equals(Qx, Not(Rx)), domain = X), Equals(ProbOfAll(instance_param_or_params = sub_expr1, instance_element = fx, domain = X, condition = Qx), subtract(one, ProbOfAll(instance_param_or_params = [y], instance_element = fy, domain = X, condition = Ry))).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())

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