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, Ry, X, f, fx, fy, x, y
from proveit.logic import Bijections, 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 = [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 = 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)), domain = Bijections(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())

\forall_{X}~\left[\forall_{f \in \left[X \xrightarrow[\text{onto}]{\text{1-to-1}} \Omega\right]}~\left[\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)\right]\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: 72 body: 4
3	ExprTuple	72
4	Operation	operator: 24 operand: 6
5	ExprTuple	6
6	Lambda	parameter: 63 body: 8
7	ExprTuple	63
8	Conditional	value: 9 condition: 10
9	Operation	operator: 24 operand: 13
10	Operation	operator: 68 operands: 12
11	ExprTuple	13
12	ExprTuple	63, 14
13	Lambda	parameters: 15 body: 16
14	Operation	operator: 17 operands: 18
15	ExprTuple	57, 70
16	Operation	operator: 19 operands: 20
17	Literal
18	ExprTuple	72, 21
19	Literal
20	ExprTuple	22, 23
21	Variable
22	Operation	operator: 24 operand: 27
23	Operation	operator: 38 operands: 26
24	Literal
25	ExprTuple	27
26	ExprTuple	28, 29
27	Lambda	parameter: 60 body: 30
28	Operation	operator: 50 operand: 35
29	Operation	operator: 32 operands: 33
30	Conditional	value: 34 condition: 53
31	ExprTuple	35
32	Literal
33	ExprTuple	36, 37
34	Operation	operator: 38 operands: 39
35	Lambda	parameter: 60 body: 40
36	Literal
37	Operation	operator: 41 operand: 46
38	Literal
39	ExprTuple	54, 43
40	Conditional	value: 44 condition: 45
41	Literal
42	ExprTuple	46
43	Operation	operator: 47 operand: 52
44	Operation	operator: 63 operand: 60
45	Operation	operator: 64 operands: 49
46	Operation	operator: 50 operand: 55
47	Literal
48	ExprTuple	52
49	ExprTuple	53, 54
50	Literal
51	ExprTuple	55
52	Operation	operator: 70 operand: 60
53	Operation	operator: 68 operands: 56
54	Operation	operator: 57 operand: 60
55	Lambda	parameter: 73 body: 59
56	ExprTuple	60, 72
57	Variable
58	ExprTuple	60
59	Conditional	value: 61 condition: 62
60	Variable
61	Operation	operator: 63 operand: 73
62	Operation	operator: 64 operands: 65
63	Variable
64	Literal
65	ExprTuple	66, 67
66	Operation	operator: 68 operands: 69
67	Operation	operator: 70 operand: 73
68	Literal
69	ExprTuple	73, 72
70	Variable
71	ExprTuple	73
72	Variable
73	Variable