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 = [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_{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]

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: 20 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 59 body: 4
3	ExprTuple	59
4	Conditional	value: 5 condition: 6
5	Operation	operator: 20 operand: 9
6	Operation	operator: 64 operands: 8
7	ExprTuple	9
8	ExprTuple	59, 10
9	Lambda	parameters: 11 body: 12
10	Operation	operator: 13 operands: 14
11	ExprTuple	53, 66
12	Operation	operator: 15 operands: 16
13	Literal
14	ExprTuple	68, 17
15	Literal
16	ExprTuple	18, 19
17	Variable
18	Operation	operator: 20 operand: 23
19	Operation	operator: 34 operands: 22
20	Literal
21	ExprTuple	23
22	ExprTuple	24, 25
23	Lambda	parameter: 56 body: 26
24	Operation	operator: 46 operand: 31
25	Operation	operator: 28 operands: 29
26	Conditional	value: 30 condition: 49
27	ExprTuple	31
28	Literal
29	ExprTuple	32, 33
30	Operation	operator: 34 operands: 35
31	Lambda	parameter: 56 body: 36
32	Literal
33	Operation	operator: 37 operand: 42
34	Literal
35	ExprTuple	50, 39
36	Conditional	value: 40 condition: 41
37	Literal
38	ExprTuple	42
39	Operation	operator: 43 operand: 48
40	Operation	operator: 59 operand: 56
41	Operation	operator: 60 operands: 45
42	Operation	operator: 46 operand: 51
43	Literal
44	ExprTuple	48
45	ExprTuple	49, 50
46	Literal
47	ExprTuple	51
48	Operation	operator: 66 operand: 56
49	Operation	operator: 64 operands: 52
50	Operation	operator: 53 operand: 56
51	Lambda	parameter: 69 body: 55
52	ExprTuple	56, 68
53	Variable
54	ExprTuple	56
55	Conditional	value: 57 condition: 58
56	Variable
57	Operation	operator: 59 operand: 69
58	Operation	operator: 60 operands: 61
59	Variable
60	Literal
61	ExprTuple	62, 63
62	Operation	operator: 64 operands: 65
63	Operation	operator: 66 operand: 69
64	Literal
65	ExprTuple	69, 68
66	Variable
67	ExprTuple	69
68	Variable
69	Variable