Expression of type Lambda

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 Lambda, 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 = Lambda(X, 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())

X \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 70 body: 2
1	ExprTuple	70
2	Operation	operator: 22 operand: 4
3	ExprTuple	4
4	Lambda	parameter: 61 body: 6
5	ExprTuple	61
6	Conditional	value: 7 condition: 8
7	Operation	operator: 22 operand: 11
8	Operation	operator: 66 operands: 10
9	ExprTuple	11
10	ExprTuple	61, 12
11	Lambda	parameters: 13 body: 14
12	Operation	operator: 15 operands: 16
13	ExprTuple	55, 68
14	Operation	operator: 17 operands: 18
15	Literal
16	ExprTuple	70, 19
17	Literal
18	ExprTuple	20, 21
19	Variable
20	Operation	operator: 22 operand: 25
21	Operation	operator: 36 operands: 24
22	Literal
23	ExprTuple	25
24	ExprTuple	26, 27
25	Lambda	parameter: 58 body: 28
26	Operation	operator: 48 operand: 33
27	Operation	operator: 30 operands: 31
28	Conditional	value: 32 condition: 51
29	ExprTuple	33
30	Literal
31	ExprTuple	34, 35
32	Operation	operator: 36 operands: 37
33	Lambda	parameter: 58 body: 38
34	Literal
35	Operation	operator: 39 operand: 44
36	Literal
37	ExprTuple	52, 41
38	Conditional	value: 42 condition: 43
39	Literal
40	ExprTuple	44
41	Operation	operator: 45 operand: 50
42	Operation	operator: 61 operand: 58
43	Operation	operator: 62 operands: 47
44	Operation	operator: 48 operand: 53
45	Literal
46	ExprTuple	50
47	ExprTuple	51, 52
48	Literal
49	ExprTuple	53
50	Operation	operator: 68 operand: 58
51	Operation	operator: 66 operands: 54
52	Operation	operator: 55 operand: 58
53	Lambda	parameter: 71 body: 57
54	ExprTuple	58, 70
55	Variable
56	ExprTuple	58
57	Conditional	value: 59 condition: 60
58	Variable
59	Operation	operator: 61 operand: 71
60	Operation	operator: 62 operands: 63
61	Variable
62	Literal
63	ExprTuple	64, 65
64	Operation	operator: 66 operands: 67
65	Operation	operator: 68 operand: 71
66	Literal
67	ExprTuple	71, 70
68	Variable
69	ExprTuple	71
70	Variable
71	Variable