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 Conditional, Lambda, Omega, Q, Qx, R, Rx, X, f, fx, x
from proveit.logic import Forall, Functions, Implies, InClass
from proveit.numbers import LessEq
from proveit.statistics import ProbOfAll, SampleSpaces

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Lambda(Omega, Conditional(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 = 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)), domain = Functions(X, Omega)).with_wrapping()), InClass(Omega, SampleSpaces)))

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())

\Omega \mapsto \left\{\forall_{X}~\left[\begin{array}{l}\forall_{f \in \left[X \rightarrow \Omega\right]}~\\
\left[\forall_{Q, R}~\left(\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}\right)\right]\end{array}\right] \textrm{ if } \Omega \underset{{\scriptscriptstyle c}}{\in} \textrm{SampleSpaces}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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