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, X, f, fx, x
from proveit.logic import Equals, Forall, Functions, SetOfAll
from proveit.statistics import Prob, 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], instance_expr = Equals(ProbOfAll(instance_param_or_params = sub_expr1, instance_element = fx, domain = X, condition = Qx), Prob(SetOfAll(instance_param_or_params = sub_expr1, instance_element = fx, domain = X, condition = Qx))).with_wrapping_at(2)), domain = Functions(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 \rightarrow \Omega\right]}~\left[\forall_{Q}~\left(\begin{array}{c} \begin{array}{l} \left[\textrm{Prob}_{x \in X~|~Q\left(x\right)}~f\left(x\right)\right] =  \\ \textrm{Pr}\left(\left\{f\left(x\right)~|~Q\left(x\right)\right\}_{x \in X}\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: 42 body: 2
1	ExprTuple	42
2	Operation	operator: 9 operand: 4
3	ExprTuple	4
4	Lambda	parameter: 33 body: 6
5	ExprTuple	33
6	Conditional	value: 7 condition: 8
7	Operation	operator: 9 operand: 12
8	Operation	operator: 38 operands: 11
9	Literal
10	ExprTuple	12
11	ExprTuple	33, 13
12	Lambda	parameter: 40 body: 15
13	Operation	operator: 16 operands: 17
14	ExprTuple	40
15	Operation	operator: 18 operands: 19
16	Literal
17	ExprTuple	42, 20
18	Literal
19	ExprTuple	21, 22
20	Variable
21	Operation	operator: 23 operand: 29
22	Operation	operator: 24 operand: 26
23	Literal
24	Literal
25	ExprTuple	26
26	Operation	operator: 27 operand: 29
27	Literal
28	ExprTuple	29
29	Lambda	parameter: 43 body: 30
30	Conditional	value: 31 condition: 32
31	Operation	operator: 33 operand: 43
32	Operation	operator: 34 operands: 35
33	Variable
34	Literal
35	ExprTuple	36, 37
36	Operation	operator: 38 operands: 39
37	Operation	operator: 40 operand: 43
38	Literal
39	ExprTuple	43, 42
40	Variable
41	ExprTuple	43
42	Variable
43	Variable