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, InSet
from proveit.numbers import LessEq
from proveit.statistics import ProbOfAll

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Lambda(f, Conditional(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)), InSet(f, 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())

f \mapsto \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) \textrm{ if } f \in \left[X \rightarrow \Omega\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: 37 body: 2
1	ExprTuple	37
2	Conditional	value: 3 condition: 4
3	Operation	operator: 18 operand: 7
4	Operation	operator: 44 operands: 6
5	ExprTuple	7
6	ExprTuple	37, 8
7	Lambda	parameters: 9 body: 10
8	Operation	operator: 11 operands: 12
9	ExprTuple	43, 46
10	Operation	operator: 13 operands: 14
11	Literal
12	ExprTuple	48, 15
13	Literal
14	ExprTuple	16, 17
15	Variable
16	Operation	operator: 18 operand: 22
17	Operation	operator: 20 operands: 21
18	Literal
19	ExprTuple	22
20	Literal
21	ExprTuple	23, 24
22	Lambda	parameter: 49 body: 25
23	Operation	operator: 27 operand: 29
24	Operation	operator: 27 operand: 30
25	Conditional	value: 42 condition: 33
26	ExprTuple	29
27	Literal
28	ExprTuple	30
29	Lambda	parameter: 49 body: 31
30	Lambda	parameter: 49 body: 32
31	Conditional	value: 34 condition: 33
32	Conditional	value: 34 condition: 35
33	Operation	operator: 38 operands: 36
34	Operation	operator: 37 operand: 49
35	Operation	operator: 38 operands: 39
36	ExprTuple	41, 40
37	Variable
38	Literal
39	ExprTuple	41, 42
40	Operation	operator: 43 operand: 49
41	Operation	operator: 44 operands: 45
42	Operation	operator: 46 operand: 49
43	Variable
44	Literal
45	ExprTuple	49, 48
46	Variable
47	ExprTuple	49
48	Variable
49	Variable