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, X, f, fx, x
from proveit.logic import Forall, Functions, Implies
from proveit.numbers import LessEq
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 = 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())

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[\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]

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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