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, Q, Qx, R, Rx, Ry, X, fx, fy, x, y
from proveit.logic import 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([Q, R], 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))

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())

\left(Q, R\right) \mapsto \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)

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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