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 A, B, C, Conditional, Lambda, Omega, Qx, X, f, fx, x
from proveit.logic import Equals, InSet, Injections
from proveit.numbers import Add
from proveit.statistics import ProbOfAll

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Lambda(f, Conditional(Equals(ProbOfAll(instance_param_or_params = sub_expr1, instance_element = fx, domain = C, condition = Qx), Add(ProbOfAll(instance_param_or_params = sub_expr1, instance_element = fx, domain = A, condition = Qx), ProbOfAll(instance_param_or_params = sub_expr1, instance_element = fx, domain = B, condition = Qx))), InSet(f, Injections(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\{\left[\textrm{Prob}_{x \in C~|~Q\left(x\right)}~f\left(x\right)\right] = \left(\left[\textrm{Prob}_{x \in A~|~Q\left(x\right)}~f\left(x\right)\right] + \left[\textrm{Prob}_{x \in B~|~Q\left(x\right)}~f\left(x\right)\right]\right) \textrm{ if } f \in \left[X \xrightarrow[]{\text{1-to-1}} \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: 5 operands: 6
4	Operation	operator: 45 operands: 7
5	Literal
6	ExprTuple	8, 9
7	ExprTuple	37, 10
8	Operation	operator: 23 operand: 16
9	Operation	operator: 12 operands: 13
10	Operation	operator: 14 operands: 15
11	ExprTuple	16
12	Literal
13	ExprTuple	17, 18
14	Literal
15	ExprTuple	19, 20
16	Lambda	parameter: 51 body: 21
17	Operation	operator: 23 operand: 26
18	Operation	operator: 23 operand: 27
19	Variable
20	Variable
21	Conditional	value: 33 condition: 25
22	ExprTuple	26
23	Literal
24	ExprTuple	27
25	Operation	operator: 38 operands: 28
26	Lambda	parameter: 51 body: 29
27	Lambda	parameter: 51 body: 30
28	ExprTuple	31, 43
29	Conditional	value: 33 condition: 32
30	Conditional	value: 33 condition: 34
31	Operation	operator: 45 operands: 35
32	Operation	operator: 38 operands: 36
33	Operation	operator: 37 operand: 51
34	Operation	operator: 38 operands: 39
35	ExprTuple	51, 40
36	ExprTuple	41, 43
37	Variable
38	Literal
39	ExprTuple	42, 43
40	Variable
41	Operation	operator: 45 operands: 44
42	Operation	operator: 45 operands: 46
43	Operation	operator: 47 operand: 51
44	ExprTuple	51, 49
45	Literal
46	ExprTuple	51, 50
47	Variable
48	ExprTuple	51
49	Variable
50	Variable
51	Variable