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, Conditional, Function, Lambda, Omega, Q, Qx, f, fx, g, gy, x, y
from proveit.logic import Bijections, Equals, Forall, InClass, Injections
from proveit.statistics import ProbOfAll, SampleSpaces

# build up the expression from sub-expressions
sub_expr1 = [gy]
expr = Lambda(Omega, Conditional(Forall(instance_param_or_params = [A, B, Q], instance_expr = Forall(instance_param_or_params = [f, g], instance_expr = Equals(ProbOfAll(instance_param_or_params = [x], instance_element = fx, domain = A, condition = Qx), ProbOfAll(instance_param_or_params = [y], instance_element = Function(f, sub_expr1), domain = B, condition = Function(Q, sub_expr1))), domains = [Injections(A, Omega), Bijections(B, A)]).with_wrapping()), InClass(Omega, SampleSpaces)))

# 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())

\Omega \mapsto \left\{\forall_{A, B, Q}~\left[\begin{array}{l}\forall_{f \in \left[A \xrightarrow[]{\text{1-to-1}} \Omega\right], g \in \left[B \xrightarrow[\text{onto}]{\text{1-to-1}} A\right]}~\\
\left(\left[\textrm{Prob}_{x \in A~|~Q\left(x\right)}~f\left(x\right)\right] = \left[\textrm{Prob}_{y \in B~|~Q\left(g\left(y\right)\right)}~f\left(g\left(y\right)\right)\right]\right)\end{array}\right] \textrm{ if } \Omega \underset{{\scriptscriptstyle c}}{\in} \textrm{SampleSpaces}\right..

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions
0	Lambda	parameter: 45 body: 2
1	ExprTuple	45
2	Conditional	value: 3 condition: 4
3	Operation	operator: 12 operand: 8
4	Operation	operator: 6 operands: 7
5	ExprTuple	8
6	Literal
7	ExprTuple	45, 9
8	Lambda	parameters: 10 body: 11
9	Literal
10	ExprTuple	60, 62, 58
11	Operation	operator: 12 operand: 14
12	Literal
13	ExprTuple	14
14	Lambda	parameters: 15 body: 16
15	ExprTuple	47, 64
16	Conditional	value: 17 condition: 18
17	Operation	operator: 19 operands: 20
18	Operation	operator: 48 operands: 21
19	Literal
20	ExprTuple	22, 23
21	ExprTuple	24, 25
22	Operation	operator: 27 operand: 31
23	Operation	operator: 27 operand: 32
24	Operation	operator: 56 operands: 29
25	Operation	operator: 56 operands: 30
26	ExprTuple	31
27	Literal
28	ExprTuple	32
29	ExprTuple	47, 33
30	ExprTuple	64, 34
31	Lambda	parameter: 61 body: 35
32	Lambda	parameter: 66 body: 36
33	Operation	operator: 37 operands: 38
34	Operation	operator: 39 operands: 40
35	Conditional	value: 41 condition: 42
36	Conditional	value: 43 condition: 44
37	Literal
38	ExprTuple	60, 45
39	Literal
40	ExprTuple	62, 60
41	Operation	operator: 47 operand: 61
42	Operation	operator: 48 operands: 46
43	Operation	operator: 47 operand: 63
44	Operation	operator: 48 operands: 49
45	Variable
46	ExprTuple	50, 51
47	Variable
48	Literal
49	ExprTuple	52, 53
50	Operation	operator: 56 operands: 54
51	Operation	operator: 58 operand: 61
52	Operation	operator: 56 operands: 57
53	Operation	operator: 58 operand: 63
54	ExprTuple	61, 60
55	ExprTuple	61
56	Literal
57	ExprTuple	66, 62
58	Variable
59	ExprTuple	63
60	Variable
61	Variable
62	Variable
63	Operation	operator: 64 operand: 66
64	Variable
65	ExprTuple	66
66	Variable

Expression of type Lambda¶

from the theory of proveit.statistics¶

Expression of type Lambda ¶

from the theory of proveit.statistics ¶