Expression of type Forall

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, Omega, Qx, X, f, fx, x
from proveit.logic import Equals, Forall, Injections
from proveit.numbers import Add
from proveit.statistics import ProbOfAll

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Forall(instance_param_or_params = [f], instance_expr = 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))), domain = 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())

\forall_{f \in \left[X \xrightarrow[]{\text{1-to-1}} \Omega\right]}~\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)\right)

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

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