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, Q, Qx, X, f, fx, x
from proveit.logic import Disjoint, Equals, Forall, Injections, SubsetEq, Union
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 = [A, B, C, X, Q], instance_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)), conditions = [SubsetEq(C, X), Equals(C, Union(A, B)), Disjoint(A, B)]).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())

\begin{array}{l}\forall_{A, B, C, X, Q~|~C \subseteq X, C = \left(A \cup B\right), \textrm{disjoint}\left(A, B\right)}~\\
\left[\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)\right]\end{array}

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	True	('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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	69, 70, 60, 39, 67
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 58 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	11, 12, 13
10	Lambda	parameter: 57 body: 15
11	Operation	operator: 16 operands: 17
12	Operation	operator: 23 operands: 18
13	Operation	operator: 19 operands: 27
14	ExprTuple	57
15	Conditional	value: 20 condition: 21
16	Literal
17	ExprTuple	60, 39
18	ExprTuple	60, 22
19	Literal
20	Operation	operator: 23 operands: 24
21	Operation	operator: 65 operands: 25
22	Operation	operator: 26 operands: 27
23	Literal
24	ExprTuple	28, 29
25	ExprTuple	57, 30
26	Literal
27	ExprTuple	69, 70
28	Operation	operator: 43 operand: 36
29	Operation	operator: 32 operands: 33
30	Operation	operator: 34 operands: 35
31	ExprTuple	36
32	Literal
33	ExprTuple	37, 38
34	Literal
35	ExprTuple	39, 40
36	Lambda	parameter: 71 body: 41
37	Operation	operator: 43 operand: 46
38	Operation	operator: 43 operand: 47
39	Variable
40	Variable
41	Conditional	value: 53 condition: 45
42	ExprTuple	46
43	Literal
44	ExprTuple	47
45	Operation	operator: 58 operands: 48
46	Lambda	parameter: 71 body: 49
47	Lambda	parameter: 71 body: 50
48	ExprTuple	51, 63
49	Conditional	value: 53 condition: 52
50	Conditional	value: 53 condition: 54
51	Operation	operator: 65 operands: 55
52	Operation	operator: 58 operands: 56
53	Operation	operator: 57 operand: 71
54	Operation	operator: 58 operands: 59
55	ExprTuple	71, 60
56	ExprTuple	61, 63
57	Variable
58	Literal
59	ExprTuple	62, 63
60	Variable
61	Operation	operator: 65 operands: 64
62	Operation	operator: 65 operands: 66
63	Operation	operator: 67 operand: 71
64	ExprTuple	71, 69
65	Literal
66	ExprTuple	71, 70
67	Variable
68	ExprTuple	71
69	Variable
70	Variable
71	Variable