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

# build up the expression from sub-expressions
sub_expr1 = [gy]
expr = 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())

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_{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]

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