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 = [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())

\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}

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