Expression of type Implies

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 Qx, Rx, Ry, X, fx, fy, x, y
from proveit.logic import Equals, Forall, Implies, Not
from proveit.numbers import one, subtract
from proveit.statistics import ProbOfAll

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = Equals(Qx, Not(Rx)), domain = X), Equals(ProbOfAll(instance_param_or_params = sub_expr1, instance_element = fx, domain = X, condition = Qx), subtract(one, ProbOfAll(instance_param_or_params = [y], instance_element = fy, domain = X, condition = Ry))).with_wrapping_at(2)).with_wrapping_at(2)

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}{c} \begin{array}{l} \left[\forall_{x \in X}~\left(Q\left(x\right) = (\lnot R\left(x\right))\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[\textrm{Prob}_{x \in X~|~Q\left(x\right)}~f\left(x\right)\right] =  \\ \left(1 - \left[\textrm{Prob}_{y \in X~|~R\left(y\right)}~f\left(y\right)\right]\right) \end{array} \end{array}\right) \end{array} \end{array}

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	(2)	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

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