Expression of type Forall

from the theory of proveit.logic.sets.enumeration

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 n, x
from proveit.core_expr_types import y_1_to_n
from proveit.logic import Equals, Forall, InSet, Set
from proveit.logic.sets import x_equals_any_y
from proveit.numbers import Natural

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [x, y_1_to_n], instance_expr = Equals(InSet(x, Set(y_1_to_n)), x_equals_any_y).with_wrapping_at(2)), domain = Natural)

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_{n \in \mathbb{N}}~\left[\forall_{x, y_{1}, y_{2}, \ldots, y_{n}}~\left(\begin{array}{c} \begin{array}{l} \left(x \in \left\{y_{1}, y_{2}, \ldots, y_{n}\right\}\right) =  \\ \left(\left(x = y_{1}\right) \lor  \left(x = y_{2}\right) \lor  \ldots \lor  \left(x = y_{n}\right)\right) \end{array} \end{array}\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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 30 body: 4
3	ExprTuple	30
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 17 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	30, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	33, 26
13	Operation	operator: 31 operands: 14
14	ExprTuple	15, 16
15	Operation	operator: 17 operands: 18
16	Operation	operator: 19 operands: 20
17	Literal
18	ExprTuple	33, 21
19	Literal
20	ExprTuple	22
21	Operation	operator: 23 operands: 24
22	ExprRange	lambda_map: 25 start_index: 29 end_index: 30
23	Literal
24	ExprTuple	26
25	Lambda	parameter: 37 body: 27
26	ExprRange	lambda_map: 28 start_index: 29 end_index: 30
27	Operation	operator: 31 operands: 32
28	Lambda	parameter: 37 body: 34
29	Literal
30	Variable
31	Literal
32	ExprTuple	33, 34
33	Variable
34	IndexedVar	variable: 35 index: 37
35	Variable
36	ExprTuple	37
37	Variable