Expression of type Forall

from the theory of proveit.logic.sets.intersection

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 m, x
from proveit.core_expr_types import A_1_to_m
from proveit.logic import Forall, Intersect, NotInSet
from proveit.logic.sets import x_notin_some_A
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [m], instance_expr = Forall(instance_param_or_params = [x, A_1_to_m], instance_expr = NotInSet(x, Intersect(A_1_to_m)), condition = x_notin_some_A), domain = NaturalPos)

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