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, InSet, Intersect
from proveit.logic.sets import x_in_every_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 = InSet(x, Intersect(A_1_to_m)), condition = x_in_every_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 \in A_{1}\right), \left(x \in A_{2}\right), \ldots, \left(x \in A_{m}\right)}~\left(x \in \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: 28 body: 4
3	ExprTuple	28
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 29 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	28, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	31, 24
13	Conditional	value: 14 condition: 15
14	Operation	operator: 29 operands: 16
15	Operation	operator: 17 operands: 18
16	ExprTuple	31, 19
17	Literal
18	ExprTuple	20
19	Operation	operator: 21 operands: 22
20	ExprRange	lambda_map: 23 start_index: 27 end_index: 28
21	Literal
22	ExprTuple	24
23	Lambda	parameter: 35 body: 25
24	ExprRange	lambda_map: 26 start_index: 27 end_index: 28
25	Operation	operator: 29 operands: 30
26	Lambda	parameter: 35 body: 32
27	Literal
28	Variable
29	Literal
30	ExprTuple	31, 32
31	Variable
32	IndexedVar	variable: 33 index: 35
33	Variable
34	ExprTuple	35
35	Variable