# from the theory of proveit.logic.sets.intersection¶

In [1]:
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.
# 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 Equals, Forall, InSet, Intersect
from proveit.logic.sets import x_in_every_A
from proveit.numbers import NaturalPos

In [2]:
# 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 = Equals(InSet(x, Intersect(A_1_to_m)), x_in_every_A).with_wrapping_at(2)), domain = NaturalPos)

expr:
In [3]:
# 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

In [4]:
# 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(\begin{array}{c} \begin{array}{l} \left(x \in \left(A_{1} \cap  A_{2} \cap  \ldots \cap  A_{m}\right)\right) =  \\ \left(\left(x \in A_{1}\right) \land  \left(x \in A_{2}\right) \land  \ldots \land  \left(x \in A_{m}\right)\right) \end{array} \end{array}\right)\right]

In [5]:
stored_expr.style_options()

namedescriptiondefaultcurrent valuerelated methods
with_wrappingIf 'True', wrap the Expression after the parametersNoneNone/False('with_wrapping',)
condition_wrappingWrap 'before' or 'after' the condition (or None).NoneNone/False('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_paramsIf 'True', wraps every two parameters AND wraps the Expression after the parametersNoneNone/False('with_params',)
justificationjustify to the 'left', 'center', or 'right' in the array cellscentercenter('with_justification',)
In [6]:
# display the expression information
stored_expr.expr_info()

core typesub-expressionsexpression
0Operationoperator: 7
operand: 2
1ExprTuple2
2Lambdaparameter: 30
body: 4
3ExprTuple30
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 10
6Operationoperator: 31
operands: 9
7Literal
8ExprTuple10
9ExprTuple30, 11
10Lambdaparameters: 12
body: 13
11Literal
12ExprTuple33, 26
13Operationoperator: 14
operands: 15
14Literal
15ExprTuple16, 17
16Operationoperator: 31
operands: 18
17Operationoperator: 19
operands: 20
18ExprTuple33, 21
19Literal
20ExprTuple22
21Operationoperator: 23
operands: 24
22ExprRangelambda_map: 25
start_index: 29
end_index: 30
23Literal
24ExprTuple26
25Lambdaparameter: 37
body: 27
26ExprRangelambda_map: 28
start_index: 29
end_index: 30
27Operationoperator: 31
operands: 32
28Lambdaparameter: 37
body: 34
29Literal
30Variable
31Literal
32ExprTuple33, 34
33Variable
34IndexedVarvariable: 35
index: 37
35Variable
36ExprTuple37
37Variable