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

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 n
from proveit.core_expr_types import x_1_to_n
from proveit.logic import Disjoint, Distinct, Equals, Forall
from proveit.logic.sets import x_singletons_range
from proveit.numbers import NaturalPos

In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [x_1_to_n], instance_expr = Equals(Distinct(x_1_to_n), Disjoint(x_singletons_range))), 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_{n \in \mathbb{N}^+}~\left[\forall_{x_{1}, x_{2}, \ldots, x_{n}}~\left(\textrm{distinct}\left(x_{1}, x_{2}, \ldots, x_{n}\right) = \textrm{disjoint}\left(\left\{x_{1}\right\}, \left\{x_{2}\right\}, \ldots, \left\{x_{n}\right\}\right)\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: 27
body: 4
3ExprTuple27
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 11
6Operationoperator: 9
operands: 10
7Literal
8ExprTuple11
9Literal
10ExprTuple27, 12
11Lambdaparameters: 19
body: 13
12Literal
13Operationoperator: 14
operands: 15
14Literal
15ExprTuple16, 17
16Operationoperator: 18
operands: 19
17Operationoperator: 20
operands: 21
18Literal
19ExprTuple22
20Literal
21ExprTuple23
22ExprRangelambda_map: 24
start_index: 26
end_index: 27
23ExprRangelambda_map: 25
start_index: 26
end_index: 27
24Lambdaparameter: 34
body: 31
25Lambdaparameter: 34
body: 28
26Literal
27Variable
28Operationoperator: 29
operand: 31
29Literal
30ExprTuple31
31IndexedVarvariable: 32
index: 34
32Variable
33ExprTuple34
34Variable