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Expression of type Forall

from the theory of proveit.logic.sets.enumeration

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.
%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 Forall, NotInSet, Set
from proveit.logic.sets import x_notequals_all_y
from proveit.numbers import Natural
In [2]:
# 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 = NotInSet(x, Set(y_1_to_n)), condition = x_notequals_all_y), domain = Natural)
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, y_{1}, y_{2}, \ldots, y_{n}~|~\left(x \neq y_{1}\right), \left(x \neq y_{2}\right), \ldots, \left(x \neq y_{n}\right)}~\left(x \notin \left\{y_{1}, y_{2}, \ldots, y_{n}\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: 30
body: 4
3ExprTuple30
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 11
6Operationoperator: 9
operands: 10
7Literal
8ExprTuple11
9Literal
10ExprTuple30, 12
11Lambdaparameters: 13
body: 14
12Literal
13ExprTuple33, 26
14Conditionalvalue: 15
condition: 16
15Operationoperator: 17
operands: 18
16Operationoperator: 19
operands: 20
17Literal
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