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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 Equals, 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 = Equals(NotInSet(x, Set(y_1_to_n)), 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(\left(x \notin \left\{y_{1}, y_{2}, \ldots, y_{n}\right\}\right) = \left(\left(x \neq y_{1}\right) \land  \left(x \neq y_{2}\right) \land  \ldots \land  \left(x \neq y_{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: 32
body: 4
3ExprTuple32
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 11
6Operationoperator: 9
operands: 10
7Literal
8ExprTuple11
9Literal
10ExprTuple32, 12
11Lambdaparameters: 13
body: 14
12Literal
13ExprTuple35, 28
14Operationoperator: 15
operands: 16
15Literal
16ExprTuple17, 18
17Operationoperator: 19
operands: 20
18Operationoperator: 21
operands: 22
19Literal
20ExprTuple35, 23
21Literal
22ExprTuple24
23Operationoperator: 25
operands: 26
24ExprRangelambda_map: 27
start_index: 31
end_index: 32
25Literal
26ExprTuple28
27Lambdaparameter: 39
body: 29
28ExprRangelambda_map: 30
start_index: 31
end_index: 32
29Operationoperator: 33
operands: 34
30Lambdaparameter: 39
body: 36
31Literal
32Variable
33Literal
34ExprTuple35, 36
35Variable
36IndexedVarvariable: 37
index: 39
37Variable
38ExprTuple39
39Variable