# from the theory of proveit.logic.booleans.quantification.existence¶

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 P, Q, n
from proveit.logic import Equals, Forall, Not
from proveit.logic.booleans.quantification import general_exists_Py_st_Qy, general_notexists_Px_st_Qx
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 = [P, Q], instance_expr = Equals(general_notexists_Px_st_Qx, Not(general_exists_Py_st_Qy)).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_{n \in \mathbb{N}^+}~\left[\forall_{P, Q}~\left(\begin{array}{c} \begin{array}{l} \left[\nexists_{x_{1}, x_{2}, \ldots, x_{n}~|~Q\left(x_{1}, x_{2}, \ldots, x_{n}\right)}~P\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right] =  \\ (\lnot \left[\exists_{y_{1}, y_{2}, \ldots, y_{n}~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~P\left(y_{1}, y_{2}, \ldots, y_{n}\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: 43
body: 4
3ExprTuple43
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 11
6Operationoperator: 9
operands: 10
7Literal
8ExprTuple11
9Literal
10ExprTuple43, 12
11Lambdaparameters: 13
body: 14
12Literal
13ExprTuple35, 36
14Operationoperator: 15
operands: 16
15Literal
16ExprTuple17, 18
17Operationoperator: 19
operand: 28
18Operationoperator: 20
operand: 22
19Literal
20Literal
21ExprTuple22
22Operationoperator: 24
operand: 28
23Conditionalvalue: 26
condition: 27
24Literal
25ExprTuple28
26Operationoperator: 35
operands: 29
27Operationoperator: 36
operands: 29
28Lambdaparameters: 37
body: 30
29ExprTuple31
30Conditionalvalue: 32
condition: 33
31ExprRangelambda_map: 34
start_index: 42
end_index: 43
32Operationoperator: 35
operands: 37
33Operationoperator: 36
operands: 37
34Lambdaparameter: 47
body: 38
35Variable
36Variable
37ExprTuple39
38IndexedVarvariable: 40
index: 47
39ExprRangelambda_map: 41
start_index: 42
end_index: 43
40Variable
41Lambdaparameter: 47
body: 44
42Literal
43Variable
44IndexedVarvariable: 45
index: 47
45Variable
46ExprTuple47
47Variable