# 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_Px_st_Qx, general_forall__Py_not_T__st_Qy
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_exists_Px_st_Qx, Not(general_forall__Py_not_T__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[\exists_{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[\forall_{y_{1}, y_{2}, \ldots, y_{n}~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~\left(P\left(y_{1}, y_{2}, \ldots, y_{n}\right) \neq \top\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: 25
operand: 2
1ExprTuple2
2Lambdaparameter: 48
body: 4
3ExprTuple48
4Conditionalvalue: 5
condition: 6
5Operationoperator: 25
operand: 10
6Operationoperator: 8
operands: 9
7ExprTuple10
8Literal
9ExprTuple48, 11
10Lambdaparameters: 12
body: 13
11Literal
12ExprTuple43, 38
13Operationoperator: 14
operands: 15
14Literal
15ExprTuple16, 17
16Operationoperator: 18
operand: 22
17Operationoperator: 20
operand: 23
18Literal
19ExprTuple22
20Literal
21ExprTuple23
22Lambdaparameters: 30
body: 24
23Operationoperator: 25
operand: 29
24Conditionalvalue: 27
condition: 28
25Literal
26ExprTuple29
27Operationoperator: 43
operands: 30
28Operationoperator: 38
operands: 30
29Lambdaparameters: 44
body: 31
30ExprTuple32
31Conditionalvalue: 33
condition: 34
32ExprRangelambda_map: 35
start_index: 47
end_index: 48
33Operationoperator: 36
operands: 37
34Operationoperator: 38
operands: 44
35Lambdaparameter: 52
body: 39
36Literal
37ExprTuple40, 41
38Variable
39IndexedVarvariable: 42
index: 52
40Operationoperator: 43
operands: 44
41Literal
42Variable
43Variable
44ExprTuple45
45ExprRangelambda_map: 46
start_index: 47
end_index: 48
46Lambdaparameter: 52
body: 49
47Literal
48Variable
49IndexedVarvariable: 50
index: 52
50Variable
51ExprTuple52
52Variable