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Expression of type Forall

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.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import P, n
from proveit.core_expr_types import P__y_1_to_n, y_1_to_n
from proveit.logic import Forall
from proveit.logic.booleans.quantification import general_exists_Px
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], instance_expr = Forall(instance_param_or_params = [y_1_to_n], instance_expr = general_exists_Px, condition = P__y_1_to_n)), 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}~\left[\forall_{y_{1}, y_{2}, \ldots, y_{n}~|~P\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~\left[\exists_{x_{1}, x_{2}, \ldots, x_{n}}~P\left(x_{1}, x_{2}, \ldots, 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: 14
operand: 2
1ExprTuple2
2Lambdaparameter: 34
body: 4
3ExprTuple34
4Conditionalvalue: 5
condition: 6
5Operationoperator: 14
operand: 10
6Operationoperator: 8
operands: 9
7ExprTuple10
8Literal
9ExprTuple34, 11
10Lambdaparameter: 27
body: 13
11Literal
12ExprTuple27
13Operationoperator: 14
operand: 16
14Literal
15ExprTuple16
16Lambdaparameters: 22
body: 17
17Conditionalvalue: 18
condition: 19
18Operationoperator: 20
operand: 23
19Operationoperator: 27
operands: 22
20Literal
21ExprTuple23
22ExprTuple24
23Lambdaparameters: 28
body: 25
24ExprRangelambda_map: 26
start_index: 33
end_index: 34
25Operationoperator: 27
operands: 28
26Lambdaparameter: 38
body: 29
27Variable
28ExprTuple30
29IndexedVarvariable: 31
index: 38
30ExprRangelambda_map: 32
start_index: 33
end_index: 34
31Variable
32Lambdaparameter: 38
body: 35
33Literal
34Variable
35IndexedVarvariable: 36
index: 38
36Variable
37ExprTuple38
38Variable