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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, Q, n
from proveit.core_expr_types import P__y_1_to_n, Q__y_1_to_n, y_1_to_n
from proveit.logic import Forall
from proveit.logic.booleans.quantification import general_exists_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 = Forall(instance_param_or_params = [y_1_to_n], instance_expr = general_exists_Px_st_Qx, conditions = [P__y_1_to_n, Q__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, Q}~\left[\forall_{y_{1}, y_{2}, \ldots, y_{n}~|~P\left(y_{1}, y_{2}, \ldots, y_{n}\right), Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~\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]\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: 40
body: 4
3ExprTuple40
4Conditionalvalue: 5
condition: 6
5Operationoperator: 14
operand: 10
6Operationoperator: 8
operands: 9
7ExprTuple10
8Literal
9ExprTuple40, 11
10Lambdaparameters: 12
body: 13
11Literal
12ExprTuple32, 33
13Operationoperator: 14
operand: 16
14Literal
15ExprTuple16
16Lambdaparameters: 28
body: 17
17Conditionalvalue: 18
condition: 19
18Operationoperator: 20
operand: 24
19Operationoperator: 22
operands: 23
20Literal
21ExprTuple24
22Literal
23ExprTuple25, 26
24Lambdaparameters: 34
body: 27
25Operationoperator: 32
operands: 28
26Operationoperator: 33
operands: 28
27Conditionalvalue: 29
condition: 30
28ExprTuple31
29Operationoperator: 32
operands: 34
30Operationoperator: 33
operands: 34
31ExprRangelambda_map: 35
start_index: 39
end_index: 40
32Variable
33Variable
34ExprTuple36
35Lambdaparameter: 45
body: 37
36ExprRangelambda_map: 38
start_index: 39
end_index: 40
37IndexedVarvariable: 41
index: 45
38Lambdaparameter: 45
body: 42
39Literal
40Variable
41Variable
42IndexedVarvariable: 43
index: 45
43Variable
44ExprTuple45
45Variable