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Expression of type Forall

from the theory of proveit.logic.sets.intersection

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 m, x
from proveit.core_expr_types import A_1_to_m
from proveit.logic import Forall, Intersect, NotInSet
from proveit.logic.sets import x_notin_some_A
from proveit.numbers import NaturalPos
In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [m], instance_expr = Forall(instance_param_or_params = [x, A_1_to_m], instance_expr = NotInSet(x, Intersect(A_1_to_m)), condition = x_notin_some_A), 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_{m \in \mathbb{N}^+}~\left[\forall_{x, A_{1}, A_{2}, \ldots, A_{m}~|~\left(x \notin A_{1}\right) \lor  \left(x \notin A_{2}\right) \lor  \ldots \lor  \left(x \notin A_{m}\right)}~\left(x \notin \left(A_{1} \cap  A_{2} \cap  \ldots \cap  A_{m}\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: 29
body: 4
3ExprTuple29
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 11
6Operationoperator: 9
operands: 10
7Literal
8ExprTuple11
9Literal
10ExprTuple29, 12
11Lambdaparameters: 13
body: 14
12Literal
13ExprTuple32, 25
14Conditionalvalue: 15
condition: 16
15Operationoperator: 30
operands: 17
16Operationoperator: 18
operands: 19
17ExprTuple32, 20
18Literal
19ExprTuple21
20Operationoperator: 22
operands: 23
21ExprRangelambda_map: 24
start_index: 28
end_index: 29
22Literal
23ExprTuple25
24Lambdaparameter: 36
body: 26
25ExprRangelambda_map: 27
start_index: 28
end_index: 29
26Operationoperator: 30
operands: 31
27Lambdaparameter: 36
body: 33
28Literal
29Variable
30Literal
31ExprTuple32, 33
32Variable
33IndexedVarvariable: 34
index: 36
34Variable
35ExprTuple36
36Variable