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Expression of type Forall

from the theory of proveit.logic.sets.unification

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 Equals, Forall, NotInSet, Union
from proveit.logic.sets import x_notin_all_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 = Equals(NotInSet(x, Union(A_1_to_m)), x_notin_all_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(\left(x \notin \left(A_{1} \cup  A_{2} \cup  \ldots \cup  A_{m}\right)\right) = \left(\left(x \notin A_{1}\right) \land  \left(x \notin A_{2}\right) \land  \ldots \land  \left(x \notin A_{m}\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: 7
operand: 2
1ExprTuple2
2Lambdaparameter: 31
body: 4
3ExprTuple31
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 11
6Operationoperator: 9
operands: 10
7Literal
8ExprTuple11
9Literal
10ExprTuple31, 12
11Lambdaparameters: 13
body: 14
12Literal
13ExprTuple34, 27
14Operationoperator: 15
operands: 16
15Literal
16ExprTuple17, 18
17Operationoperator: 32
operands: 19
18Operationoperator: 20
operands: 21
19ExprTuple34, 22
20Literal
21ExprTuple23
22Operationoperator: 24
operands: 25
23ExprRangelambda_map: 26
start_index: 30
end_index: 31
24Literal
25ExprTuple27
26Lambdaparameter: 38
body: 28
27ExprRangelambda_map: 29
start_index: 30
end_index: 31
28Operationoperator: 32
operands: 33
29Lambdaparameter: 38
body: 35
30Literal
31Variable
32Literal
33ExprTuple34, 35
34Variable
35IndexedVarvariable: 36
index: 38
36Variable
37ExprTuple38
38Variable