# from the theory of proveit.linear_algebra.vector_sets¶

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 K, V, n
from proveit.core_expr_types import x_1_to_n
from proveit.linear_algebra import Bases, Dim, VecSpaces
from proveit.logic import Equals, Forall, InSet, Set
from proveit.numbers import NaturalPos

In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [V], instance_expr = Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [x_1_to_n], instance_expr = Equals(Dim(V), n), condition = InSet(Set(x_1_to_n), Bases(V))), domain = NaturalPos), domain = VecSpaces(K))

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_{V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left[\forall_{n \in \mathbb{N}^+}~\left[\forall_{x_{1}, x_{2}, \ldots, x_{n}~|~\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \in \textrm{Bases}\left(V\right)}~\left(\textrm{Dim}\left(V\right) = n\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: 18
operand: 2
1ExprTuple2
2Lambdaparameter: 39
body: 3
3Conditionalvalue: 4
condition: 5
4Operationoperator: 18
operand: 9
5Operationoperator: 7
operands: 8
6ExprTuple9
7Literal
8ExprTuple39, 10
9Lambdaparameter: 42
body: 12
10Operationoperator: 13
operand: 17
11ExprTuple42
12Conditionalvalue: 15
condition: 16
13Literal
14ExprTuple17
15Operationoperator: 18
operand: 21
16Operationoperator: 28
operands: 20
17Variable
18Literal
19ExprTuple21
20ExprTuple42, 22
21Lambdaparameters: 35
body: 23
22Literal
23Conditionalvalue: 24
condition: 25
24Operationoperator: 26
operands: 27
25Operationoperator: 28
operands: 29
26Literal
27ExprTuple30, 42
28Literal
29ExprTuple31, 32
30Operationoperator: 33
operand: 39
31Operationoperator: 34
operands: 35
32Operationoperator: 36
operand: 39
33Literal
34Literal
35ExprTuple38
36Literal
37ExprTuple39
38ExprRangelambda_map: 40
start_index: 41
end_index: 42
39Variable
40Lambdaparameter: 46
body: 43
41Literal
42Variable
43IndexedVarvariable: 44
index: 46
44Variable
45ExprTuple46
46Variable