# 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 Conditional, 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, InClass, InSet, Set
from proveit.numbers import NaturalPos

In [2]:
# build up the expression from sub-expressions
expr = Conditional(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), InClass(V, 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())

\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] \textrm{ if } V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)\right..

In [5]:
stored_expr.style_options()

namedescriptiondefaultcurrent valuerelated methods
condition_delimiter'comma' or 'and'commacomma('with_comma_delimiter', 'with_conjunction_delimiter')
In [6]:
# display the expression information
stored_expr.expr_info()

core typesub-expressionsexpression
0Conditionalvalue: 1
condition: 2
1Operationoperator: 15
operand: 6
2Operationoperator: 4
operands: 5
3ExprTuple6
4Literal
5ExprTuple36, 7
6Lambdaparameter: 39
body: 9
7Operationoperator: 10
operand: 14
8ExprTuple39
9Conditionalvalue: 12
condition: 13
10Literal
11ExprTuple14
12Operationoperator: 15
operand: 18
13Operationoperator: 25
operands: 17
14Variable
15Literal
16ExprTuple18
17ExprTuple39, 19
18Lambdaparameters: 32
body: 20
19Literal
20Conditionalvalue: 21
condition: 22
21Operationoperator: 23
operands: 24
22Operationoperator: 25
operands: 26
23Literal
24ExprTuple27, 39
25Literal
26ExprTuple28, 29
27Operationoperator: 30
operand: 36
28Operationoperator: 31
operands: 32
29Operationoperator: 33
operand: 36
30Literal
31Literal
32ExprTuple35
33Literal
34ExprTuple36
35ExprRangelambda_map: 37
start_index: 38
end_index: 39
36Variable
37Lambdaparameter: 43
body: 40
38Literal
39Variable
40IndexedVarvariable: 41
index: 43
41Variable
42ExprTuple43
43Variable