# 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 = [K], instance_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_{K}~\left[\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]\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: 21
operand: 2
1ExprTuple2
2Lambdaparameter: 20
body: 3
3Operationoperator: 21
operand: 5
4ExprTuple5
5Lambdaparameter: 42
body: 6
6Conditionalvalue: 7
condition: 8
7Operationoperator: 21
operand: 12
8Operationoperator: 10
operands: 11
9ExprTuple12
10Literal
11ExprTuple42, 13
12Lambdaparameter: 45
body: 15
13Operationoperator: 16
operand: 20
14ExprTuple45
15Conditionalvalue: 18
condition: 19
16Literal
17ExprTuple20
18Operationoperator: 21
operand: 24
19Operationoperator: 31
operands: 23
20Variable
21Literal
22ExprTuple24
23ExprTuple45, 25
24Lambdaparameters: 38
body: 26
25Literal
26Conditionalvalue: 27
condition: 28
27Operationoperator: 29
operands: 30
28Operationoperator: 31
operands: 32
29Literal
30ExprTuple33, 45
31Literal
32ExprTuple34, 35
33Operationoperator: 36
operand: 42
34Operationoperator: 37
operands: 38
35Operationoperator: 39
operand: 42
36Literal
37Literal
38ExprTuple41
39Literal
40ExprTuple42
41ExprRangelambda_map: 43
start_index: 44
end_index: 45
42Variable
43Lambdaparameter: 49
body: 46
44Literal
45Variable
46IndexedVarvariable: 47
index: 49
47Variable
48ExprTuple49
49Variable