# from the theory of proveit.linear_algebra.scalar_multiplication¶

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, a, x
from proveit.linear_algebra import ScalarMult, VecSpaces
from proveit.logic import Forall, InSet

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 = [a], instance_expr = Forall(instance_param_or_params = [x], instance_expr = InSet(ScalarMult(a, x), V), domain = V), domain = K), 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_{a \in K}~\left[\forall_{x \in V}~\left(\left(a \cdot x\right) \in V\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: 25
body: 3
3Operationoperator: 21
operand: 5
4ExprTuple5
5Lambdaparameter: 34
body: 7
6ExprTuple34
7Conditionalvalue: 8
condition: 9
8Operationoperator: 21
operand: 13
9Operationoperator: 11
operands: 12
10ExprTuple13
11Literal
12ExprTuple34, 14
13Lambdaparameter: 37
body: 16
14Operationoperator: 17
operand: 25
15ExprTuple37
16Conditionalvalue: 19
condition: 20
17Literal
18ExprTuple25
19Operationoperator: 21
operand: 24
20Operationoperator: 31
operands: 23
21Literal
22ExprTuple24
23ExprTuple37, 25
24Lambdaparameter: 38
body: 27
25Variable
26ExprTuple38
27Conditionalvalue: 28
condition: 29
28Operationoperator: 31
operands: 30
29Operationoperator: 31
operands: 32
30ExprTuple33, 34
31Literal
32ExprTuple38, 34
33Operationoperator: 35
operands: 36
34Variable
35Literal
36ExprTuple37, 38
37Variable
38Variable