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Expression of type Forall

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.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import K, V, k
from proveit.core_expr_types import Q__b_1_to_j, b_1_to_j, f__b_1_to_j
from proveit.linear_algebra import ScalarMult, VecSpaces, VecSum
from proveit.linear_algebra.addition import vec_summation_b1toj_fQ
from proveit.logic import Equals, Forall, Implies, InSet
In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [V], instance_expr = Forall(instance_param_or_params = [k], instance_expr = Implies(InSet(vec_summation_b1toj_fQ, V), Equals(ScalarMult(k, vec_summation_b1toj_fQ), VecSum(index_or_indices = [b_1_to_j], summand = ScalarMult(k, f__b_1_to_j), condition = Q__b_1_to_j)).with_wrapping_at(1)).with_wrapping_at(2), 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_{V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left[\forall_{k \in K}~\left(\begin{array}{c} \begin{array}{l} \left(\left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right] \in V\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left(k \cdot \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right]\right) \\  = \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(k \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right)\right] \end{array} \end{array}\right) \end{array} \end{array}\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: 29
body: 4
3ExprTuple29
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 11
6Operationoperator: 9
operands: 10
7Literal
8ExprTuple11
9Literal
10ExprTuple29, 12
11Lambdaparameter: 45
body: 14
12Operationoperator: 15
operand: 24
13ExprTuple45
14Conditionalvalue: 17
condition: 18
15Literal
16ExprTuple24
17Operationoperator: 19
operands: 20
18Operationoperator: 25
operands: 21
19Literal
20ExprTuple22, 23
21ExprTuple45, 24
22Operationoperator: 25
operands: 26
23Operationoperator: 27
operands: 28
24Variable
25Literal
26ExprTuple34, 29
27Literal
28ExprTuple30, 31
29Variable
30Operationoperator: 42
operands: 32
31Operationoperator: 36
operand: 35
32ExprTuple45, 34
33ExprTuple35
34Operationoperator: 36
operand: 39
35Lambdaparameters: 49
body: 38
36Literal
37ExprTuple39
38Conditionalvalue: 40
condition: 44
39Lambdaparameters: 49
body: 41
40Operationoperator: 42
operands: 43
41Conditionalvalue: 46
condition: 44
42Literal
43ExprTuple45, 46
44Operationoperator: 47
operands: 49
45Variable
46Operationoperator: 48
operands: 49
47Variable
48Variable
49ExprTuple50
50ExprRangelambda_map: 51
start_index: 52
end_index: 53
51Lambdaparameter: 57
body: 54
52Literal
53Variable
54IndexedVarvariable: 55
index: 57
55Variable
56ExprTuple57
57Variable