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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, j, 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
from proveit.numbers import NaturalPos
In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [j], instance_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)), domain = NaturalPos)
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_{j \in \mathbb{N}^+}~\left[\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]\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: 15
operand: 2
1ExprTuple2
2Lambdaparameter: 61
body: 4
3ExprTuple61
4Conditionalvalue: 5
condition: 6
5Operationoperator: 15
operand: 9
6Operationoperator: 33
operands: 8
7ExprTuple9
8ExprTuple61, 10
9Lambdaparameter: 37
body: 12
10Literal
11ExprTuple37
12Conditionalvalue: 13
condition: 14
13Operationoperator: 15
operand: 19
14Operationoperator: 17
operands: 18
15Literal
16ExprTuple19
17Literal
18ExprTuple37, 20
19Lambdaparameter: 53
body: 22
20Operationoperator: 23
operand: 32
21ExprTuple53
22Conditionalvalue: 25
condition: 26
23Literal
24ExprTuple32
25Operationoperator: 27
operands: 28
26Operationoperator: 33
operands: 29
27Literal
28ExprTuple30, 31
29ExprTuple53, 32
30Operationoperator: 33
operands: 34
31Operationoperator: 35
operands: 36
32Variable
33Literal
34ExprTuple42, 37
35Literal
36ExprTuple38, 39
37Variable
38Operationoperator: 50
operands: 40
39Operationoperator: 44
operand: 43
40ExprTuple53, 42
41ExprTuple43
42Operationoperator: 44
operand: 47
43Lambdaparameters: 57
body: 46
44Literal
45ExprTuple47
46Conditionalvalue: 48
condition: 52
47Lambdaparameters: 57
body: 49
48Operationoperator: 50
operands: 51
49Conditionalvalue: 54
condition: 52
50Literal
51ExprTuple53, 54
52Operationoperator: 55
operands: 57
53Variable
54Operationoperator: 56
operands: 57
55Variable
56Variable
57ExprTuple58
58ExprRangelambda_map: 59
start_index: 60
end_index: 61
59Lambdaparameter: 65
body: 62
60Literal
61Variable
62IndexedVarvariable: 63
index: 65
63Variable
64ExprTuple65
65Variable