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Expression of type Forall

from the theory of proveit.numbers.summation

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 fi, i, j, k, x
from proveit.logic import Equals, Forall
from proveit.numbers import Complex, Integer, Interval, Mult, Sum
In [2]:
# build up the expression from sub-expressions
sub_expr1 = [i]
sub_expr2 = Interval(j, k)
expr = Forall(instance_param_or_params = [x], instance_expr = Forall(instance_param_or_params = [j, k], instance_expr = Equals(Sum(index_or_indices = sub_expr1, summand = Mult(x, fi), domain = sub_expr2), Mult(x, Sum(index_or_indices = sub_expr1, summand = fi, domain = sub_expr2))), domain = Integer), domain = Complex)
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_{x \in \mathbb{C}}~\left[\forall_{j, k \in \mathbb{Z}}~\left(\left(\sum_{i = j}^{k} \left(x \cdot f\left(i\right)\right)\right) = \left(x \cdot \left(\sum_{i = j}^{k} f\left(i\right)\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: 7
operand: 2
1ExprTuple2
2Lambdaparameter: 38
body: 4
3ExprTuple38
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 10
6Operationoperator: 43
operands: 9
7Literal
8ExprTuple10
9ExprTuple38, 11
10Lambdaparameters: 48
body: 12
11Literal
12Conditionalvalue: 13
condition: 14
13Operationoperator: 15
operands: 16
14Operationoperator: 17
operands: 18
15Literal
16ExprTuple19, 20
17Literal
18ExprTuple21, 22
19Operationoperator: 31
operand: 27
20Operationoperator: 35
operands: 24
21Operationoperator: 43
operands: 25
22Operationoperator: 43
operands: 26
23ExprTuple27
24ExprTuple38, 28
25ExprTuple49, 29
26ExprTuple50, 29
27Lambdaparameter: 45
body: 30
28Operationoperator: 31
operand: 34
29Literal
30Conditionalvalue: 33
condition: 40
31Literal
32ExprTuple34
33Operationoperator: 35
operands: 36
34Lambdaparameter: 45
body: 37
35Literal
36ExprTuple38, 39
37Conditionalvalue: 39
condition: 40
38Variable
39Operationoperator: 41
operand: 45
40Operationoperator: 43
operands: 44
41Variable
42ExprTuple45
43Literal
44ExprTuple45, 46
45Variable
46Operationoperator: 47
operands: 48
47Literal
48ExprTuple49, 50
49Variable
50Variable