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Expression of type Forall

from the theory of proveit.numbers.absolute_value

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 ExprRange, IndexedVar, Variable, n, x
from proveit.core_expr_types import x_1_to_n
from proveit.logic import Equals, Forall
from proveit.numbers import Abs, Complex, Mult, one
In [2]:
# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [x_1_to_n], instance_expr = Equals(Abs(Mult(x_1_to_n)), Mult(ExprRange(sub_expr1, Abs(IndexedVar(x, sub_expr1)), one, n))), 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_{1}, x_{2}, \ldots, x_{n} \in \mathbb{C}}~\left(\left|x_{1} \cdot  x_{2} \cdot  \ldots \cdot  x_{n}\right| = \left(\left|x_{1}\right| \cdot  \left|x_{2}\right| \cdot  \ldots \cdot  \left|x_{n}\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: 1
operand: 3
1Literal
2ExprTuple3
3Lambdaparameters: 21
body: 4
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operands: 8
6Operationoperator: 9
operands: 10
7Literal
8ExprTuple11, 12
9Literal
10ExprTuple13
11Operationoperator: 31
operand: 17
12Operationoperator: 20
operands: 15
13ExprRangelambda_map: 16
start_index: 29
end_index: 30
14ExprTuple17
15ExprTuple18
16Lambdaparameter: 36
body: 19
17Operationoperator: 20
operands: 21
18ExprRangelambda_map: 22
start_index: 29
end_index: 30
19Operationoperator: 23
operands: 24
20Literal
21ExprTuple25
22Lambdaparameter: 36
body: 26
23Literal
24ExprTuple33, 27
25ExprRangelambda_map: 28
start_index: 29
end_index: 30
26Operationoperator: 31
operand: 33
27Literal
28Lambdaparameter: 36
body: 33
29Literal
30Variable
31Literal
32ExprTuple33
33IndexedVarvariable: 34
index: 36
34Variable
35ExprTuple36
36Variable