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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 Forall
from proveit.numbers import Abs, Add, Complex, LessEq, NaturalPos, one
In [2]:
# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [x_1_to_n], instance_expr = LessEq(Abs(Add(x_1_to_n)), Add(ExprRange(sub_expr1, Abs(IndexedVar(x, sub_expr1)), one, n))), domain = Complex), 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_{n \in \mathbb{N}^+}~\left[\forall_{x_{1}, x_{2}, \ldots, x_{n} \in \mathbb{C}}~\left(\left|x_{1} +  x_{2} +  \ldots +  x_{n}\right| \leq \left(\left|x_{1}\right| +  \left|x_{2}\right| +  \ldots +  \left|x_{n}\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: 31
operands: 9
7Literal
8ExprTuple10
9ExprTuple38, 11
10Lambdaparameters: 29
body: 12
11Literal
12Conditionalvalue: 13
condition: 14
13Operationoperator: 15
operands: 16
14Operationoperator: 17
operands: 18
15Literal
16ExprTuple19, 20
17Literal
18ExprTuple21
19Operationoperator: 39
operand: 25
20Operationoperator: 28
operands: 23
21ExprRangelambda_map: 24
start_index: 37
end_index: 38
22ExprTuple25
23ExprTuple26
24Lambdaparameter: 44
body: 27
25Operationoperator: 28
operands: 29
26ExprRangelambda_map: 30
start_index: 37
end_index: 38
27Operationoperator: 31
operands: 32
28Literal
29ExprTuple33
30Lambdaparameter: 44
body: 34
31Literal
32ExprTuple41, 35
33ExprRangelambda_map: 36
start_index: 37
end_index: 38
34Operationoperator: 39
operand: 41
35Literal
36Lambdaparameter: 44
body: 41
37Literal
38Variable
39Literal
40ExprTuple41
41IndexedVarvariable: 42
index: 44
42Variable
43ExprTuple44
44Variable