# from the theory of proveit.numbers.number_sets.real_numbers¶

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.
# import Expression classes needed to build the expression
from proveit import Conditional, a, b, x
from proveit.logic import And, Equals, Forall, InSet
from proveit.numbers import IntervalOO, Less, Real

In [2]:
# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [x], instance_expr = Equals(InSet(x, IntervalOO(a, b)), And(InSet(x, Real), And(Less(a, x), Less(x, b)).with_total_ordering_style()))), And(InSet(a, Real), InSet(b, Real)))

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())

\left\{\forall_{x}~\left(\left(x \in \left(a,b\right)\right) = \left(\left(x \in \mathbb{R}\right) \land \left(a < x < b\right)\right)\right) \textrm{ if } a \in \mathbb{R} ,  b \in \mathbb{R}\right..

In [5]:
stored_expr.style_options()

namedescriptiondefaultcurrent valuerelated methods
condition_delimiter'comma' or 'and'commacomma('with_comma_delimiter', 'with_conjunction_delimiter')
In [6]:
# display the expression information
stored_expr.expr_info()

core typesub-expressionsexpression
0Conditionalvalue: 1
condition: 2
1Operationoperator: 3
operand: 6
2Operationoperator: 26
operands: 5
3Literal
4ExprTuple6
5ExprTuple7, 8
6Lambdaparameter: 35
body: 10
7Operationoperator: 24
operands: 11
8Operationoperator: 24
operands: 12
9ExprTuple35
10Operationoperator: 13
operands: 14
11ExprTuple34, 28
12ExprTuple36, 28
13Literal
14ExprTuple15, 16
15Operationoperator: 24
operands: 17
16Operationoperator: 26
operands: 18
17ExprTuple35, 19
18ExprTuple20, 21
19Operationoperator: 22
operands: 23
20Operationoperator: 24
operands: 25
21Operationoperator: 26
operands: 27
22Literal
23ExprTuple34, 36
24Literal
25ExprTuple35, 28
26Literal
27ExprTuple29, 30
28Literal
29Operationoperator: 32
operands: 31
30Operationoperator: 32
operands: 33
31ExprTuple34, 35
32Literal
33ExprTuple35, 36
34Variable
35Variable
36Variable