# 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 a, b, x
from proveit.logic import And, Equals, Forall, InSet
from proveit.numbers import IntervalCC, LessEq, Real

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

\forall_{a, b \in \mathbb{R}}~\left[\forall_{x}~\left(\left(x \in \left[a,b\right]\right) = \left(\left(x \in \mathbb{R}\right) \land \left(a \leq x \leq b\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: 6
operand: 2
1ExprTuple2
2Lambdaparameters: 26
body: 3
3Conditionalvalue: 4
condition: 5
4Operationoperator: 6
operand: 9
5Operationoperator: 29
operands: 8
6Literal
7ExprTuple9
8ExprTuple10, 11
9Lambdaparameter: 38
body: 13
10Operationoperator: 27
operands: 14
11Operationoperator: 27
operands: 15
12ExprTuple38
13Operationoperator: 16
operands: 17
14ExprTuple37, 31
15ExprTuple39, 31
16Literal
17ExprTuple18, 19
18Operationoperator: 27
operands: 20
19Operationoperator: 29
operands: 21
20ExprTuple38, 22
21ExprTuple23, 24
22Operationoperator: 25
operands: 26
23Operationoperator: 27
operands: 28
24Operationoperator: 29
operands: 30
25Literal
26ExprTuple37, 39
27Literal
28ExprTuple38, 31
29Literal
30ExprTuple32, 33
31Literal
32Operationoperator: 35
operands: 34
33Operationoperator: 35
operands: 36
34ExprTuple37, 38
35Literal
36ExprTuple38, 39
37Variable
38Variable
39Variable