# from the theory of proveit.numbers.integration¶

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 e, l, t
from proveit.logic import Forall, Implies, InSet
from proveit.numbers import Exp, Integrate, IntervalCC, Real, frac, one, subtract, two

In [2]:
# build up the expression from sub-expressions
sub_expr1 = [l]
sub_expr2 = frac(one, Exp(l, two))
sub_expr3 = IntervalCC(e, subtract(Exp(two, subtract(t, one)), one))
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(sub_expr2, Real), domain = sub_expr3), InSet(Integrate(index_or_indices = sub_expr1, integrand = sub_expr2, domain = sub_expr3), 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_{l \in \left[e,2^{t - 1} - 1\right]}~\left(\frac{1}{l^{2}} \in \mathbb{R}\right)\right] \Rightarrow \left(\left(\int_{e}^{2^{t - 1} - 1} \frac{1}{l^{2}}\,dl\right) \in \mathbb{R}\right)

In [5]:
stored_expr.style_options()

namedescriptiondefaultcurrent valuerelated methods
operation'infix' or 'function' style formattinginfixinfix
wrap_positionsposition(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.()()('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justificationif any wrap positions are set, justify to the 'left', 'center', or 'right'centercenter('with_justification',)
directionDirection of the relation (normal or reversed)normalnormal('with_direction_reversed', 'is_reversed')
In [6]:
# display the expression information
stored_expr.expr_info()

core typesub-expressionsexpression
0Operationoperator: 1
operands: 2
1Literal
2ExprTuple3, 4
3Operationoperator: 5
operand: 8
4Operationoperator: 23
operands: 7
5Literal
6ExprTuple8
7ExprTuple9, 18
8Lambdaparameter: 30
body: 10
9Operationoperator: 11
operand: 14
10Conditionalvalue: 13
condition: 20
11Literal
12ExprTuple14
13Operationoperator: 23
operands: 15
14Lambdaparameter: 30
body: 17
15ExprTuple19, 18
16ExprTuple30
17Conditionalvalue: 19
condition: 20
18Literal
19Operationoperator: 21
operands: 22
20Operationoperator: 23
operands: 24
21Literal
22ExprTuple45, 25
23Literal
24ExprTuple30, 26
25Operationoperator: 35
operands: 27
26Operationoperator: 28
operands: 29
27ExprTuple30, 37
28Literal
29ExprTuple31, 32
30Variable
31Variable
32Operationoperator: 39
operands: 33
33ExprTuple34, 42
34Operationoperator: 35
operands: 36
35Literal
36ExprTuple37, 38
37Literal
38Operationoperator: 39
operands: 40
39Literal
40ExprTuple41, 42
41Variable
42Operationoperator: 43
operand: 45
43Literal
44ExprTuple45
45Literal