Expression of type Lambda

from the theory of proveit.numbers.summation

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 Conditional, Lambda, S, a, b, fa, fx, x
from proveit.logic import And, InSet, SubsetEq
from proveit.numbers import Add, Integer, Integrate, Interval, IntervalCC, LessEq, Sum

# build up the expression from sub-expressions
sub_expr1 = [x]
sub_expr2 = IntervalCC(a, b)
expr = Lambda([a, b], Conditional(LessEq(Sum(index_or_indices = sub_expr1, summand = fx, domain = Interval(a, b)), Add(fa, Integrate(index_or_indices = sub_expr1, integrand = fx, domain = sub_expr2))), And(InSet(a, Integer), InSet(b, Integer), LessEq(a, b), SubsetEq(sub_expr2, S))))

expr:

# 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

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left(a, b\right) \mapsto \left\{\left(\sum_{x = a}^{b} f\left(x\right)\right) \leq \left(f\left(a\right) + \left(\int_{a}^{b} f\left(x\right)\,dx\right)\right) \textrm{ if } a \in \mathbb{Z} ,  b \in \mathbb{Z} ,  a \leq b ,  \left[a,b\right] \subseteq S\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 46 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 19 operands: 4
3	Operation	operator: 5 operands: 6
4	ExprTuple	7, 8
5	Literal
6	ExprTuple	9, 10, 11, 12
7	Operation	operator: 13 operand: 22
8	Operation	operator: 15 operands: 16
9	Operation	operator: 41 operands: 17
10	Operation	operator: 41 operands: 18
11	Operation	operator: 19 operands: 46
12	Operation	operator: 20 operands: 21
13	Literal
14	ExprTuple	22
15	Literal
16	ExprTuple	23, 24
17	ExprTuple	47, 25
18	ExprTuple	48, 25
19	Literal
20	Literal
21	ExprTuple	44, 26
22	Lambda	parameter: 43 body: 27
23	Operation	operator: 39 operand: 47
24	Operation	operator: 29 operand: 32
25	Literal
26	Variable
27	Conditional	value: 36 condition: 31
28	ExprTuple	47
29	Literal
30	ExprTuple	32
31	Operation	operator: 41 operands: 33
32	Lambda	parameter: 43 body: 34
33	ExprTuple	43, 35
34	Conditional	value: 36 condition: 37
35	Operation	operator: 38 operands: 46
36	Operation	operator: 39 operand: 43
37	Operation	operator: 41 operands: 42
38	Literal
39	Variable
40	ExprTuple	43
41	Literal
42	ExprTuple	43, 44
43	Variable
44	Operation	operator: 45 operands: 46
45	Literal
46	ExprTuple	47, 48
47	Variable
48	Variable