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, fx, x
from proveit.logic import And, InSet, SubsetEq
from proveit.numbers import Integer, Integrate, Interval, IntervalCC, LessEq, Sum, one, subtract

# build up the expression from sub-expressions
sub_expr1 = [x]
sub_expr2 = IntervalCC(subtract(a, one), b)
expr = Lambda([a, b], Conditional(LessEq(Sum(index_or_indices = sub_expr1, summand = fx, domain = Interval(a, b)), 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(\int_{a - 1}^{b} f\left(x\right)\,dx\right) \textrm{ if } a \in \mathbb{Z} ,  b \in \mathbb{Z} ,  a \leq b ,  \left[a - 1,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: 40 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 operand: 23
9	Operation	operator: 34 operands: 17
10	Operation	operator: 34 operands: 18
11	Operation	operator: 19 operands: 40
12	Operation	operator: 20 operands: 21
13	Literal
14	ExprTuple	22
15	Literal
16	ExprTuple	23
17	ExprTuple	47, 24
18	ExprTuple	44, 24
19	Literal
20	Literal
21	ExprTuple	38, 25
22	Lambda	parameter: 37 body: 26
23	Lambda	parameter: 37 body: 27
24	Literal
25	Variable
26	Conditional	value: 29 condition: 28
27	Conditional	value: 29 condition: 30
28	Operation	operator: 34 operands: 31
29	Operation	operator: 32 operand: 37
30	Operation	operator: 34 operands: 35
31	ExprTuple	37, 36
32	Variable
33	ExprTuple	37
34	Literal
35	ExprTuple	37, 38
36	Operation	operator: 39 operands: 40
37	Variable
38	Operation	operator: 41 operands: 42
39	Literal
40	ExprTuple	47, 44
41	Literal
42	ExprTuple	43, 44
43	Operation	operator: 45 operands: 46
44	Variable
45	Literal
46	ExprTuple	47, 48
47	Variable
48	Operation	operator: 49 operand: 51
49	Literal
50	ExprTuple	51
51	Literal