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, f, fa, fx, x
from proveit.logic import And, Forall, InSet, SubsetEq
from proveit.numbers import Add, Integer, Integrate, Interval, IntervalCC, LessEq, MonDecFuncs, Real, Sum

# build up the expression from sub-expressions
sub_expr1 = [x]
sub_expr2 = IntervalCC(a, b)
expr = Lambda(f, Conditional(Forall(instance_param_or_params = [a, b], instance_expr = 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))), domain = Integer, conditions = [LessEq(a, b), SubsetEq(sub_expr2, S)]), And(InSet(f, MonDecFuncs(S)), SubsetEq(S, Real))))

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())

f \mapsto \left\{\forall_{a, b \in \mathbb{Z}~|~a \leq b, \left[a,b\right] \subseteq S}~\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)\right) \textrm{ if } f \in \textrm{MonDecFuncs}\left(S\right) ,  S \subseteq \mathbb{R}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 55 body: 2
1	ExprTuple	55
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 19 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameters: 62 body: 11
9	Operation	operator: 57 operands: 12
10	Operation	operator: 36 operands: 13
11	Conditional	value: 14 condition: 15
12	ExprTuple	55, 16
13	ExprTuple	42, 17
14	Operation	operator: 35 operands: 18
15	Operation	operator: 19 operands: 20
16	Operation	operator: 21 operand: 42
17	Literal
18	ExprTuple	23, 24
19	Literal
20	ExprTuple	25, 26, 27, 28
21	Literal
22	ExprTuple	42
23	Operation	operator: 29 operand: 38
24	Operation	operator: 31 operands: 32
25	Operation	operator: 57 operands: 33
26	Operation	operator: 57 operands: 34
27	Operation	operator: 35 operands: 62
28	Operation	operator: 36 operands: 37
29	Literal
30	ExprTuple	38
31	Literal
32	ExprTuple	39, 40
33	ExprTuple	63, 41
34	ExprTuple	64, 41
35	Literal
36	Literal
37	ExprTuple	60, 42
38	Lambda	parameter: 59 body: 43
39	Operation	operator: 55 operand: 63
40	Operation	operator: 45 operand: 48
41	Literal
42	Variable
43	Conditional	value: 52 condition: 47
44	ExprTuple	63
45	Literal
46	ExprTuple	48
47	Operation	operator: 57 operands: 49
48	Lambda	parameter: 59 body: 50
49	ExprTuple	59, 51
50	Conditional	value: 52 condition: 53
51	Operation	operator: 54 operands: 62
52	Operation	operator: 55 operand: 59
53	Operation	operator: 57 operands: 58
54	Literal
55	Variable
56	ExprTuple	59
57	Literal
58	ExprTuple	59, 60
59	Variable
60	Operation	operator: 61 operands: 62
61	Literal
62	ExprTuple	63, 64
63	Variable
64	Variable