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 Lambda, S, a, b, f, fa, fx, x
from proveit.logic import Forall, 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(S, Forall(instance_param_or_params = [f], instance_expr = 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)]), domain = MonDecFuncs(S), condition = 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())

S \mapsto \left[\forall_{f \in \textrm{MonDecFuncs}\left(S\right)~|~S \subseteq \mathbb{R}}~\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)\right]\right]

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 45 body: 1
1	Operation	operator: 8 operand: 3
2	ExprTuple	3
3	Lambda	parameter: 58 body: 5
4	ExprTuple	58
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operand: 11
7	Operation	operator: 22 operands: 10
8	Literal
9	ExprTuple	11
10	ExprTuple	12, 13
11	Lambda	parameters: 65 body: 14
12	Operation	operator: 60 operands: 15
13	Operation	operator: 39 operands: 16
14	Conditional	value: 17 condition: 18
15	ExprTuple	58, 19
16	ExprTuple	45, 20
17	Operation	operator: 38 operands: 21
18	Operation	operator: 22 operands: 23
19	Operation	operator: 24 operand: 45
20	Literal
21	ExprTuple	26, 27
22	Literal
23	ExprTuple	28, 29, 30, 31
24	Literal
25	ExprTuple	45
26	Operation	operator: 32 operand: 41
27	Operation	operator: 34 operands: 35
28	Operation	operator: 60 operands: 36
29	Operation	operator: 60 operands: 37
30	Operation	operator: 38 operands: 65
31	Operation	operator: 39 operands: 40
32	Literal
33	ExprTuple	41
34	Literal
35	ExprTuple	42, 43
36	ExprTuple	66, 44
37	ExprTuple	67, 44
38	Literal
39	Literal
40	ExprTuple	63, 45
41	Lambda	parameter: 62 body: 46
42	Operation	operator: 58 operand: 66
43	Operation	operator: 48 operand: 51
44	Literal
45	Variable
46	Conditional	value: 55 condition: 50
47	ExprTuple	66
48	Literal
49	ExprTuple	51
50	Operation	operator: 60 operands: 52
51	Lambda	parameter: 62 body: 53
52	ExprTuple	62, 54
53	Conditional	value: 55 condition: 56
54	Operation	operator: 57 operands: 65
55	Operation	operator: 58 operand: 62
56	Operation	operator: 60 operands: 61
57	Literal
58	Variable
59	ExprTuple	62
60	Literal
61	ExprTuple	62, 63
62	Variable
63	Operation	operator: 64 operands: 65
64	Literal
65	ExprTuple	66, 67
66	Variable
67	Variable