Expression of type Lambda

from the theory of proveit.numbers.integration

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, f
from proveit.core_expr_types import f__x_1_to_n, x_1_to_n
from proveit.logic import Forall, Implies, InSet
from proveit.numbers import Integrate, RealNonPos

# build up the expression from sub-expressions
sub_expr1 = [x_1_to_n]
expr = Lambda([f, S], Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(f__x_1_to_n, RealNonPos), domain = S), InSet(Integrate(index_or_indices = sub_expr1, integrand = f__x_1_to_n, domain = S), RealNonPos)))

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(f, S\right) \mapsto \left(\left[\forall_{x_{1}, x_{2}, \ldots, x_{n} \in S}~\left(f\left(x_{1}, x_{2}, \ldots, x_{n}\right) \in \mathbb{R}^{\le 0}\right)\right] \Rightarrow \left(\left[\int_{x_{1}, x_{2}, \ldots, x_{n} \in S}~f\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right] \in \mathbb{R}^{\le 0}\right)\right)

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 1 body: 2
1	ExprTuple	22, 36
2	Operation	operator: 3 operands: 4
3	Literal
4	ExprTuple	5, 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 33 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	11, 19
10	Lambda	parameters: 23 body: 12
11	Operation	operator: 13 operand: 16
12	Conditional	value: 15 condition: 21
13	Literal
14	ExprTuple	16
15	Operation	operator: 33 operands: 17
16	Lambda	parameters: 23 body: 18
17	ExprTuple	20, 19
18	Conditional	value: 20 condition: 21
19	Literal
20	Operation	operator: 22 operands: 23
21	Operation	operator: 24 operands: 25
22	Variable
23	ExprTuple	26
24	Literal
25	ExprTuple	27
26	ExprRange	lambda_map: 28 start_index: 30 end_index: 31
27	ExprRange	lambda_map: 29 start_index: 30 end_index: 31
28	Lambda	parameter: 39 body: 35
29	Lambda	parameter: 39 body: 32
30	Literal
31	Variable
32	Operation	operator: 33 operands: 34
33	Literal
34	ExprTuple	35, 36
35	IndexedVar	variable: 37 index: 39
36	Variable
37	Variable
38	ExprTuple	39
39	Variable