Expression of type Conditional

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 Conditional, S, f, n
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, NaturalPos, RealNeg

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

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\{\forall_{f, S}~\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}^-\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}^-\right)\right) \textrm{ if } n \in \mathbb{N}^+\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

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