Expression of type Forall

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 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, RealNonNeg

# build up the expression from sub-expressions
sub_expr1 = [x_1_to_n]
expr = 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, RealNonNeg), domain = S), InSet(Integrate(index_or_indices = sub_expr1, integrand = f__x_1_to_n, domain = S), RealNonNeg)))

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())

\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}^{\ge 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}^{\ge 0}\right)\right)

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

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