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, 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, RealNonNeg

# build up the expression from sub-expressions
sub_expr1 = [x_1_to_n]
expr = Forall(instance_param_or_params = [n], instance_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))), domain = 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())

\forall_{n \in \mathbb{N}^+}~\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}^{\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)\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: 17 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 41 body: 4
3	ExprTuple	41
4	Conditional	value: 5 condition: 6
5	Operation	operator: 17 operand: 9
6	Operation	operator: 43 operands: 8
7	ExprTuple	9
8	ExprTuple	41, 10
9	Lambda	parameters: 11 body: 12
10	Literal
11	ExprTuple	32, 46
12	Operation	operator: 13 operands: 14
13	Literal
14	ExprTuple	15, 16
15	Operation	operator: 17 operand: 20
16	Operation	operator: 43 operands: 19
17	Literal
18	ExprTuple	20
19	ExprTuple	21, 29
20	Lambda	parameters: 33 body: 22
21	Operation	operator: 23 operand: 26
22	Conditional	value: 25 condition: 31
23	Literal
24	ExprTuple	26
25	Operation	operator: 43 operands: 27
26	Lambda	parameters: 33 body: 28
27	ExprTuple	30, 29
28	Conditional	value: 30 condition: 31
29	Literal
30	Operation	operator: 32 operands: 33
31	Operation	operator: 34 operands: 35
32	Variable
33	ExprTuple	36
34	Literal
35	ExprTuple	37
36	ExprRange	lambda_map: 38 start_index: 40 end_index: 41
37	ExprRange	lambda_map: 39 start_index: 40 end_index: 41
38	Lambda	parameter: 49 body: 45
39	Lambda	parameter: 49 body: 42
40	Literal
41	Variable
42	Operation	operator: 43 operands: 44
43	Literal
44	ExprTuple	45, 46
45	IndexedVar	variable: 47 index: 49
46	Variable
47	Variable
48	ExprTuple	49
49	Variable