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 Conditional, Lambda, P, S, n
from proveit.core_expr_types import P__x_1_to_n, x_1_to_n
from proveit.logic import Forall, Implies, InSet
from proveit.numbers import Integrate, NaturalPos, Real

# build up the expression from sub-expressions
sub_expr1 = [x_1_to_n]
expr = Lambda(n, Conditional(Forall(instance_param_or_params = [P, S], instance_expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(P__x_1_to_n, Real), domain = S), InSet(Integrate(index_or_indices = sub_expr1, integrand = P__x_1_to_n, domain = S), Real))), 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())

n \mapsto \left\{\forall_{P, S}~\left(\left[\forall_{x_{1}, x_{2}, \ldots, x_{n} \in S}~\left(P\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}~P\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()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 39 body: 2
1	ExprTuple	39
2	Conditional	value: 3 condition: 4
3	Operation	operator: 15 operand: 7
4	Operation	operator: 41 operands: 6
5	ExprTuple	7
6	ExprTuple	39, 8
7	Lambda	parameters: 9 body: 10
8	Literal
9	ExprTuple	30, 44
10	Operation	operator: 11 operands: 12
11	Literal
12	ExprTuple	13, 14
13	Operation	operator: 15 operand: 18
14	Operation	operator: 41 operands: 17
15	Literal
16	ExprTuple	18
17	ExprTuple	19, 27
18	Lambda	parameters: 31 body: 20
19	Operation	operator: 21 operand: 24
20	Conditional	value: 23 condition: 29
21	Literal
22	ExprTuple	24
23	Operation	operator: 41 operands: 25
24	Lambda	parameters: 31 body: 26
25	ExprTuple	28, 27
26	Conditional	value: 28 condition: 29
27	Literal
28	Operation	operator: 30 operands: 31
29	Operation	operator: 32 operands: 33
30	Variable
31	ExprTuple	34
32	Literal
33	ExprTuple	35
34	ExprRange	lambda_map: 36 start_index: 38 end_index: 39
35	ExprRange	lambda_map: 37 start_index: 38 end_index: 39
36	Lambda	parameter: 47 body: 43
37	Lambda	parameter: 47 body: 40
38	Literal
39	Variable
40	Operation	operator: 41 operands: 42
41	Literal
42	ExprTuple	43, 44
43	IndexedVar	variable: 45 index: 47
44	Variable
45	Variable
46	ExprTuple	47
47	Variable