Expression of type Implies

from the theory of proveit.numbers.summation

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 a, b, j
from proveit.logic import Forall, Implies, InSet
from proveit.numbers import Interval, Natural, Sum

# build up the expression from sub-expressions
sub_expr1 = [j]
sub_expr2 = Interval(a, b)
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(j, Natural), domain = sub_expr2), InSet(Sum(index_or_indices = sub_expr1, summand = j, domain = sub_expr2), Natural))

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_{j \in \{a~\ldotp \ldotp~b\}}~\left(j \in \mathbb{N}\right)\right] \Rightarrow \left(\left(\sum_{j = a}^{b} j\right) \in \mathbb{N}\right)

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	()	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 20 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 18
8	Lambda	parameter: 22 body: 10
9	Operation	operator: 11 operand: 14
10	Conditional	value: 13 condition: 19
11	Literal
12	ExprTuple	14
13	Operation	operator: 20 operands: 15
14	Lambda	parameter: 22 body: 17
15	ExprTuple	22, 18
16	ExprTuple	22
17	Conditional	value: 22 condition: 19
18	Literal
19	Operation	operator: 20 operands: 21
20	Literal
21	ExprTuple	22, 23
22	Variable
23	Operation	operator: 24 operands: 25
24	Literal
25	ExprTuple	26, 27
26	Variable
27	Variable