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 S, fx, gx, x
from proveit.logic import Forall, Implies
from proveit.numbers import LessEq, Sum

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = LessEq(fx, gx), domain = S), LessEq(Sum(index_or_indices = sub_expr1, summand = fx, domain = S), Sum(index_or_indices = sub_expr1, summand = gx, domain = S)))

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_{x \in S}~\left(f\left(x\right) \leq g\left(x\right)\right)\right] \Rightarrow \left(\left[\sum_{x \in S}~f\left(x\right)\right] \leq \left[\sum_{x \in S}~g\left(x\right)\right]\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: 18 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameter: 30 body: 11
9	Operation	operator: 13 operand: 16
10	Operation	operator: 13 operand: 17
11	Conditional	value: 15 condition: 24
12	ExprTuple	16
13	Literal
14	ExprTuple	17
15	Operation	operator: 18 operands: 19
16	Lambda	parameter: 30 body: 20
17	Lambda	parameter: 30 body: 21
18	Literal
19	ExprTuple	22, 23
20	Conditional	value: 22 condition: 24
21	Conditional	value: 23 condition: 24
22	Operation	operator: 25 operand: 30
23	Operation	operator: 26 operand: 30
24	Operation	operator: 28 operands: 29
25	Variable
26	Variable
27	ExprTuple	30
28	Literal
29	ExprTuple	30, 31
30	Variable
31	Variable