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, k, l
from proveit.logic import Forall, Implies
from proveit.numbers import Less, Sum
from proveit.numbers.summation import ak, al, bk, bl

# build up the expression from sub-expressions
sub_expr1 = [l]
expr = Implies(Forall(instance_param_or_params = [k], instance_expr = Less(ak, bk), domain = S), Less(Sum(index_or_indices = sub_expr1, summand = al, domain = S), Sum(index_or_indices = sub_expr1, summand = bl, 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_{k \in S}~\left(a\left(k\right) < b\left(k\right)\right)\right] \Rightarrow \left(\left[\sum_{l \in S}~a\left(l\right)\right] < \left[\sum_{l \in S}~b\left(l\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: 19 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameter: 35 body: 11
9	Operation	operator: 13 operand: 17
10	Operation	operator: 13 operand: 18
11	Conditional	value: 15 condition: 16
12	ExprTuple	17
13	Literal
14	ExprTuple	18
15	Operation	operator: 19 operands: 20
16	Operation	operator: 33 operands: 21
17	Lambda	parameter: 36 body: 22
18	Lambda	parameter: 36 body: 23
19	Literal
20	ExprTuple	24, 25
21	ExprTuple	35, 37
22	Conditional	value: 26 condition: 28
23	Conditional	value: 27 condition: 28
24	Operation	operator: 30 operand: 35
25	Operation	operator: 31 operand: 35
26	Operation	operator: 30 operand: 36
27	Operation	operator: 31 operand: 36
28	Operation	operator: 33 operands: 34
29	ExprTuple	35
30	Variable
31	Variable
32	ExprTuple	36
33	Literal
34	ExprTuple	36, 37
35	Variable
36	Variable
37	Variable