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 Conditional, Lambda, a, b, l
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Add, Integer, Mult, one

# build up the expression from sub-expressions
sub_expr1 = Add(l, one)
sub_expr2 = InSet(l, Integer)
sub_expr3 = Mult(a, b, sub_expr1)
sub_expr4 = Mult(sub_expr1, Mult(a, b))
expr = Implies(Forall(instance_param_or_params = [l], instance_expr = Equals(sub_expr3, sub_expr4), domain = Integer), Equals(Lambda(l, Conditional(sub_expr3, sub_expr2)), Lambda(l, Conditional(sub_expr4, sub_expr2))).with_wrapping_at(2)).with_wrapping_at(2)

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())

\begin{array}{c} \begin{array}{l} \left[\forall_{l \in \mathbb{Z}}~\left(\left(a \cdot b \cdot \left(l + 1\right)\right) = \left(\left(l + 1\right) \cdot \left(a \cdot b\right)\right)\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[l \mapsto \left\{a \cdot b \cdot \left(l + 1\right) \textrm{ if } l \in \mathbb{Z}\right..\right] =  \\ \left[l \mapsto \left\{\left(l + 1\right) \cdot \left(a \cdot b\right) \textrm{ if } l \in \mathbb{Z}\right..\right] \end{array} \end{array}\right) \end{array} \end{array}

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.	()	(2)	('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: 17 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameter: 32 body: 11
9	Lambda	parameter: 32 body: 12
10	Lambda	parameter: 32 body: 14
11	Conditional	value: 15 condition: 16
12	Conditional	value: 21 condition: 16
13	ExprTuple	32
14	Conditional	value: 22 condition: 16
15	Operation	operator: 17 operands: 18
16	Operation	operator: 19 operands: 20
17	Literal
18	ExprTuple	21, 22
19	Literal
20	ExprTuple	32, 23
21	Operation	operator: 30 operands: 24
22	Operation	operator: 30 operands: 25
23	Literal
24	ExprTuple	34, 35, 26
25	ExprTuple	26, 27
26	Operation	operator: 28 operands: 29
27	Operation	operator: 30 operands: 31
28	Literal
29	ExprTuple	32, 33
30	Literal
31	ExprTuple	34, 35
32	Variable
33	Literal
34	Variable
35	Variable