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 ExprRange, IndexedVar, Variable, a, c, l
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Add, Complex, Integer, Mult, Sum, one, zero

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = [l]
sub_expr3 = Add(l, one)
sub_expr4 = IndexedVar(c, one)
sub_expr5 = ExprRange(sub_expr1, IndexedVar(a, sub_expr1), one, zero)
expr = Implies(Forall(instance_param_or_params = sub_expr2, instance_expr = InSet(sub_expr3, Complex), domain = Integer), Forall(instance_param_or_params = [sub_expr5, sub_expr4], instance_expr = Equals(Mult(sub_expr5, Sum(index_or_indices = sub_expr2, summand = sub_expr3, domain = Integer), sub_expr4), Sum(index_or_indices = sub_expr2, summand = Mult(sub_expr5, sub_expr3, sub_expr4), domain = Integer)).with_wrapping_at(2), domain = Complex).with_wrapping()).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(l + 1\right) \in \mathbb{C}\right)\right] \Rightarrow  \\ \left[\begin{array}{l}\forall_{a_{1}, a_{2}, \ldots, a_{0}, c_{1} \in \mathbb{C}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{0} \cdot \left[\sum_{l \in \mathbb{Z}}~\left(l + 1\right)\right] \cdot c_{1}\right) =  \\ \left[\sum_{l \in \mathbb{Z}}~\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{0} \cdot \left(l + 1\right) \cdot c_{1}\right)\right] \end{array} \end{array}\right)\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: 6 operand: 8
4	Operation	operator: 6 operand: 9
5	ExprTuple	8
6	Literal
7	ExprTuple	9
8	Lambda	parameter: 57 body: 10
9	Lambda	parameters: 11 body: 12
10	Conditional	value: 13 condition: 43
11	ExprTuple	44, 46
12	Conditional	value: 14 condition: 15
13	Operation	operator: 47 operands: 16
14	Operation	operator: 17 operands: 18
15	Operation	operator: 19 operands: 20
16	ExprTuple	45, 38
17	Literal
18	ExprTuple	21, 22
19	Literal
20	ExprTuple	23, 24
21	Operation	operator: 41 operands: 25
22	Operation	operator: 32 operand: 30
23	ExprRange	lambda_map: 27 start_index: 58 end_index: 50
24	Operation	operator: 47 operands: 28
25	ExprTuple	44, 29, 46
26	ExprTuple	30
27	Lambda	parameter: 61 body: 31
28	ExprTuple	46, 38
29	Operation	operator: 32 operand: 36
30	Lambda	parameter: 57 body: 34
31	Operation	operator: 47 operands: 35
32	Literal
33	ExprTuple	36
34	Conditional	value: 37 condition: 43
35	ExprTuple	56, 38
36	Lambda	parameter: 57 body: 40
37	Operation	operator: 41 operands: 42
38	Literal
39	ExprTuple	57
40	Conditional	value: 45 condition: 43
41	Literal
42	ExprTuple	44, 45, 46
43	Operation	operator: 47 operands: 48
44	ExprRange	lambda_map: 49 start_index: 58 end_index: 50
45	Operation	operator: 51 operands: 52
46	IndexedVar	variable: 53 index: 58
47	Literal
48	ExprTuple	57, 55
49	Lambda	parameter: 61 body: 56
50	Literal
51	Literal
52	ExprTuple	57, 58
53	Variable
54	ExprTuple	58
55	Literal
56	IndexedVar	variable: 59 index: 61
57	Variable
58	Literal
59	Variable
60	ExprTuple	61
61	Variable