Expression of type Forall

from the theory of proveit.numbers.multiplication

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 Function, Q, f
from proveit.core_expr_types import a_1_to_i, b_1_to_j, c_1_to_k
from proveit.logic import Equals, Forall
from proveit.numbers import Complex, Mult, Sum
from proveit.numbers.summation import summation_b1toj_fQ

# build up the expression from sub-expressions
sub_expr1 = [b_1_to_j]
expr = Forall(instance_param_or_params = [a_1_to_i, c_1_to_k], instance_expr = Equals(Mult(a_1_to_i, summation_b1toj_fQ, c_1_to_k), Sum(index_or_indices = sub_expr1, summand = Mult(a_1_to_i, Function(f, sub_expr1), c_1_to_k), condition = Function(Q, sub_expr1))).with_wrapping_at(2), domain = Complex).with_wrapping()

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}{l}\forall_{a_{1}, a_{2}, \ldots, a_{i}, c_{1}, c_{2}, \ldots, c_{k} \in \mathbb{C}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right]\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) =  \\ \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right)\right] \end{array} \end{array}\right)\end{array}

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	True	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameters: 4 body: 5
4	ExprTuple	37, 39
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 10 operands: 11
8	Literal
9	ExprTuple	12, 13
10	Literal
11	ExprTuple	14, 15
12	Operation	operator: 34 operands: 16
13	Operation	operator: 24 operand: 21
14	ExprRange	lambda_map: 18 start_index: 52 end_index: 42
15	ExprRange	lambda_map: 19 start_index: 52 end_index: 46
16	ExprTuple	37, 20, 39
17	ExprTuple	21
18	Lambda	parameter: 58 body: 22
19	Lambda	parameter: 58 body: 23
20	Operation	operator: 24 operand: 30
21	Lambda	parameters: 44 body: 26
22	Operation	operator: 28 operands: 27
23	Operation	operator: 28 operands: 29
24	Literal
25	ExprTuple	30
26	Conditional	value: 31 condition: 36
27	ExprTuple	47, 32
28	Literal
29	ExprTuple	49, 32
30	Lambda	parameters: 44 body: 33
31	Operation	operator: 34 operands: 35
32	Literal
33	Conditional	value: 38 condition: 36
34	Literal
35	ExprTuple	37, 38, 39
36	Operation	operator: 40 operands: 44
37	ExprRange	lambda_map: 41 start_index: 52 end_index: 42
38	Operation	operator: 43 operands: 44
39	ExprRange	lambda_map: 45 start_index: 52 end_index: 46
40	Variable
41	Lambda	parameter: 58 body: 47
42	Variable
43	Variable
44	ExprTuple	48
45	Lambda	parameter: 58 body: 49
46	Variable
47	IndexedVar	variable: 50 index: 58
48	ExprRange	lambda_map: 51 start_index: 52 end_index: 53
49	IndexedVar	variable: 54 index: 58
50	Variable
51	Lambda	parameter: 58 body: 55
52	Literal
53	Variable
54	Variable
55	IndexedVar	variable: 56 index: 58
56	Variable
57	ExprTuple	58
58	Variable