Expression of type Lambda

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, Lambda, 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, Implies, InSet
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]
sub_expr2 = Function(f, sub_expr1)
sub_expr3 = Function(Q, sub_expr1)
expr = Lambda([f, Q], Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(sub_expr2, Complex), condition = sub_expr3), 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, sub_expr2, c_1_to_k), condition = sub_expr3)).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())

\left(f, Q\right) \mapsto \left(\begin{array}{c} \begin{array}{l} \left[\forall_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(f\left(b_{1}, b_{2}, \ldots, b_{j}\right) \in \mathbb{C}\right)\right] \Rightarrow  \\ \left[\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}\right] \end{array} \end{array}\right)

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 1 body: 2
1	ExprTuple	54, 51
2	Operation	operator: 3 operands: 4
3	Literal
4	ExprTuple	5, 6
5	Operation	operator: 8 operand: 10
6	Operation	operator: 8 operand: 11
7	ExprTuple	10
8	Literal
9	ExprTuple	11
10	Lambda	parameters: 55 body: 12
11	Lambda	parameters: 13 body: 14
12	Conditional	value: 15 condition: 47
13	ExprTuple	48, 50
14	Conditional	value: 16 condition: 17
15	Operation	operator: 39 operands: 18
16	Operation	operator: 19 operands: 20
17	Operation	operator: 21 operands: 22
18	ExprTuple	49, 43
19	Literal
20	ExprTuple	23, 24
21	Literal
22	ExprTuple	25, 26
23	Operation	operator: 45 operands: 27
24	Operation	operator: 35 operand: 32
25	ExprRange	lambda_map: 29 start_index: 63 end_index: 53
26	ExprRange	lambda_map: 30 start_index: 63 end_index: 57
27	ExprTuple	48, 31, 50
28	ExprTuple	32
29	Lambda	parameter: 69 body: 33
30	Lambda	parameter: 69 body: 34
31	Operation	operator: 35 operand: 41
32	Lambda	parameters: 55 body: 37
33	Operation	operator: 39 operands: 38
34	Operation	operator: 39 operands: 40
35	Literal
36	ExprTuple	41
37	Conditional	value: 42 condition: 47
38	ExprTuple	58, 43
39	Literal
40	ExprTuple	60, 43
41	Lambda	parameters: 55 body: 44
42	Operation	operator: 45 operands: 46
43	Literal
44	Conditional	value: 49 condition: 47
45	Literal
46	ExprTuple	48, 49, 50
47	Operation	operator: 51 operands: 55
48	ExprRange	lambda_map: 52 start_index: 63 end_index: 53
49	Operation	operator: 54 operands: 55
50	ExprRange	lambda_map: 56 start_index: 63 end_index: 57
51	Variable
52	Lambda	parameter: 69 body: 58
53	Variable
54	Variable
55	ExprTuple	59
56	Lambda	parameter: 69 body: 60
57	Variable
58	IndexedVar	variable: 61 index: 69
59	ExprRange	lambda_map: 62 start_index: 63 end_index: 64
60	IndexedVar	variable: 65 index: 69
61	Variable
62	Lambda	parameter: 69 body: 66
63	Literal
64	Variable
65	Variable
66	IndexedVar	variable: 67 index: 69
67	Variable
68	ExprTuple	69
69	Variable