Expression of type Conditional

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 Conditional, Function, Q, f, i, j, k
from proveit.core_expr_types import a_1_to_i, b_1_to_j, c_1_to_k
from proveit.logic import And, Equals, Forall, Implies, InSet
from proveit.numbers import Complex, Mult, Natural, NaturalPos, 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 = Conditional(Forall(instance_param_or_params = [f, Q], instance_expr = 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)), And(InSet(i, Natural), InSet(j, NaturalPos), InSet(k, Natural)))

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_{f, Q}~\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) \textrm{ if } i \in \mathbb{N} ,  j \in \mathbb{N}^+ ,  k \in \mathbb{N}\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 21 operand: 5
2	Operation	operator: 34 operands: 4
3	ExprTuple	5
4	ExprTuple	6, 7, 8
5	Lambda	parameters: 9 body: 10
6	Operation	operator: 52 operands: 11
7	Operation	operator: 52 operands: 12
8	Operation	operator: 52 operands: 13
9	ExprTuple	67, 64
10	Operation	operator: 14 operands: 15
11	ExprTuple	66, 17
12	ExprTuple	77, 16
13	ExprTuple	70, 17
14	Literal
15	ExprTuple	18, 19
16	Literal
17	Literal
18	Operation	operator: 21 operand: 23
19	Operation	operator: 21 operand: 24
20	ExprTuple	23
21	Literal
22	ExprTuple	24
23	Lambda	parameters: 68 body: 25
24	Lambda	parameters: 26 body: 27
25	Conditional	value: 28 condition: 60
26	ExprTuple	61, 63
27	Conditional	value: 29 condition: 30
28	Operation	operator: 52 operands: 31
29	Operation	operator: 32 operands: 33
30	Operation	operator: 34 operands: 35
31	ExprTuple	62, 56
32	Literal
33	ExprTuple	36, 37
34	Literal
35	ExprTuple	38, 39
36	Operation	operator: 58 operands: 40
37	Operation	operator: 48 operand: 45
38	ExprRange	lambda_map: 42 start_index: 76 end_index: 66
39	ExprRange	lambda_map: 43 start_index: 76 end_index: 70
40	ExprTuple	61, 44, 63
41	ExprTuple	45
42	Lambda	parameter: 82 body: 46
43	Lambda	parameter: 82 body: 47
44	Operation	operator: 48 operand: 54
45	Lambda	parameters: 68 body: 50
46	Operation	operator: 52 operands: 51
47	Operation	operator: 52 operands: 53
48	Literal
49	ExprTuple	54
50	Conditional	value: 55 condition: 60
51	ExprTuple	71, 56
52	Literal
53	ExprTuple	73, 56
54	Lambda	parameters: 68 body: 57
55	Operation	operator: 58 operands: 59
56	Literal
57	Conditional	value: 62 condition: 60
58	Literal
59	ExprTuple	61, 62, 63
60	Operation	operator: 64 operands: 68
61	ExprRange	lambda_map: 65 start_index: 76 end_index: 66
62	Operation	operator: 67 operands: 68
63	ExprRange	lambda_map: 69 start_index: 76 end_index: 70
64	Variable
65	Lambda	parameter: 82 body: 71
66	Variable
67	Variable
68	ExprTuple	72
69	Lambda	parameter: 82 body: 73
70	Variable
71	IndexedVar	variable: 74 index: 82
72	ExprRange	lambda_map: 75 start_index: 76 end_index: 77
73	IndexedVar	variable: 78 index: 82
74	Variable
75	Lambda	parameter: 82 body: 79
76	Literal
77	Variable
78	Variable
79	IndexedVar	variable: 80 index: 82
80	Variable
81	ExprTuple	82
82	Variable