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, ExprRange, IndexedVar, Variable, a, b, c, 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, InSet
from proveit.numbers import Abs, Add, Complex, Mult, RealNonNeg, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_b", latex_format = r"{_{-}b}")
sub_expr2 = Variable("_a", latex_format = r"{_{-}a}")
expr = Conditional(Forall(instance_param_or_params = [b_1_to_j], instance_expr = Equals(Mult(a_1_to_i, Abs(Add(b_1_to_j)), c_1_to_k), Abs(Add(ExprRange(sub_expr1, Mult(a_1_to_i, IndexedVar(b, sub_expr1), c_1_to_k), one, j)))).with_wrapping_at(2), domain = Complex).with_wrapping(), And(ExprRange(sub_expr2, InSet(IndexedVar(a, sub_expr2), RealNonNeg), one, i), ExprRange(sub_expr2, InSet(IndexedVar(c, sub_expr2), RealNonNeg), one, k)))

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\{\begin{array}{l}\forall_{b_{1}, b_{2}, \ldots, b_{j} \in \mathbb{C}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot \left|b_{1} +  b_{2} +  \ldots +  b_{j}\right|\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) =  \\ \left|\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot b_{1}\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) +  \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot b_{2}\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) +  \ldots +  \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot b_{j}\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right)\right| \end{array} \end{array}\right)\end{array} \textrm{ if } \left(a_{1} \in \mathbb{R}^{\ge 0}\right) ,  \left(a_{2} \in \mathbb{R}^{\ge 0}\right) ,  \ldots ,  \left(a_{i} \in \mathbb{R}^{\ge 0}\right), \left(c_{1} \in \mathbb{R}^{\ge 0}\right) ,  \left(c_{2} \in \mathbb{R}^{\ge 0}\right) ,  \ldots ,  \left(c_{k} \in \mathbb{R}^{\ge 0}\right)\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: 3 operand: 6
2	Operation	operator: 18 operands: 5
3	Literal
4	ExprTuple	6
5	ExprTuple	7, 8
6	Lambda	parameters: 41 body: 9
7	ExprRange	lambda_map: 10 start_index: 58 end_index: 54
8	ExprRange	lambda_map: 11 start_index: 58 end_index: 59
9	Conditional	value: 12 condition: 13
10	Lambda	parameter: 66 body: 14
11	Lambda	parameter: 66 body: 15
12	Operation	operator: 16 operands: 17
13	Operation	operator: 18 operands: 19
14	Operation	operator: 35 operands: 20
15	Operation	operator: 35 operands: 21
16	Literal
17	ExprTuple	22, 23
18	Literal
19	ExprTuple	24
20	ExprTuple	60, 25
21	ExprTuple	62, 25
22	Operation	operator: 47 operands: 26
23	Operation	operator: 32 operand: 30
24	ExprRange	lambda_map: 28 start_index: 58 end_index: 46
25	Literal
26	ExprTuple	50, 29, 52
27	ExprTuple	30
28	Lambda	parameter: 66 body: 31
29	Operation	operator: 32 operand: 37
30	Operation	operator: 40 operands: 34
31	Operation	operator: 35 operands: 36
32	Literal
33	ExprTuple	37
34	ExprTuple	38
35	Literal
36	ExprTuple	49, 39
37	Operation	operator: 40 operands: 41
38	ExprRange	lambda_map: 42 start_index: 58 end_index: 46
39	Literal
40	Literal
41	ExprTuple	43
42	Lambda	parameter: 61 body: 44
43	ExprRange	lambda_map: 45 start_index: 58 end_index: 46
44	Operation	operator: 47 operands: 48
45	Lambda	parameter: 66 body: 49
46	Variable
47	Literal
48	ExprTuple	50, 51, 52
49	IndexedVar	variable: 55 index: 66
50	ExprRange	lambda_map: 53 start_index: 58 end_index: 54
51	IndexedVar	variable: 55 index: 61
52	ExprRange	lambda_map: 57 start_index: 58 end_index: 59
53	Lambda	parameter: 66 body: 60
54	Variable
55	Variable
56	ExprTuple	61
57	Lambda	parameter: 66 body: 62
58	Literal
59	Variable
60	IndexedVar	variable: 63 index: 66
61	Variable
62	IndexedVar	variable: 64 index: 66
63	Variable
64	Variable
65	ExprTuple	66
66	Variable