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, b, 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, Natural, RealNonNeg, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_b", latex_format = r"{_{-}b}")
expr = Conditional(Forall(instance_param_or_params = [a_1_to_i, c_1_to_k], instance_expr = 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(), domain = RealNonNeg).with_wrapping(), And(InSet(i, Natural), InSet(j, Natural), 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\{\begin{array}{l}\forall_{a_{1}, a_{2}, \ldots, a_{i}, c_{1}, c_{2}, \ldots, c_{k} \in \mathbb{R}^{\ge 0}}~\\
\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}\right]\end{array} \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: 17 operand: 5
2	Operation	operator: 32 operands: 4
3	ExprTuple	5
4	ExprTuple	6, 7, 8
5	Lambda	parameters: 9 body: 10
6	Operation	operator: 49 operands: 11
7	Operation	operator: 49 operands: 12
8	Operation	operator: 49 operands: 13
9	ExprTuple	64, 66
10	Conditional	value: 14 condition: 15
11	ExprTuple	68, 16
12	ExprTuple	60, 16
13	ExprTuple	73, 16
14	Operation	operator: 17 operand: 20
15	Operation	operator: 32 operands: 19
16	Literal
17	Literal
18	ExprTuple	20
19	ExprTuple	21, 22
20	Lambda	parameters: 55 body: 23
21	ExprRange	lambda_map: 24 start_index: 72 end_index: 68
22	ExprRange	lambda_map: 25 start_index: 72 end_index: 73
23	Conditional	value: 26 condition: 27
24	Lambda	parameter: 80 body: 28
25	Lambda	parameter: 80 body: 29
26	Operation	operator: 30 operands: 31
27	Operation	operator: 32 operands: 33
28	Operation	operator: 49 operands: 34
29	Operation	operator: 49 operands: 35
30	Literal
31	ExprTuple	36, 37
32	Literal
33	ExprTuple	38
34	ExprTuple	74, 39
35	ExprTuple	76, 39
36	Operation	operator: 61 operands: 40
37	Operation	operator: 46 operand: 44
38	ExprRange	lambda_map: 42 start_index: 72 end_index: 60
39	Literal
40	ExprTuple	64, 43, 66
41	ExprTuple	44
42	Lambda	parameter: 80 body: 45
43	Operation	operator: 46 operand: 51
44	Operation	operator: 54 operands: 48
45	Operation	operator: 49 operands: 50
46	Literal
47	ExprTuple	51
48	ExprTuple	52
49	Literal
50	ExprTuple	63, 53
51	Operation	operator: 54 operands: 55
52	ExprRange	lambda_map: 56 start_index: 72 end_index: 60
53	Literal
54	Literal
55	ExprTuple	57
56	Lambda	parameter: 75 body: 58
57	ExprRange	lambda_map: 59 start_index: 72 end_index: 60
58	Operation	operator: 61 operands: 62
59	Lambda	parameter: 80 body: 63
60	Variable
61	Literal
62	ExprTuple	64, 65, 66
63	IndexedVar	variable: 69 index: 80
64	ExprRange	lambda_map: 67 start_index: 72 end_index: 68
65	IndexedVar	variable: 69 index: 75
66	ExprRange	lambda_map: 71 start_index: 72 end_index: 73
67	Lambda	parameter: 80 body: 74
68	Variable
69	Variable
70	ExprTuple	75
71	Lambda	parameter: 80 body: 76
72	Literal
73	Variable
74	IndexedVar	variable: 77 index: 80
75	Variable
76	IndexedVar	variable: 78 index: 80
77	Variable
78	Variable
79	ExprTuple	80
80	Variable