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 ExprRange, IndexedVar, Variable, b, j
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 Abs, Add, Complex, Mult, RealNonNeg, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_b", latex_format = r"{_{-}b}")
expr = 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()

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{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}

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