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, one

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

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_{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}

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