Expression of type Forall

from the theory of proveit.numbers.absolute_value

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, n, x
from proveit.core_expr_types import x_1_to_n
from proveit.logic import Equals, Forall
from proveit.numbers import Abs, Complex, Mult, NaturalPos, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [x_1_to_n], instance_expr = Equals(Abs(Mult(x_1_to_n)), Mult(ExprRange(sub_expr1, Abs(IndexedVar(x, sub_expr1)), one, n))), domain = Complex), domain = NaturalPos)

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())

\forall_{n \in \mathbb{N}^+}~\left[\forall_{x_{1}, x_{2}, \ldots, x_{n} \in \mathbb{C}}~\left(\left|x_{1} \cdot  x_{2} \cdot  \ldots \cdot  x_{n}\right| = \left(\left|x_{1}\right| \cdot  \left|x_{2}\right| \cdot  \ldots \cdot  \left|x_{n}\right|\right)\right)\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('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	parameter: 38 body: 4
3	ExprTuple	38
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 31 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	38, 11
10	Lambda	parameters: 29 body: 12
11	Literal
12	Conditional	value: 13 condition: 14
13	Operation	operator: 15 operands: 16
14	Operation	operator: 17 operands: 18
15	Literal
16	ExprTuple	19, 20
17	Literal
18	ExprTuple	21
19	Operation	operator: 39 operand: 25
20	Operation	operator: 28 operands: 23
21	ExprRange	lambda_map: 24 start_index: 37 end_index: 38
22	ExprTuple	25
23	ExprTuple	26
24	Lambda	parameter: 44 body: 27
25	Operation	operator: 28 operands: 29
26	ExprRange	lambda_map: 30 start_index: 37 end_index: 38
27	Operation	operator: 31 operands: 32
28	Literal
29	ExprTuple	33
30	Lambda	parameter: 44 body: 34
31	Literal
32	ExprTuple	41, 35
33	ExprRange	lambda_map: 36 start_index: 37 end_index: 38
34	Operation	operator: 39 operand: 41
35	Literal
36	Lambda	parameter: 44 body: 41
37	Literal
38	Variable
39	Literal
40	ExprTuple	41
41	IndexedVar	variable: 42 index: 44
42	Variable
43	ExprTuple	44
44	Variable