Expression of type Lambda

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 Conditional, ExprRange, IndexedVar, Lambda, Variable, n, x
from proveit.core_expr_types import x_1_to_n
from proveit.logic import Equals, Forall, InSet
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 = Lambda(n, Conditional(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), InSet(n, 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())

n \mapsto \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) \textrm{ if } n \in \mathbb{N}^+\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 36 body: 2
1	ExprTuple	36
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 29 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	36, 9
8	Lambda	parameters: 27 body: 10
9	Literal
10	Conditional	value: 11 condition: 12
11	Operation	operator: 13 operands: 14
12	Operation	operator: 15 operands: 16
13	Literal
14	ExprTuple	17, 18
15	Literal
16	ExprTuple	19
17	Operation	operator: 37 operand: 23
18	Operation	operator: 26 operands: 21
19	ExprRange	lambda_map: 22 start_index: 35 end_index: 36
20	ExprTuple	23
21	ExprTuple	24
22	Lambda	parameter: 42 body: 25
23	Operation	operator: 26 operands: 27
24	ExprRange	lambda_map: 28 start_index: 35 end_index: 36
25	Operation	operator: 29 operands: 30
26	Literal
27	ExprTuple	31
28	Lambda	parameter: 42 body: 32
29	Literal
30	ExprTuple	39, 33
31	ExprRange	lambda_map: 34 start_index: 35 end_index: 36
32	Operation	operator: 37 operand: 39
33	Literal
34	Lambda	parameter: 42 body: 39
35	Literal
36	Variable
37	Literal
38	ExprTuple	39
39	IndexedVar	variable: 40 index: 42
40	Variable
41	ExprTuple	42
42	Variable