Expression of type Forall

from the theory of proveit.numbers.exponentiation

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 m, n
from proveit.core_expr_types import a_1_to_m
from proveit.logic import Equals, Forall
from proveit.numbers import Complex, Exp, Mult, NaturalPos
from proveit.numbers.exponentiation import prod_ai_raise_n__1_to_m

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [m, n], instance_expr = Forall(instance_param_or_params = [a_1_to_m], instance_expr = Equals(Exp(Mult(a_1_to_m), n), prod_ai_raise_n__1_to_m), 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_{m, n \in \mathbb{N}^+}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{m} \in \mathbb{C}}~\left(\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{m}\right)^{n} = \left(\left(a_{1}\right)^{n} \cdot  \left(a_{2}\right)^{n} \cdot  \ldots \cdot  \left(a_{m}\right)^{n}\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	parameters: 3 body: 4
3	ExprTuple	42, 46
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 21 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	11, 12
10	Lambda	parameters: 33 body: 13
11	Operation	operator: 35 operands: 14
12	Operation	operator: 35 operands: 15
13	Conditional	value: 16 condition: 17
14	ExprTuple	42, 18
15	ExprTuple	46, 18
16	Operation	operator: 19 operands: 20
17	Operation	operator: 21 operands: 22
18	Literal
19	Literal
20	ExprTuple	23, 24
21	Literal
22	ExprTuple	25
23	Operation	operator: 43 operands: 26
24	Operation	operator: 32 operands: 27
25	ExprRange	lambda_map: 28 start_index: 41 end_index: 42
26	ExprTuple	29, 46
27	ExprTuple	30
28	Lambda	parameter: 49 body: 31
29	Operation	operator: 32 operands: 33
30	ExprRange	lambda_map: 34 start_index: 41 end_index: 42
31	Operation	operator: 35 operands: 36
32	Literal
33	ExprTuple	37
34	Lambda	parameter: 49 body: 38
35	Literal
36	ExprTuple	45, 39
37	ExprRange	lambda_map: 40 start_index: 41 end_index: 42
38	Operation	operator: 43 operands: 44
39	Literal
40	Lambda	parameter: 49 body: 45
41	Literal
42	Variable
43	Literal
44	ExprTuple	45, 46
45	IndexedVar	variable: 47 index: 49
46	Variable
47	Variable
48	ExprTuple	49
49	Variable