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 b, m
from proveit.core_expr_types import a_1_to_m
from proveit.logic import Equals, Forall
from proveit.numbers import Complex, Exp, Mult, NaturalPos, RealPos
from proveit.numbers.exponentiation import prod_ai_raise_b__1_to_m

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [m], instance_expr = Forall(instance_param_or_params = [a_1_to_m, b], instance_expr = Equals(Exp(Mult(a_1_to_m), b), prod_ai_raise_b__1_to_m), domains = [RealPos, 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 \in \mathbb{N}^+}~\left[\forall_{\left(a_{1} \in \mathbb{R}^+\right), \left(a_{2} \in \mathbb{R}^+\right), \ldots, \left(a_{m} \in \mathbb{R}^+\right), b \in \mathbb{C}}~\left(\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{m}\right)^{b} = \left(\left(a_{1}\right)^{b} \cdot  \left(a_{2}\right)^{b} \cdot  \ldots \cdot  \left(a_{m}\right)^{b}\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: 42 body: 4
3	ExprTuple	42
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 35 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	42, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	37, 46
13	Conditional	value: 14 condition: 15
14	Operation	operator: 16 operands: 17
15	Operation	operator: 18 operands: 19
16	Literal
17	ExprTuple	20, 21
18	Literal
19	ExprTuple	22, 23
20	Operation	operator: 43 operands: 24
21	Operation	operator: 32 operands: 25
22	ExprRange	lambda_map: 26 start_index: 41 end_index: 42
23	Operation	operator: 35 operands: 27
24	ExprTuple	28, 46
25	ExprTuple	29
26	Lambda	parameter: 49 body: 30
27	ExprTuple	46, 31
28	Operation	operator: 32 operands: 33
29	ExprRange	lambda_map: 34 start_index: 41 end_index: 42
30	Operation	operator: 35 operands: 36
31	Literal
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