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 a, m
from proveit.core_expr_types import b_1_to_m
from proveit.logic import Equals, Forall
from proveit.numbers import Add, Complex, Exp, NaturalPos, RealPos
from proveit.numbers.exponentiation import prod_a_raise_bi__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, b_1_to_m], instance_expr = Equals(prod_a_raise_bi__1_to_m, Exp(a, Add(b_1_to_m))), domains = [Complex, RealPos]), 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_{a \in \mathbb{C},\left(b_{1} \in \mathbb{R}^+\right), \left(b_{2} \in \mathbb{R}^+\right), \ldots, \left(b_{m} \in \mathbb{R}^+\right)}~\left(\left(a^{b_{1}} \cdot  a^{b_{2}} \cdot  \ldots \cdot  a^{b_{m}}\right) = a^{b_{1} +  b_{2} +  \ldots +  b_{m}}\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: 45 body: 4
3	ExprTuple	45
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 36 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	45, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	46, 39
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: 24 operands: 25
21	Operation	operator: 41 operands: 26
22	Operation	operator: 36 operands: 27
23	ExprRange	lambda_map: 28 start_index: 44 end_index: 45
24	Literal
25	ExprTuple	29
26	ExprTuple	46, 30
27	ExprTuple	46, 31
28	Lambda	parameter: 50 body: 32
29	ExprRange	lambda_map: 33 start_index: 44 end_index: 45
30	Operation	operator: 34 operands: 35
31	Literal
32	Operation	operator: 36 operands: 37
33	Lambda	parameter: 50 body: 38
34	Literal
35	ExprTuple	39
36	Literal
37	ExprTuple	47, 40
38	Operation	operator: 41 operands: 42
39	ExprRange	lambda_map: 43 start_index: 44 end_index: 45
40	Literal
41	Literal
42	ExprTuple	46, 47
43	Lambda	parameter: 50 body: 47
44	Literal
45	Variable
46	Variable
47	IndexedVar	variable: 48 index: 50
48	Variable
49	ExprTuple	50
50	Variable