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, NotEquals
from proveit.numbers import Add, Complex, Exp, NaturalPos, Real, zero
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, Real], condition = NotEquals(a, zero)), 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)~|~a \neq 0}~\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: 49 body: 4
3	ExprTuple	49
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 40 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	49, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	50, 43
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, 24
20	Operation	operator: 25 operands: 26
21	Operation	operator: 45 operands: 27
22	Operation	operator: 40 operands: 28
23	ExprRange	lambda_map: 29 start_index: 48 end_index: 49
24	Operation	operator: 30 operands: 31
25	Literal
26	ExprTuple	32
27	ExprTuple	50, 33
28	ExprTuple	50, 34
29	Lambda	parameter: 54 body: 35
30	Literal
31	ExprTuple	50, 36
32	ExprRange	lambda_map: 37 start_index: 48 end_index: 49
33	Operation	operator: 38 operands: 39
34	Literal
35	Operation	operator: 40 operands: 41
36	Literal
37	Lambda	parameter: 54 body: 42
38	Literal
39	ExprTuple	43
40	Literal
41	ExprTuple	51, 44
42	Operation	operator: 45 operands: 46
43	ExprRange	lambda_map: 47 start_index: 48 end_index: 49
44	Literal
45	Literal
46	ExprTuple	50, 51
47	Lambda	parameter: 54 body: 51
48	Literal
49	Variable
50	Variable
51	IndexedVar	variable: 52 index: 54
52	Variable
53	ExprTuple	54
54	Variable