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 k_1_to_m
from proveit.logic import Equals, Forall
from proveit.numbers import Add, Complex, Exp, NaturalPos
from proveit.numbers.exponentiation import prod_a_raise_ki__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, k_1_to_m], instance_expr = Equals(prod_a_raise_ki__1_to_m, Exp(a, Add(k_1_to_m))), domains = [Complex, NaturalPos]), 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(k_{1} \in \mathbb{N}^+\right), \left(k_{2} \in \mathbb{N}^+\right), \ldots, \left(k_{m} \in \mathbb{N}^+\right)}~\left(\left(a^{k_{1}} \cdot  a^{k_{2}} \cdot  \ldots \cdot  a^{k_{m}}\right) = a^{k_{1} +  k_{2} +  \ldots +  k_{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: 44 body: 4
3	ExprTuple	44
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	44, 39
10	Lambda	parameters: 11 body: 12
11	ExprTuple	45, 38
12	Conditional	value: 13 condition: 14
13	Operation	operator: 15 operands: 16
14	Operation	operator: 17 operands: 18
15	Literal
16	ExprTuple	19, 20
17	Literal
18	ExprTuple	21, 22
19	Operation	operator: 23 operands: 24
20	Operation	operator: 40 operands: 25
21	Operation	operator: 35 operands: 26
22	ExprRange	lambda_map: 27 start_index: 43 end_index: 44
23	Literal
24	ExprTuple	28
25	ExprTuple	45, 29
26	ExprTuple	45, 30
27	Lambda	parameter: 49 body: 31
28	ExprRange	lambda_map: 32 start_index: 43 end_index: 44
29	Operation	operator: 33 operands: 34
30	Literal
31	Operation	operator: 35 operands: 36
32	Lambda	parameter: 49 body: 37
33	Literal
34	ExprTuple	38
35	Literal
36	ExprTuple	46, 39
37	Operation	operator: 40 operands: 41
38	ExprRange	lambda_map: 42 start_index: 43 end_index: 44
39	Literal
40	Literal
41	ExprTuple	45, 46
42	Lambda	parameter: 49 body: 46
43	Literal
44	Variable
45	Variable
46	IndexedVar	variable: 47 index: 49
47	Variable
48	ExprTuple	49
49	Variable