Expression of type Lambda

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 Conditional, Lambda, b, m
from proveit.core_expr_types import a_1_to_m
from proveit.logic import Equals, Forall, InSet
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 = Lambda(m, Conditional(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]), InSet(m, 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())

m \mapsto \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) \textrm{ if } m \in \mathbb{N}^+\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 40 body: 2
1	ExprTuple	40
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 33 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	40, 9
8	Lambda	parameters: 10 body: 11
9	Literal
10	ExprTuple	35, 44
11	Conditional	value: 12 condition: 13
12	Operation	operator: 14 operands: 15
13	Operation	operator: 16 operands: 17
14	Literal
15	ExprTuple	18, 19
16	Literal
17	ExprTuple	20, 21
18	Operation	operator: 41 operands: 22
19	Operation	operator: 30 operands: 23
20	ExprRange	lambda_map: 24 start_index: 39 end_index: 40
21	Operation	operator: 33 operands: 25
22	ExprTuple	26, 44
23	ExprTuple	27
24	Lambda	parameter: 47 body: 28
25	ExprTuple	44, 29
26	Operation	operator: 30 operands: 31
27	ExprRange	lambda_map: 32 start_index: 39 end_index: 40
28	Operation	operator: 33 operands: 34
29	Literal
30	Literal
31	ExprTuple	35
32	Lambda	parameter: 47 body: 36
33	Literal
34	ExprTuple	43, 37
35	ExprRange	lambda_map: 38 start_index: 39 end_index: 40
36	Operation	operator: 41 operands: 42
37	Literal
38	Lambda	parameter: 47 body: 43
39	Literal
40	Variable
41	Literal
42	ExprTuple	43, 44
43	IndexedVar	variable: 45 index: 47
44	Variable
45	Variable
46	ExprTuple	47
47	Variable