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, a, m
from proveit.core_expr_types import k_1_to_m
from proveit.logic import Equals, Forall, InSet
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 = Lambda(m, Conditional(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]), 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_{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) \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: 42 body: 2
1	ExprTuple	42
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	42, 37
8	Lambda	parameters: 9 body: 10
9	ExprTuple	43, 36
10	Conditional	value: 11 condition: 12
11	Operation	operator: 13 operands: 14
12	Operation	operator: 15 operands: 16
13	Literal
14	ExprTuple	17, 18
15	Literal
16	ExprTuple	19, 20
17	Operation	operator: 21 operands: 22
18	Operation	operator: 38 operands: 23
19	Operation	operator: 33 operands: 24
20	ExprRange	lambda_map: 25 start_index: 41 end_index: 42
21	Literal
22	ExprTuple	26
23	ExprTuple	43, 27
24	ExprTuple	43, 28
25	Lambda	parameter: 47 body: 29
26	ExprRange	lambda_map: 30 start_index: 41 end_index: 42
27	Operation	operator: 31 operands: 32
28	Literal
29	Operation	operator: 33 operands: 34
30	Lambda	parameter: 47 body: 35
31	Literal
32	ExprTuple	36
33	Literal
34	ExprTuple	44, 37
35	Operation	operator: 38 operands: 39
36	ExprRange	lambda_map: 40 start_index: 41 end_index: 42
37	Literal
38	Literal
39	ExprTuple	43, 44
40	Lambda	parameter: 47 body: 44
41	Literal
42	Variable
43	Variable
44	IndexedVar	variable: 45 index: 47
45	Variable
46	ExprTuple	47
47	Variable