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 b_1_to_m
from proveit.logic import Equals, Forall, InSet, 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 = Lambda(m, Conditional(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)), 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(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) \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: 47 body: 2
1	ExprTuple	47
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 38 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	47, 9
8	Lambda	parameters: 10 body: 11
9	Literal
10	ExprTuple	48, 41
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, 22
18	Operation	operator: 23 operands: 24
19	Operation	operator: 43 operands: 25
20	Operation	operator: 38 operands: 26
21	ExprRange	lambda_map: 27 start_index: 46 end_index: 47
22	Operation	operator: 28 operands: 29
23	Literal
24	ExprTuple	30
25	ExprTuple	48, 31
26	ExprTuple	48, 32
27	Lambda	parameter: 52 body: 33
28	Literal
29	ExprTuple	48, 34
30	ExprRange	lambda_map: 35 start_index: 46 end_index: 47
31	Operation	operator: 36 operands: 37
32	Literal
33	Operation	operator: 38 operands: 39
34	Literal
35	Lambda	parameter: 52 body: 40
36	Literal
37	ExprTuple	41
38	Literal
39	ExprTuple	49, 42
40	Operation	operator: 43 operands: 44
41	ExprRange	lambda_map: 45 start_index: 46 end_index: 47
42	Literal
43	Literal
44	ExprTuple	48, 49
45	Lambda	parameter: 52 body: 49
46	Literal
47	Variable
48	Variable
49	IndexedVar	variable: 50 index: 52
50	Variable
51	ExprTuple	52
52	Variable