Expression of type Lambda

from the theory of proveit.numbers.number_sets.complex_numbers

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, r, theta
from proveit.logic import And, Equals, InSet
from proveit.numbers import Add, Exp, Mult, Neg, Real, e, i, pi

# build up the expression from sub-expressions
expr = Lambda([r, theta], Conditional(Equals(Mult(Neg(r), Exp(e, Mult(i, Add(theta, pi)))), Mult(r, Exp(e, Mult(i, theta)))), And(InSet(r, Real), InSet(theta, Real))))

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())

\left(r, \theta\right) \mapsto \left\{\left(\left(-r\right) \cdot \mathsf{e}^{\mathsf{i} \cdot \left(\theta + \pi\right)}\right) = \left(r \cdot \mathsf{e}^{\mathsf{i} \cdot \theta}\right) \textrm{ if } r \in \mathbb{R} ,  \theta \in \mathbb{R}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 1 body: 2
1	ExprTuple	27, 38
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operands: 6
4	Operation	operator: 7 operands: 8
5	Literal
6	ExprTuple	9, 10
7	Literal
8	ExprTuple	11, 12
9	Operation	operator: 32 operands: 13
10	Operation	operator: 32 operands: 14
11	Operation	operator: 16 operands: 15
12	Operation	operator: 16 operands: 17
13	ExprTuple	18, 19
14	ExprTuple	27, 20
15	ExprTuple	27, 21
16	Literal
17	ExprTuple	38, 21
18	Operation	operator: 22 operand: 27
19	Operation	operator: 25 operands: 24
20	Operation	operator: 25 operands: 26
21	Literal
22	Literal
23	ExprTuple	27
24	ExprTuple	29, 28
25	Literal
26	ExprTuple	29, 30
27	Variable
28	Operation	operator: 32 operands: 31
29	Literal
30	Operation	operator: 32 operands: 33
31	ExprTuple	35, 34
32	Literal
33	ExprTuple	35, 38
34	Operation	operator: 36 operands: 37
35	Literal
36	Literal
37	ExprTuple	38, 39
38	Variable
39	Literal